The Spectral Shift Function for Non-Self-Adjoint Perturbations

THE SPECTRAL SHIFT FUNCTION F OR NON-SELF-ADJOI NT PER TURBA TIONS VINCENT BRUNEA U, NICOLAS FRANTZ, A ND FR A NC ¸ OIS N ICOLEAU Abstra ct. This pap er is devoted to the deﬁn ition and analysis of the sp ectral shift function (SSF) associated with non-self-adjoin t p erturbations of self-adjoin t operators. Motiv ated by applications in scattering th eory , w e consider b oth trace-class and relative ly trace-class p erturbations. W e extend the Lifshits– Kre ˘ ın trace formula to n on-self-adjoin t op erators un der suitable assumptions on the sp ectru m and t he b ehavior of the resolv ent. The role of sp ectral singularities is carefully analyzed, and w e provide a generalization of the SSF using functional calculus. Finally , we apply our results to Schr¨ odinger op erators with complex-v alued short-range p otentials in dimension th ree. T oy mod els illustrate prop erties that one might hop e to ex tend to general cases. In particular, they suggest t h at the SSF carries information on the presence of complex eigenv alues. Contents 1. In tro d uction 2 2. Assumptions and main resu lts 3 2.1. Abstract s etting 3 2.2. Assumptions 4 2.3. Main results 4 3. F un ctional Calculus 6 3.1. Helﬀer-Sj¨ ostrand form ula 6 3.2. Sp ectral c hanging of v ariables 13 4. SSF for T r ace class p erturb ations 16 4.1. Existence 16 4.2. A ﬁr s t representat ion formula for the S SF 17 5. The SS F for non-selfadjoint relativ ely trace class p erturbations 18 5.1. Deﬁnition of the S SF for r elativ ely trace class p ertur bations 18 5.2. Represent ation form ula in the case of r elativ ely trace class p ertu r bations 19 6. Limiting absorp tion prin ciple for n on -self-adjoint p erturbation 21 6.1. Sp ectral singularities 21 6.2. Resolv en t estimates 22 7. SSF for Sc hr¨ odinger Op erators with Comp lex-V alued Po tent ial 23 7.1. Sp ectral singularities and resonances 23 7.2. Preliminary r esults 24 7.3. Existence of the S SF 26 7.4. Regularit y of the Sp ectral S hift F unction 27 7.5. Asymptotics n ear a Sp ectral Singularity 28 7.6. High Energy Asymptotic 31 8. Explicit simp le examples 32 8.1. In ﬁn ite dimension: diagonalizable op erators 33 8.2. In ﬁn ite dimension: an un diagonalizable case 33 1 2 V. BR UNEAU, N. FRANTZ, AND F. NICOLEAU 8.3. Rank one p erturb ation: w eak in teraction with the con tin uous sp ectrum 34 8.4. Rank one p erturb ation: stronger in teraction w ith the con tin uous sp ectrum 35 Ac kno wledgment s 39 References 39 1. Introduction The Sp ectral Shift F unction (S SF), in itially introd uced b y Lifshits and K re ˘ ın [18, 19], pro vides a framew ork for the sp ectral analysis of trace -class (or relativ ely trace-class) p erturb ations of a reference op erator. It yields a sp ectral inv arian t that is often r elated to scattering quan tities suc h as the sc attering phase and the aver age time delay (see r eferences [24, 31]). More precisely , in the self-adjoin t setting, the deriv ativ e of the SSF w ith resp ect to energy coincides (up to a factor of 1 / 2 π ) w ith the s cattering ph ase shift, as established by the Birman–Kre ˘ ın formula. This connection allo w s one to inte rp ret the SSF as enco ding th e cumulati ve sp ectral eﬀect of the s cattering p ro cess. F ur thermore, the Eisenbud–Wigner form ula sho ws that the time dela y op erator (measuring the diﬀerence in the s o journ time du e to the interactio n), is also expr essed in terms of the energy deriv ativ e of th e scattering matrix, and th us r elated to the SS F. Originally deﬁn ed f or p airs of self-adjoin t or unitary op erators, the SSF h as b een extended to con traction op erators viewed as p erturbations of unitary op erators (see [27, 28]), and to d issipativ e (or accumulativ e) op erators int erpr eted as b ound ary p erturbations of self-adjoin t op erators (see [20, 21]). A Levinson’s form ula h as recen tly b een obtained in [1] for dissipative op erators. In th ese w orks, at least one side of the complex plane (the u pp er or low er half-plane, or the exterior of the unit disk) lies in the resolv en t set. The SSF for a general pair of op erators in a Banac h space is considered in [23], where it is deﬁn ed on (0 , + ∞ ), whic h is assumed to b e a subset of the resolv en t set. Our goal here is to consider a reference op erator H 0 whic h remains self-adjoin t, and to deﬁne a Sp e ctr al Shift F unction (SSF for short) for non-self-adjoint p erturb ations of H 0 . With the ob jectiv e of establishin g connections with scatte rin g theory , w e aim to deﬁne the SSF on the real line R , which con tains the essenti al sp ectrum of the op erators under consideration. Th at is, for a general relativ ely compact p erturbation H of a self-adjoin t op erator H 0 (in p articular, H ma y ha ve eigen v alues on b oth s ides of the real axis), we seek to deﬁne and analyze a function ξ := ξ ( · ; H , H 0 ) suc h that for an y f ∈ D ( R ), the space of s mo oth functions with compact supp ort, T r ( f ( H ) − f ( H 0 )) = Z R ξ ( λ ) f ′ ( λ ) dλ. (1.1) A particularly relev ant mo del in this cont ext is the Schr¨ odinger op erator H = H 0 + V , w h ere H 0 = − ∆ on L 2 ( R d ) and V is a complex-v alued elect ric p oten tial v anishing at inﬁnity . As in [5], the op erator f ( H ) for f ∈ D ( R ) is deﬁ n ed via the Helﬀer–Sj¨ ostrand f orm ula (3.1), using an almost analytic extension of f . This construction is extend to our non-self-adjoin t setting in section 3. W e then stud y the left-hand side of (1.1), ﬁrst f or trace-c lass p erturbations and subsequently for relativ ely trace-class p erturbations. W e sh o w that it deﬁn es the deriv ativ e of the SSF in the sense of distribu tions, and that for lo wer-boun ded op erators, the S SF m a y b e c hosen to v anish n ear −∞ . In the original works of Lifs hits and Kre ˘ ın [18, 19], in the self-adjoint setting, the existence of the SS F and its inte grabilit y prop er ties were rigorously justiﬁed via its relation to the p ertu rbation THE SPECTRAL SHIFT FUNCTION FOR NON- SELF-ADJOINT PER TURBA TIONS 3 determinan t: ξ ( λ ) = 1 π lim ε → 0 + arg D ( λ + iε ); D ( z ) := Det  I + V ( H 0 − z ) − 1  . (1.2) F or non-self-adjoin t op erators H , this formula no longer h olds since the p er tu rbation determinant ma y v anish and the prop ert y D ( z ) = D ( z ) is no longer true. Ho w ev er, using f actorizat ion th eorems for holomorphic functions on C + , where C ± = { z ∈ C | ± Im z > 0 } (see for instance [20]), one can still d eriv e r epresen tation r esu lts for the SSF in terms of measur es, but as we will see in the toy mo dels, these measures are no longer necessarily r eal and nor absolutely con tin uous. Throughout the pap er, we will use the follo win g n otatio ns : Giv en a Banac h space A , we denote b y B ( A ) the set of b ounded linear op erators on A . F or a Hilb ert space H , the S c hatten class of order p is denoted by L p ( H ). If A is a closed op erator, its sp ectrum is denoted b y σ ( A ) and its resolve nt set by ρ ( A ). The k ernel and range of A are denoted resp ectiv ely b y Ker( A ) an d Ran( A ). Giv en a self-adjoin t op erator H 0 , w e den ote by R 0 ( z ) := ( H 0 − z ) − 1 its resolv en t at z ∈ ρ ( H 0 ), and b y R ( z ) := ( H − z ) − 1 the resolv ent of another (p ossibly non-self-adjoint ) op erator H . F or an subset I ⊂ R , w e denote by D ( I ) th e space of smo oth fun ctions with compact supp ort in I and v alues in C . Its top ological d u al is denoted b y D ′ ( I ). F or in terv als I ⊂ R and a ∈ (0 , ∞ ], we in tro du ce the follo win g subset of the complex plane: S a ( I ) := { z ∈ C   Re( z ) ∈ I and 0 < | Im ( z ) | < a } , S ± a ( I ) := S a ( I ) ∩ C ± . 2. Assump tions and main res ul ts 2.1. Abstract sett ing. W e consider an op erator H acting on a complex separable Hilb ert space H . W e assume that H is of the form H := H 0 + V , where ( H 0 , D ( H 0 )) is a self-adjoin t op erator b oun ded from b elo w, and V is b oun ded and r elativ ely compact w ith r esp ect to H 0 . In p articular, H is closed with domain D ( H ) = D ( H 0 ), and its adjoin t is giv en b y H ∗ = H 0 + V ∗ , D ( H ∗ ) = D ( H 0 ) . W e deﬁn e the p oint s p ectrum of H as σ p ( H ) := { λ ∈ C | Ker( H − λ ) 6 = { 0 }} , that is, the set of eigenv alues of H . F or an isolated eigenv alue λ ∈ σ p ( H ), the corresp onding Riesz pro jection is giv en b y Π λ ( H ) := 1 2 π i I Γ ( z − H ) − 1 d z , (2.1) where Γ = C ( λ, r ) is a p ositiv ely orient ed circle cen tered at λ with radius r > 0 small enough so that λ is the only p oint of the sp ectrum σ ( H ) lying insid e the disk D ( λ, r ). An isolated eigen v alue is called discr e te if the range of the corresp onding Riesz pro jection is of ﬁnite dimension. W e denote by σ disc ( H ) the set of d iscrete eigen v alues of H . T h e essential sp e ctrum of H , den oted by σ ess ( H ), is deﬁned as the complemen t of the discrete s p ectrum in the full sp ectrum of H : σ ess ( H ) := σ ( H ) \ σ disc ( H ) . With this deﬁnition, and since V is relativ ely compact with resp ect to H 0 , it follo ws from W eyl’s theorem (see e.g. [25 , Theorem XI I I.14]) that σ ess ( H ) = σ ess ( H 0 ) . (2.2) 4 V. BR UNEAU, N. FRANTZ, AND F. NICOLEAU Moreo ver, usin g a v ariational argu m en t and th e b oundedn ess of V , one sees that σ ( H ) lies in a horizon tal strip and that its real part is b ounded from b elo w, since H = H 0 + V is a b oun ded p ertur b ation of H 0 . 2.2. Assumptions. T hroughout this pap er, some hyp otheses are understo o d to b e stated r elativ e to a ﬁ x ed op en int erv al I ⊂ R wh ic h is c hosen according to th e sp ectral region under consideration. Whenev er a hypothesis is inv oked, the corresp onding set I is assumed to b e ﬁ xed. Hyp othesis 1. L et I ⊂ R b e an op en interval. The p e rturb e d op er ator H has only ﬁnitely many non-r e al eigenvalues whose r e al p arts lie in I . The f ollo win g assumption requires the resolv en t of H blo w up at most p olynomially in a tub ular neigh b orh o o d of th e r eal in terv al I . Hyp othesis 2. L et I ⊂ R b e an op en interval. Ther e exists a I > 0 , n I > 0 and c I > 0 , such that S a I ( I ) ⊂ ρ ( H ) and for al l z ∈ S a I ( I ) , kR H ( z ) k B ( H ) ≤ c I | Im( z ) | − 1  h Re( z ) i | Im( z ) |  n I . (2.3) The n ext assu mption r equire that the diﬀerence b etw een the r esolv en ts (or of the op erators) is a trace class op erator. Hyp othesis 3. Ther e exists c ∈ R and m ∈ Z , such that ( H − c ) − m − ( H 0 − c ) − m ∈ L 1 ( H ) Remark 2.1. L e t us c omment these hyp otheses in p articular c ases. • If Hyp othesis 1 holds on I = R , then H has a ﬁnite numb er of non-r e al eigenvalues. • If I is a b ounde d interval, H yp othesis 1 me ans ther e is no ac cumulation of discr ete eige n- values to I × { 0 } . • If I is a b ounde d interval, the estimate (2.3) in Hyp othesis 2 is e quivalent to kR H ( z ) k B ( H ) ≤ c I | Im( z ) | − n I − 1 , ∀ z ∈ S a I ( I ) . • If I ⊂ ( −∞ , c ) with c < inf σ ess ( H 0 ) then b oth Hyp othesis 1 and 2 hold on I . • The c ase m = − 1 in Hyp othesis 3 me ans that V = H − H 0 b elongs to L 1 ( H ) . 2.3. Main results. W e no w summarize the main resu lts of the pap er. Ou r ﬁrs t result concerns the constru ction of a functional calculus for the n on-self-adjoin t op erator H = H 0 + V on the real line. Since H ma y p ossess n on-real eigenv alues, we separate the sp ectral con tribution asso ciated with these eigen v alues and apply the Helﬀer–Sj¨ ostrand fun ctional calculus, follo w ing the approac h of Da vies [5, 6], to the part of the op erator whose sp ectrum lies on the real axis. Under Hyp otheses 1 and 2 on an op en in terv al I ⊂ R , this construction allo ws u s to deﬁ n e the op erator f ( H ) for suitable functions f sup p orted in I through a Helﬀer-Sj¨ ostrand type formula. Moreo ver, th e map f 7− → f ( H ) deﬁnes a con tin uous algebra morp hism for the p oin t wise m ultiplication, (see Section 3). Our second resu lt establishes the existence of the sp ectral shift function for tr ace-cl ass p er tu rbations and is pro ve d in Section 4. THE SPECTRAL SHIFT FUNCTION FOR NON- SELF-ADJOINT PER TURBA TIONS 5 Theorem 2.2 (Existence of the S SF) . Assu me that Hyp otheses 1, 2, and 3 hold on an op en interval I with m = − 1 . Then for every f ∈ D ( I ) the op er ator diﬀe r enc e f ( H ) − f ( H 0 ) b elongs to L 1 ( H ) and the map f 7→ T r( f ( H ) − f ( H 0 )) deﬁnes a distribution on I . The sp e ctr al shift function ξ ( · ; H , H 0 ) i s deﬁne d, up to an additive c onstant, b y h ξ ′ , f i = T r( f ( H ) − f ( H 0 )) . Mor e over, it satisﬁes ξ ′ ( · ; H , H 0 ) = 1 2 π i lim ε → 0 + ( σ ( · + iε ) − σ ( · − iε )) i n D ′ ( I ) , (2.4) wher e for z ∈ ρ ( H ) ∩ ρ ( H 0 ) , σ ( z ) := T r  R H ( z ) − R H 0 ( z )  . Note th at unlike the self-adjoin tness case where σ ( · − iε ) = σ ( · + iε ), the righ t hand side of (2.4) is not n ecessarily real. Ho w ev er, as in the self-adjoin t setting, in Section 5, we then extend this construction to relativ ely trace-cl ass p erturb ations by using a sp ectral c hange of v ariables. Theorem 2.3 (Relativ ely trace-class p erturbations) . Su pp ose that Hyp otheses 1, 2 and 3 hold on I for some m ∈ N ∗ . Then the sp e ctr al shift function for the p air ( H , H 0 ) c an b e deﬁne d by ξ ( λ ; H , H 0 ) := ξ  ( λ + c ) − m ; ( H + c ) − m , ( H 0 + c ) − m  , wher e ξ ( · ; ( H + c ) − m , ( H 0 + c ) − m ) denotes the SSF asso ci ate d with the tr ac e-class p air (( H + c ) − m , ( H 0 + c ) − m ) , with some c ≫ 1 . Under the assump tions of the previous theorem, together with an ad d itional tec hn ical condition, one can derive a form ula similar to (2.4). This representat ion will b e the k ey to ol in the analysis of the sp ectral shift function for Sc hr¨ odinger op erators. In the en d , w e fo cus on S chr¨ odinger op erators with (smo oth) complex-v alued compactly supp orted p oten tials in dimension three. In this framew ork w e introd uce the notions of outgoing and incoming sp ectral singularities λ 0 > 0 (see Section 6), corresp ondin g to real resonances ± √ λ 0 . W e pr o v e that the sp ectral sh ift fun ction is regular a wa y fr om these singularities. More p recisely , outside the set of sp ectral singularities, the deriv ativ e of the SSF is a smo oth function of the energy . Near a sp ectral sin gularit y λ 0 > 0 of ﬁnite order ν 0 , we obtain a precise distrib u tional asymptotic expansion. In the outgoing case (for instance), one has (see Section 7), ξ ′ ( λ ; H , H 0 ) = ν 0 X k =1 α k ( λ 0 ) ( λ − λ 0 + i 0) k + H ( λ ) in D ′ ( I ) , where H ( λ ) is smo oth n ear λ 0 . Equiv alentl y , the sin gular part of ξ ′ ( λ ; H , H 0 ) is a ﬁnite linear com bination of p rincipal v alue distr ibutions p . v . ( λ − λ 0 ) − k and deriv ativ es of the Dirac m ass at λ 0 . Finally , in Section 8, we conclude the p ap er with the study of sev eral simp le and explicit examples illustrating the general r esults obtained ab o ve . 6 V. BR UNEAU, N. FRANTZ, AND F. NICOLEAU 3. Functiona l Calculus 3.1. Helﬀer-Sj¨ ostrand form ula. In what follo ws, we iden tify R 2 with C , and for z ∈ C we denote b y ( x, y ) its co ordinates in R 2 . If f : R 2 → C is d iﬀerentiable, w e set ∂ ¯ z f := 1 2  ∂ x f + i∂ y f  . F or an y β ∈ R , we deﬁne S β the space of smo oth functions f : R → C suc h that f ( r ) ( x ) = O  h x i β − r  , | x | → ∞ , for all r ≥ 0 and w h ere h x i = (1 + x 2 ) 1 2 . W e deﬁne A := [ β < 0 S β . Then A is an algebra und er p oin twise multiplica tion, it con tains the smo oth compactly supp orted functions from R to C , and it is stable under p oint wise m ultiplication by functions in S 0 . O n A w e consider the f amily of norms k f k n := n X k =0 Z R | f ( k ) ( x ) |h x i k − 1 d x. In particular, D ( R ) is dense in A for eac h norm k · k n . In [5], Davie s pr o v es that for an y closed op erator L acting on a Banac h sp ace B with r e al sp ectrum satisfying an assump tion similar to (2.3 ), the op erator f ( L ) acting on B is well deﬁned by the Helﬀer–Sj¨ ostrand form ula for an y f ∈ A : f ( L ) = 1 π Z C ∂ ¯ z ˜ f ( z ) ( L − z ) − 1 d x d y , (3.1) where ˜ f is a su itable almost-analytic extension of f . Moreo v er, the map Ψ : A − → B ( B ) f 7→ f ( L ) is linear, con tin uou s , and a morphism with resp ect to p oint wise m ultiplication. By follo wing the pro of of [5], we c heek easily th at for f compactly supp orted in an op en inte rv al I ⊂ R , f ( L ) is still w ell deﬁned if th e sp ectrum of L is real only in I × R and f ( L ∗ ) = ( ¯ f ( L )) ∗ . Let us recall the deﬁnition of a su c h almost-analytic extension of a function f ∈ C ∞ ( R , C ) : Deﬁnition 3.1 (Almost-analytic extension) . L et f ∈ C ∞ ( R ) and let N ∈ N . An almost-analytic extension of order N of f is a func tion ˜ f : C → C of the form ˜ f ( z ) = N X k =0 f ( k ) ( x ) k ! ( iy ) k ! χ ( x, y ) , with χ ( x, y ) = τ  y h x i  , (3.2) and τ b eing a cut-oﬀ function deﬁne d on R . The almost-analytic extension ˜ f satisﬁes ∂ ¯ z ˜ f ( z ) = O ( | Im z | N ) as Im z → 0 . In th is section, we aim to give a meaning to (3.1) by replacing L with H , and B by H , taking in to accoun t that H ma y ha v e n on-real eigen v alues. The idea is to decomp ose H as a d irect sum of an op erator with r eal s p ectrum, denoted H r , and another one with non-real sp ectrum, and then to THE SPECTRAL SHIFT FUNCTION FOR NON- SELF-ADJOINT PER TURBA TIONS 7 deﬁne f ( H ) in terms of f ( H r ). S ince our assu m ptions are made lo cally in nature, we introdu ce the follo w ing n otations : f or an y op en interv al I ⊂ R , w e s et A ( I ) := { f ∈ A | s upp( f ) ⊂ I } . In particular, • if I = R , then A ( I ) = A ; • if I is b ounded, then A ( I ) = D ( I ); • for all I ⊂ R op en interv al, D ( I ) is dense in A ( I ) for eac h n orm k · k n . F or the rest of the Subsection 3.1, we ﬁx I ⊂ R to b e an op en in terv al where Hyp otheses 1 and 2 hold. 3.1.1. De c omp osition. In order to use the functional calculus dev elop ed in [5 ], let us d ecomp ose H in to a ”real part” (i.e. an op erator with real sp ectrum) and th e ”non-real p art” (the restriction of H to the eigenspaces asso ciated with non-real eigen v alues). This decomp osition is based on the follo w ing lemma. Lemma 3.2. L et P ∈ B ( H ) b e a b ounde d pr oje ction (i.e. P 2 = P ) such that P ( D ( H )) ⊂ D ( H ) and [ P , H ] u = 0 ∀ u ∈ D ( H ) . (3.3) Deﬁne F := Ran( P ) , G := Ran(Id H − P ) . Then: (1) F and G ar e invariant under H , i. e ., H ( F ∩ D ( H )) ⊂ F , H ( G ∩ D ( H )) ⊂ G. (2) ther e exists a c ontinuous isomorphism Ψ : F ⊕ G → H ( u F , u G ) 7→ u F + u G . (3) Denoting by H | F and H | G the r estrictions of H to F and G r esp e ctively, then for al l u ∈ D ( H ) , H u = Ψ  H | F 0 0 H | G  Ψ − 1 u = H | F u F + H | G u G (4) The sp e ctrum satisﬁes σ ( H ) = σ ( H | F ) ∪ σ ( H | G ) (5) F or al l z ∈ ρ ( H ) , for al l u ∈ H , R H ( z ) u = Ψ  R H | F ( z ) 0 0 R H | G ( z )  Ψ − 1 u, R • ( z ) := ( • − z ) − 1 Pr o of. Since P has closed range, w e ha v e the top ological dir ect s um H = Ran( P ) ⊕ Ran(Id H − P ) = F ⊕ G. Let u ∈ D ( H ). By (3.3 ), P u ∈ D ( H ) and H P u = P H u , which s h o ws th at F is inv arian t under H . Similarly , G is inv arian t under H . Thus, for any u = u F + u G ∈ D ( H ) with u F ∈ F , u G ∈ G , w e ha v e H u = H u F + H u G = H | F u F + H | G u G . 8 V. BR UNEAU, N. FRANTZ, AND F. NICOLEAU Finally , to r elate the sp ectra, it suﬃces to observ e that for an y z ∈ C , for an y u ∈ D ( H ), ( H − z Id H ) u = Ψ  H | F − z Id | F 0 0 H | G − z Id | G  Ψ − 1 u.  In the follo w in g and for sake of conciseness, w e will wr ite H = H | F ⊕ H | G instead of using the matrix representati on. Prop osition 3.3. Assume that Hyp otheses 1 and 2 hold on an op en interval I ⊂ R . Then ther e exist a b ounde d pr oje ction Π I ∈ B ( H ) such that, Π I ( D ( H )) ⊂ D ( H ) , and [Π I , H ] u = 0 , ∀ u ∈ D ( H ) , and two op er ators, H I , c acting on Ran(Π I ) , and H I , r acting on Ran(Id H − Π I ) such that one has (1) ther e exists an isomorp hism Ψ : Ran(Π I ( H )) ⊕ Ran(Id H − Π I ( H )) → H , such that Ψ( u, v ) = u + v (2) for al l u ∈ D ( H ) H u = Ψ  H I , c 0 0 H I , r  Ψ − 1 u (3) σ ( H I , r ) ∩ I × R ⊂ I × { 0 } and σ ( H I , c ) ∩ I × { 0 } = ∅ , (4) ther e exists a I > 0 su ch that S a I ∋ z 7→ R H I , c ( z ) ∈ B (Ran(Π I )) is holomorph ic, and the fol lowing r esolvent estimates hold : ∀ z ∈ S a I ( I ) , kR H I , r ( z ) k B (Ran (Id H − Π I )) ≤ c I | Im( z ) | − 1  h Re( z ) i | Im( z ) |  n I ; (3.4) sup z ∈ S a I kR H I , c ( z ) k B (Ran (Π I ) < ∞ . (3.5) Pr o of. By Hyp othesis 1, H has only ﬁnitely many non-real eigen v alues whose real p arts lie in I . Therefore, by setting Ω = { λ ∈ σ p ( H ) | Re( λ ) ∈ I } , the op erator Π I ( H ) := X λ ∈ Ω Π λ ( H ) , where Π λ is the Riesz pro jection asso ciated to λ (see (2.1)), is a ﬁnite sum of ﬁnite-rank p ro jections. Moreo ver as Π λ Π µ = 0 for λ 6 = µ , Π I is a pro jection and has ﬁ nite r ank. Setting F := Ran(Π I ( H )) , G := Ran(Id − Π I ( H )) , and using the notations of Lemma 3.2, we denote H I , c := H | F , H I , r := H | G ; this pr o vides the decomp osition of H . Moreo v er, b y constr u ction, σ ( H I , c ) ∩ I × { 0 } = ∅ , σ ( H I , r ) ∩ I × R ⊂ I × { 0 } . By Hyp othesis 1, w e ma y ﬁn d a I > 0 sm all enough su c h that S a I ( I ) d o es not con tain any n on-real eigen v alues of H . In particular, z 7→ R H I , c ( z ) is holomorphic on S a I and this provides (3.5) since THE SPECTRAL SHIFT FUNCTION FOR NON- SELF-ADJOINT PER TURBA TIONS 9 there are only a ﬁnite num b er of complex eigen v alues. Finally to obtain (3.4), we write for all z ∈ S a I , kR H I , r k B (Id − Ran (Π I )) = sup u ∈ Ran(Id − Π I ) \{ 0 } kR H I , r u k Ran(Id − Π I ) k u k Ran(Id − Π I ) = sup u ∈ Ran(Id − Π I ) \{ 0 } kR H u k H k u k H ≤ kR H k B (Ran (Id H − Π I )) ≤ c I | Im( z ) | − 1  h Re( z ) i | Im( z ) |  n I , where in th e last step, we use Hyp othesis 2.  3.1.2. De ﬁnition and pr op erties. Let us no w ﬁx an op en in terv al I ⊂ R , and deﬁne f ( H ) f or f supp orted in I u nder the Hyp otheses 1 and 2 on I . Deﬁnition 3.4. L et f ∈ A ( I ) and N ∈ N . We say that ˜ f : C → C is a almost-analytic extension of or der N which is admissible for H (or a H -admissible almost-anal ytic e xtension) if ther e e xists a cut-oﬀ function τ such that the supp ort of χ , c onsider e d in Deﬁnition 3.1, do es not c ontain any non-r e al eigenvalues of H whose r e al p art lies in I . Example 3.5. As an example, supp ose that H yp othesis 1 hold on R . Then ther e e xi sts a > 0 and b > 0 such that U a , b := { z ∈ C | | Re ( z ) | > a and | Im( z ) | < b } do es not c ontain any non-r e al eigenv alues of H ψ : R 2 → R ( x, y ) 7→ y b h a i h x i , and let τ ∈ C ∞ ( R , C ) a cutoﬀ f unction which satisﬁes τ ( s ) = 1 for | s | < 1 / 2 and τ ( s ) = 0 for | s | > 1 . Then, the supp ort of the function χ := τ ◦ ψ , do es non c ontain any non-r e al eigenvalues of H . Deﬁnition-Prop osition 3.6. Supp ose that Hyp otheses 1 and 2 hold on I . Then for al l f ∈ A ( I ) , the op er ator f ( H ) deﬁne d by f ( H ) := 1 π Z C ∂ ¯ z ˜ f ( z ) R H ( z )d x d y , wher e ˜ f is an almost-analytic extension of f as in D eﬁnition 3.1, is wel l-deﬁne d. Mor e over, we have the estimate k f ( H ) k B ( H ) ≤ c k f k n I +1 . (3.6) Final ly the deﬁnition of f ( H ) do es not dep end on N , or on the cut- oﬀ function τ , pr ovide d to N > n I in Deﬁnition 3.1. Mor e over, it is indep endent of the choic e of the interval I as long as I c ontains the supp ort of f and Hyp otheses 1 and 2 hold on I . T o prov e the last prop osition, the id ea is to w ork on Ran(Id − Π I ) ⊕ Ran(Π I ) instead of H , and to analyse the action of H on eac h of these s paces. F or an y in terv al I ⊂ R such that Hyp otheses 1 and 2 h old, the op erator H I , r deﬁned on Ran(Id H − Π I ) satisﬁes the assumptions required b y Da vies [5] to deﬁne f ( H I , r ) via the Helﬀer–Sj¨ ostrand form ula (3.1), for any f ∈ A ( I ). It therefore remains to pro ve th at f ( H I , c ) is w ell deﬁned on Ran(Π I ). This is the p urp ose of the f ollo wing lemma. 10 V. BR UNEAU, N. FRANTZ, AND F. NICOLEAU Lemma 3.7. Supp ose that Hyp otheses 1 and 2 hold on I . L et F = Ran(Π I ) . Then for al l f ∈ A ( I ) and ˜ f an H -admissible almost analytic extension, the inte gr al Z C ∂ ¯ z ˜ f ( z ) R H I , c ( z )d x d y , (3.7) is wel l deﬁne d on F and vanishes. Mor e over, this do es not dep end on the c onsider e d H -admissible almost analytic extension. Pr o of of L emma 3.7. Let f ∈ A ( I ). Let ˜ f b e an H -admissible almost an alytic extension giv en by (3.2). Th en for all inte ger N ≥ 1, a direct computation giv es that ∂ ¯ z ˜ f ( z ) = 1 2 f ( N +1) ( x ) ( iy ) N N ! χ ( x, y ) + 1 2 " N X k =0 f ( k ) ( x ) ( iy ) k k ! # ∂ ¯ z χ ( x, y ) . (3.8) Next a direct computation giv es that ∂ ¯ z χ ( x, y ) = 1 2  − y h a i b x h x i 3 + i h a i b h x i  τ ′ ( χ ( x, y )) . In particular if we denote V 1 =  ( x, y ) ∈ R 2     0 < | y | b < h x i h a i  , V 2 =  ( x, y ) ∈ R 2     h x i 2 h a i ≤ | y | b ≤ h x i h a i  , then su p p( χ ) ⊂ V 1 and supp( ∂ ¯ z χ ) ⊂ V 2 . Then for all z ∈ U a , b , u sing (3.8), w e ha v e    ∂ ¯ z ˜ f ( z ) R H ( z )    B (Ran (Π I )) ≤ C    f ( N +1) ( x )    | y | N 1 V 1 ( z ) + N X k =0    f ( k ) ( x )    | y | k 1 V 2 ( z ) ! ≤ C    f ( N +1) ( x )    | y | N 1 V 1 ( z ) + N X k =0    f ( k ) ( x )    h x i k − 2 1 V 2 ( z ) ! . (3.9) Finally , int egration with r esp ect to y giv es k f ( H ) k B (Ran (Π I )) ≤ C Z C k ∂ ¯ z ˜ f ( z ) R H ( z ) k B (Ran (Π I )) d x d y ≤ C Z R    f ( N +1) ( x )    h x i N d x + N X k =0 Z R    f ( k ) ( x )    h x i k − 1 d x ! ≤ C k f k N +1 . Next to p ro v e th at (3.7) v anishes, we pro ceed by dens ity . Let f ∈ C ∞ c ( I , C ) su pp orted in an op en set Ω. Then ˜ f ∈ C ∞ ( R 2 , C ) and the S tok es’s f orm ula giv es Z C R H I , c ( z ) ∂ ¯ z ˜ f ( z )d x d y = 1 2 i Z ∂ Ω R H I , c ( z ) ˜ f ( z )d z , where ∂ Ω is th e b oun d ary of Ω and the ab ov e con tour integ ral is take n coun terclo c kwise. S o as ˜ f is compactly supp orted, Z ∂ Ω R H I , c ( z ) ˜ f ( z )d z = 0 . Using the same argum en t, we can sho w as in [5, Pro of of Theorem 2], that th e v alue of (3.7) is indep end en t of the H -admissible almost analytic extension considered. Finally Lemma 3.7 follo ws b y using that D ( I ) is dense in A ( I ) and the inequalit y ∀ f ∈ A ( I ) , k f ( H ) k B (Ran (Π I )) ≤ C k f k N +1 .  THE SPECTRAL SHIFT FUNCTION FOR NON- SELF-ADJOINT PER TURBA TIONS 11 Pr o of of Pr op osition 3.6. Let f ∈ A ( I ). By Prop osition 3.3 and L emma 3.2, ther e exists an isomor- phism Ψ : Ran(Π I ( H )) ⊕ Ran(Id − Π I ( H )) → H suc h that for all z ∈ ρ ( H ), R H ( z ) = Ψ  R H I , r ( z ) 0 0 R H I , c ( z )  Ψ − 1 . So us ing Da vies’s w ork [5], f ( H I , r ) is wel l deﬁned as an op erator on Ran(Id − Π I ) and its deﬁ n ition do es not dep end on th e almost-analyt ic extension considered in (3.1) as so on as its ord er is larger than n I in Hyp othesis 2. Moreo ve r, b y Lemma 3.7 f ( H I , c ) is we ll-deﬁned indep endent ly of the H - admissible almost-analyt ic extension of f . This also give s indep endence in the c hoice of th e interv al I con taining the su p p ort of f (where Hyp otheses 1 and 2 are f u lﬁlled) b ecause for I ⊂ J , w e h a v e H J, c = H I , c ⊕ H J \ I , c ; H I , r = H J, r ⊕ H J \ I , c and then f ( H I , r ) = f ( H J, r ) ⊕ 0.  In p articular in the follo wing, we m a y write f ( H ) = Ψ  f ( H I , r ) 0 0 0  Ψ − 1 . (3.10) This repr esen tation of f ( H ) and [5, Theorem 3 and 4] implies im m ediately the follo w ing Prop osition: Prop osition 3.8. L e t I ⊂ R b e an op en interval and supp ose that Hyp otheses 1 and 2 hold on I . Then the map A ( I ) ∋ f 7→ f ( H ) ∈ B ( H ) is a morphism of algebr a for the p ointwise multiplic ation. M or e over if f ∈ C ∞ c ( I , C ) has disjoint supp ort fr om σ ( H ) , then f ( H ) = 0 . 3.1.3. Comp arison with F r antz-F aupin c alculus. In [11, P rop osition 5.2], F ran tz an d F aupin in tro- duce a functional calculus und er similar assum ptions. Their construction is insp ired by Stone’s form ula: f ( H ) = 1 2 π i lim ε → 0 + Z supp( f ) f ( λ ) ( R H ( λ + iε ) − R H ( λ − iε )) d λ. (3.11) They prov e that the formula (3.11) makes sense in the weak top ology for fu nctions f in C b ( I , C ), where I is an interv al of the essential sp ectrum without s p ectral singularities (see Section 6 for the deﬁnition) and wh ere C b ( I , C ) d enotes the space of b ounded con tin uous fun ctions on I . They dev elop also a functional calculus whic h tak es in to account sp ectral sin gularities (see [11, Prop osition 5.3]). Th ese ap p roac hes require a limiting abs orp tion principle for H 0 , and the pr o ofs are based on the interc h ange of a limit and an in tegral. I n the follo wing pr op osition, we show th at one can giv e a meaning to the formula (3.11) by means of th e Helﬀer-Sj¨ ostrand f orm ula constructed ab o v e at least for compactly sup p orted fun ctions. Prop osition 3.9. L e t I ⊂ R b e an op en interval and supp ose that Hyp otheses 1 and 2 hold on I . L et f ∈ C ∞ c ( I , C ) . Then f ( H ) = 1 2 π i lim ε → 0 + Z supp( f ) f ( λ ) ( R H ( λ + iε ) − R H ( λ − iε )) d λ. 12 V. BR UNEAU, N. FRANTZ, AND F. NICOLEAU Pr o of. W e claim th at f ( H ) = π − 1 lim ε → 0 + Z Im( z ) > 0 ∂ ¯ z ˜ f ( z ) R H ( z + iε )d z + Z Im( z ) < 0 ∂ ¯ z ˜ f ( z ) R H ( z − iε )d z ! . (3.12) Indeed, if we denote J := su pp( f ) ⊂ I , there exists a I > 0 suﬃcient ly small suc h that su pp( ˜ f ) ⊂ S a I ( J ) and S a I ( J ) do es n ot con tain any non real eigen v alue of H . S o, und er Hyp otheses 1 and 2, for ε > 0 su ﬃcien tly small the m ap z → ∂ ¯ z ˜ f ( z ) R H ( z ± iε ) is con tin uous on supp( ˜ f ) ∩ C ± and for all z ∈ S a I ( J ), w e ha v e lim ε → 0 + R H ( z ± iε ) = R H ( z ) , k ∂ ¯ z ˜ f ( z ) R H ( z ± iε ) k B ( H ) ≤ c 0 for some constant c 0 > 0, uniform ly with resp ect to ε and z . Consequently , L eb esgue’s theorem implies (3.12). No w since z 7→ R H ( z ± iε ) are holomorph ic on S a I ( J ), w e may app ly Stok es’s theorem to ob tain Z ± Im( z ) > 0 ∂ ¯ z ˜ f ( z ) R H ( z ± iε )d x d y = ± 1 2 i Z Γ ± ˜ f ( z ) R H ( z ± iε )d z , (3.13) where Γ ± denotes the b ound ary of S a I ( J ) in C ± . Finally the supp ort prop erties of ˜ f imp lies that Z Γ ± ˜ f ( z ) R H ( z ± iε )d z = Z J f ( λ ) R H ( λ ± iε )d λ. (3.14) So the conclusion follo ws by substituting ﬁ rst (3.13) and then (3.14) in (3.12).  T o conclude this introd uction to the functional calculus, observ e that if ω ∈ ρ ( H ) and r ω ( x ) = ( x − ω ) − 1 , then r ω ( H ) is precisely the r esolven t ( H − ω ) − 1 . More precisely , un der the Hyp othesis 1 on I = R , H has a ﬁnite n umb er of n o-real eigen v alues an d the p ro jection on to the n on -r eal eigen v alues of H is w ell d eﬁ ned by: Π complex ( H ) := X λ ∈ σ disc ( H ) \ R Π λ ( H ) . Then we ha v e: Prop osition 3.10. Supp ose that Hyp otheses 1 and 2 hold on R . L et ω ∈ ρ ( H ) and r ω : R → C b e deﬁne d for al l x ∈ R by r ω ( x ) = ( x − ω ) − 1 . Then r ω ( H ) = R H ( ω )(Id − Π complex ( H )) , R H ( ω ) = r ω ( H ) + X λ ∈ σ disc ( H ) \ R ( λ − ω ) − 1 Π λ ( H ) . (3.15) Pr o of. Applying P r op osition 3.3 for I = R , we introd uce the real part of H : H r the restriction of H to the range of Id − Π complex ( H ). Using (3.10) and [5, Theorem 5], f or all u ∈ H , w e h a v e : r ω ( H ) u = Ψ  ( H r − ω ) − 1 0 0 0  Ψ − 1 u = ( H r − ω ) − 1 (Id − Π complex ( H )) u = ( H − ω ) − 1 (Id − Π complex ( H )) u. Then we deduce (3.15).  THE SPECTRAL SHIFT FUNCTION FOR NON- SELF-ADJOINT PER TURBA TIONS 13 3.2. Sp ectra l changing of v ariables. In this section, for any − c ∈ R ∩ ρ ( H ) with c > 0, and m ∈ N ∗ , we consider a function f ∈ D ((0 , + ∞ )) and deﬁne g m ∈ D (( − c, + ∞ )) by the r elation g m ( λ ) := f (( λ + c ) − m ) . Our goal is to iden tify su ﬃcien t conditions on H un der wh ic h the follo wing comp osition form ula holds: f (( H + c ) − m ) = g m ( H ) . Initially , instead of H + c , we consider a general op erator L . F or in terv als I ⊂ R , R ⊂ (0 , + ∞ ), Θ 0 ⊂ [ − π / 2 , π / 2], and a > 0, w e in tro du ce the follo win g subset of the complex plane: R · e i Θ 0 := n ρe iθ ∈ C    ρ ∈ R, θ ∈ Θ 0 o . (3.16) T o establish relations b et w een the r esolven ts of L and those of L − m , we state the follo wing lemma concerning geometric sums of b oun ded op erators. Lemma 3.11. L et m ∈ N ∗ and let T ∈ B ( H ) b e a b ounde d op er ator such that the ge ometric sum G m ( T ) := m − 1 X k =0 T k (3.17) is invertible. Then ( T − Id) is invertible if and only i f ( T m − Id) is invertible, and in this c ase the fol lowing identities hold: ( T − Id) − 1 = G m ( T )( T m − Id) − 1 (3.18) = m ( T m − Id) − 1 + m − 1 X k =1 G k ( T ) G m ( T ) − 1 . (3.19) Pr o of. The ﬁ rst identit y follo ws from the factorization ( T m − Id) = ( T − Id) G m ( T ). F or the second iden tit y , w e wr ite for all k ∈ N ⋆ , T k = Id + ( T k − Id) = I + G k ( T )( T − Id ) and use the fact that ( T − Id)( T m − Id) − 1 = G m ( T ) − 1 .  W e deﬁn e C > the set of complex n umb er with p ositiv e real part : C > := { z ∈ C | Re ( z ) > 0 } . By applying the ab ov e lemma with T = z L − 1 , we deduce: Lemma 3.12. L et m ∈ N ∗ and let ( L, D ( L )) b e a close d op er ator on a c omplex sep ar able Hilb ert sp ac e H , such that σ ( L ) ⊂ (0 , + ∞ ) · e i [ − θ 0 ,θ 0 ] for some θ 0 ∈  0 , π 2 m  . (3.20) Then, (1) z ∈ ρ ( L ) \{ 0 } if and only if z − m ∈ ρ ( L − m ) , (2) the op er ator-value d map Ψ : z 7− → z − m +1 G m ( z L − 1 ) = m − 1 X k =0 z k +1 − m L − k (3.21) is biholomorphic in the se ctor (0 , + ∞ ) · e i [ − θ 0 ,θ 0 ] , 14 V. BR UNEAU, N. FRANTZ, AND F. NICOLEAU (3) for every z ∈ ρ ( L ) ∩ (0 , + ∞ ) · e i [ − θ 0 ,θ 0 ] , the fol lowing r esolvent identities hold: ( L − z ) − 1 = − z − m L − 1 G m ( z L − 1 )  L − m − z − m  − 1 , (3.22) = − mz − 1 − m  L − m − z − m  − 1 − z − 1 m − 1 X k =1 G k ( z L − 1 ) G m ( z L − 1 ) − 1 . (3.23) Pr o of. Since 0 ∈ ρ ( L ), it follo ws f r om holomorphic fun ctional calculus of closed op erator ([7], Theorem VI I.9.8.4) that for any k ∈ N , th e op erator L − k is b ound ed and its sp ectrum is conta ined in the sector σ ( L − k ) = { z − k | z ∈ σ ( L ) } ∪ { 0 } ⊂ [0 , + ∞ ) · e i [ − k θ 0 , k θ 0 ] . This p ro v e item (1). On the other h and, for an y k ∈ J 0 , m − 1 K , for any z ∈ (0 , + ∞ ) · e i [ − θ 0 , θ 0 ] , we ha v e z k +1 − m ∈ (0 , + ∞ ) · e i [ − ( k +1 − m ) θ 0 , ( k +1 − m ) θ 0 ] . Therefore, the sp ectrum of the op erator z k +1 − m L − k is con tained in the sector [0 , + ∞ ) · e i [ − ( m − 1) θ 0 , ( m − 1) θ 0 ] ⊂ [0 , + ∞ ) · e i ( − π 2 , π 2 ) . As sh o wn in ([16], P r op osition 3.2.10), if t wo b oun ded linear op erators A and B commute, th en the sp ectrum of th eir sum satisﬁes the inclus ion σ ( A + B ) ⊂ σ ( A ) + σ ( B ). S o, it follo ws that th e sp ectrum of z − m +1  G m ( z L − 1 ) − Id  = m − 1 X k =1 z k +1 − m L − k is con tained in C > . S o using one against Prop osition 3.2.10, we deduce that, for all z ∈ (0 , + ∞ ) · e i [ − θ 0 ,θ 0 ] , the sp ectrum of the op erator z − m +1 G m ( z L − 1 ) is cont ained in the op en right h alf-plane C > . In particular, th e op erator G m ( z L − 1 ) is inv ertible for all z ∈ C > . Applying (3.18) with T = z L − 1 , we obtain for ev ery z ∈ ρ ( L ) ∩ (0 , + ∞ ) · e i [ − θ 0 ,θ 0 ] : ( L − z ) − 1 = L − 1 (Id − z L − 1 ) − 1 = − L − 1 G m ( z L − 1 )  ( z L − 1 ) m − Id  − 1 = − L − 1 z − m G m ( z L − 1 )  L − m − z − m  − 1 , whic h yields identi ty (3.22). Ident it y (3.23) then follo ws from the second formula in Lemma 3.11 as follo ws: ( L − z ) − 1 = L − 1 (Id − z L − 1 ) − 1 = − z − 1 Id − z − 1 ( z L − 1 − Id) − 1 = − z − 1 Id − mz − 1 − m  L − m − z − m  − 1 − z − 1 m − 1 X k =1 G k ( z L − 1 ) G m ( z L − 1 ) − 1 . The biholomorphy of Ψ can b e prov ed by us ing Neumann series.  Prop osition 3.13. Supp ose that Hyp otheses 1 and 2 hold for H on I = ( s 0 , s 1 ) , s 1 > s 0 . Fix m ∈ N ∗ . Then ther e exists c > 0 such that for any J = ( r 0 , r 1 ) , with 0 < ( s 1 + c ) − m < r 0 < r 1 < ( s 0 + c ) − m the op er ator ( H + c ) − m satisﬁes Hyp otheses 1 and 2 on J with n J = n I . Mor e over, for any f ∈ D ( J ) , we have f (( H + c ) − m ) = g m ( H ) , THE SPECTRAL SHIFT FUNCTION FOR NON- SELF-ADJOINT PER TURBA TIONS 15 wher e g m ∈ D ( I ) is deﬁne d b y g m ( λ ) = f (( λ + c ) − m ) . Pr o of. Let u s ﬁx θ 0 ∈ (0 , π / 2). As H 0 is b ounded from b elo w and V is b ounded, there exists c > 0 suc h th at σ ( H + c ) ⊂ (0 , + ∞ ) · e i [ − θ 0 ,θ 0 ] . Moreo v er, since σ ess ( H ) = σ ess ( H 0 ), eac h n on-real num b er in σ ( H + c ) is a discrete eigen v alue. By using Lemma 3.12 with L = H + c and the biholomorphic function ϕ m : − c + (0 , + ∞ ) · e i [ − θ 0 ,θ 0 ] − → (0 , + ∞ ) · e i [ − mθ 0 ,mθ 0 ] (3.24) z = − c + ρe iθ 7− → ( z + c ) − m = ρ − m e − imθ , (3.25) w e ha v e σ (( H + c ) − m ) ⊂ (0 , + ∞ ) · e i [ − mθ 0 ,mθ 0 ] ⊂ C > and the n on-real sp ectrum of ( H + c ) − m consists of d iscrete eigen v alues Λ = ϕ m ( λ ) with λ a non-real eigen v alue of H . S o if H satisﬁes Hyp othesis 1, no p oint in the inte rv al I is an accum ulation p oint of d iscrete eigen v alues and Hyp othesis 1 holds true for ( H + c ) − m on ϕ m ( I ) = (( s 1 + c ) − m , ( s 0 + c ) − m ). Moreo ver, u nder the Hyp othesis 2 for H on I , there exists a I > 0 su ch that S a I ( I ) ⊂ ρ ( H ) and then, ϕ m ( S a I ( I )) ⊂ ρ (( H + c ) − m ). Let J = ( r 0 , r 1 ) w ith ( s 1 + c ) − m < r 0 < r 1 < ( s 0 + c ) − m . By taking a J > 0 suﬃcien tly sm all suc h th at S a J ( J ) ⊂ ϕ m ( S a I ( I )), we ha v e S a J ( J )) ⊂ ρ (( H + c ) − m ) and for Z ∈ S a J ( J ) there exists z ∈ S a I ( I ) such th at Z = ϕ m ( z ). By combining (3.22) f or L = H + c (and for z + c instead of z ) with Hyp othesis 2 for H on I , w e obtain: k  ( H + c ) − m − Z  − 1 k B ( H ) ≤ sup z ∈ S a I ( I ) k (( z + c ) − m G m (( z + c )( H + c ) − 1 )) − 1 k k ( H + c )( H − z ) − 1 k ≤ C J | Im( z ) | − n I − 1 , for some C J > 0. W riting z = − c + ρe iθ , with θ ∈ ( − π 2 m , π 2 m ) (it is p ossib le for a I suﬃcien tly small), w e hav e Z = ( z + c ) − m = ρ − m e − imθ , and Im ( Z ) = − ρ − m sin( mθ ) . Thanks to the inequalit y 2 π x ≤ sin( x ) ≤ x for x ∈ [0 , π / 2], w e deduce ρ − ( m +1) 2 π m | Im( z ) | ≤ 2 π ρ − m m | θ | ≤ | Im( Z ) | ≤ ρ − m m | θ | ≤ ρ − ( m +1) π 2 m | Im( z ) | , (3.26) and the Hyp othesis 2 holds for ( H + c ) − m on J . No w let f ∈ D ( J ) and let ˜ f ∈ C ∞ c ( C ) b e an almost analytic extension of f , supp orted in J × ( − a J , a J ). Using the c hange of v ariables Z = ϕ m ( z ) (with Z = X + iY , z = x + iy ) in the functional calculus form ula f (( H + c ) − m ) := π − 1 Z C ∂ Z ˜ f ( Z )  ( H + c ) − m − Z  − 1 d X d Y , w e obtain f (( H + c ) − m ) := π − 1 Z C ∂ Z ˜ f ( ϕ m ( z ))  ( H + c ) − m − ( z + c ) − m  − 1 | ∂ z ϕ m ( z ) | 2 d x d y . F rom (3.26), we kno w that th e function ˜ g m := ˜ f ◦ ϕ m is an almost analytic extension of g m = f ◦ ϕ m , and satisﬁes ∂ ¯ z ˜ g m ( z ) = ∂ z ϕ m ( ¯ z ) · ∂ Z ˜ f ( ϕ m ( z )) . Then, using the identit y (3.23) from Lemma 3.12, and the fact that the map z 7→ m − 1 X k =1 G k (( z + c )( H + c ) − 1 ) G m (( z + c )( H + c ) − 1 ) − 1 16 V. BR UNEAU, N. FRANTZ, AND F. NICOLEAU is holomorphic on the supp ort of ˜ g m , we obtain f (( H + c ) − m ) = π − 1 Z R 2 ∂ ¯ z ˜ g m ( z )  − ( z + c ) m +1 m ( H − z ) − 1  ∂ z ϕ m ( z ) d x d y = g m ( H ) , where we u se the iden tit y − ( z + c ) m +1 m · ∂ z ϕ m ( z ) = 1.  4. S SF for Trace cla ss p er turba tions 4.1. Existence. Th e follo w ing theorem pro vides th e k ey analytic framew ork n eeded to deﬁn e the sp ectral shift fun ction (SS F) in a general non-self-adjoin t setting. It ensures the trace-cla ss p rop erty of the diﬀerence f ( H ) − f ( H 0 ) under su itable assumptions, allo wing one to asso ciate a distrib ution to this diﬀerence. Theorem 4.1. Su pp ose that Hyp otheses 1, 2 and 3 hold on an op en interval I with m = − 1 . Then, for al l f ∈ D ( I ) the op er ator diﬀer enc e f ( H ) − f ( H 0 ) b elongs to L 1 ( H ) . Mor e over, the map f 7− → T r  f ( H ) − f ( H 0 )  (4.1) deﬁnes a distribution on I which vanishes on I ∩ ρ ( H ) ∩ ρ ( H 0 ) . Pr o of. Let ˜ f b e an almost analytic extension of f of ord er n I + 2, s upp orted in S a ( K ), with K a compact subinterv al of I con taining th e sup p ort of f an d a > 0 chosen small en ough so that σ ( H ) ∩ S a ( K ) = ∅ . Then, by d eﬁnition, we hav e f ( H ) − f ( H 0 ) = π − 1 Z C ∂ ¯ z ˜ f ( z ) ( R H ( z ) − R 0 ( z )) d x d y = − π − 1 Z C ∂ ¯ z ˜ f ( z ) R H ( z ) V R 0 ( z ) d x d y = − π − 1 Z Im( z ) > 0 ∂ ¯ z ˜ f ( z ) R H ( z ) V R 0 ( z ) d x d y + Z Im( z ) < 0 ∂ ¯ z ˜ f ( z ) R H ( z ) V R 0 ( z ) d x d y ! . F or all z ∈ ρ ( H ) ∩ ρ ( H 0 ), ∂ ¯ z ˜ f ( z ) R H ( z ) V R 0 ( z ) is of trace class and since ˜ f is compactly sup p orted, this shows that the full integral b elongs to L 1 ( H ). Finally , in the s ame wa y as (3.9), w e get | T r ( f ( H ) − f ( H 0 )) | ≤ β K k V k L 1 k f k n K +2 , whic h prov e th at (4.1) is a distribution on I . Finally Prop osition 3.8 implies that the distr ib ution v anishes on R ∩ ρ ( H ) ∩ ρ ( H 0 ).  Under the assum ptions of T heorem 4.1, the trace formula n aturally leads to the deﬁnition of the Sp ectral Sh ift F unction on I , which captures the v ariation of the sp ectrum under p erturb ation. This function is in itially d eﬁned only up to an additive constan t, whic h can b e ﬁxed b y prescrib in g its v alue on a sp ectral gap wh en suc h a gap exists in I ∩ ρ ( H ) ∩ ρ ( H 0 ). Deﬁnition 4.2. Supp ose that Hyp otheses 1, 2 and 3 hold on an op en interval I with m = − 1 . The sp ectral sh ift function ξ ( · ; H , H 0 ) on I is deﬁne d, u p to an additive c onstant, as the distribution satisfying h ξ ′ , f i := T r  f ( H ) − f ( H 0 )  , for al l f ∈ D ( I ) . When I ∩ ρ ( H ) ∩ ρ ( H 0 ) is nonempty (for example, if I = R and b oth H 0 and H ar e b ounde d or semi-b ounde d), the additive c onstant c an b e uni q uely determine d by pr escribing the value of ξ on a sp e ctr al gap, i.e., an interval c ontaine d in I ∩ ρ ( H ) ∩ ρ ( H 0 ) . THE SPECTRAL SHIFT FUNCTION FOR NON- SELF-ADJOINT PER TURBA TIONS 17 4.2. A ﬁrst representation formula for t he SSF. O nce the sp ectral shift f unction (SS F) h as b een deﬁn ed, w e aim to establish more precise p rop erties, starting w ith a ﬁrst representa tion for- m ula. F or z ∈ ρ ( H ) ∩ ρ ( H 0 ), we deﬁne σ ( z ) := T r  R H ( z ) − R 0 ( z )  . This quantit y is w ell-deﬁned whenev er H − H 0 ∈ L 1 ( H ), as ensu red by the resolv ent iden tit y . Ou r goal is to show that the deriv ativ e of the sp ectral shift fu nction, as deﬁned in Deﬁnition 4.2, admits a concrete represen tation in terms of the discon tin uity of σ ( z ) across the real axis. Prop osition 4.3. L et I ⊂ R b e an op en interval. Supp ose that H yp otheses 1, 2 and 3 hold on I with m = − 1 . Then, in the sense of distributions on I , we have ξ ′ ( · ; H , H 0 ) = 1 2 π i lim ε → 0 + ( σ ( · + iε ) − σ ( · − iε )) . (4.2) Remark 4.4. Note that the right-hand side of (4.2) vanishes on the set I ∩ ρ ( H ) ∩ ρ ( H 0 ) , sinc e σ ( · ) is wel l-deﬁne d and c ontinuous ther e. Mor e pr e cisely, for any λ ∈ I ∩ ρ ( H ) ∩ ρ ( H 0 ) , we have lim ε → 0 + σ ( λ ± iε ) = σ ( λ ) . Pr o of. Let f ∈ D ( I ) b e a test function supp orted in a compact interv al K ⊂ I , and let ˜ f b e an almost analytic extension supp orted in S a ( K ), w here a > 0 is choosen sm all enough so that σ ( H ) ∩ S a ( K ) = ∅ . Then, we w rite: h ξ ′ , f i = π − 1 Z C ∂ ¯ z ˜ f ( z ) T r ( R H ( z ) − R 0 ( z )) d x d y = π − 1 lim ε → 0 + Z Im( z ) > 0 ∂ ¯ z ˜ f ( z ) σ ( z + iε ) d x d y + lim ε → 0 + Z Im( z ) < 0 ∂ ¯ z ˜ f ( z ) σ ( z − iε ) d x d y ! , (4 .3) where σ ( z ) := T r ( R H ( z ) − R 0 ( z )). Indeed, by construction, the s u pp ort of ˜ f do es not con tain any complex eigen v alue of H . Hence, for ε > 0 small enough, the map z 7→ ∂ ¯ z ˜ f ( z ) R H ( z ± iε ) is con tin uous on supp( ˜ f ). Moreo ver, since H − H 0 ∈ L 1 ( H ), resolv ent iden tit y im p lies that R H ( z ) − R 0 ( z ) is trace-cla ss for z ∈ ρ ( H ) ∩ ρ ( H 0 ), and we h a v e lim ε → 0 + σ ( z ± iε ) = σ ( z ) , for all z ∈ S a ( K ) with ± Im( z ) > 0 . In addition, for all suﬃcien tly small ε > 0 and all z ∈ S a ( K ) with ± Im( z ) > 0, w e estimate    ∂ ¯ z ˜ f ( z ) σ ( z ± iε )    =    ∂ ¯ z ˜ f ( z ) T r ( R H ( z ± iε ) V R 0 ( z ± iε ))    ≤    ∂ ¯ z ˜ f ( z )    · k V k L 1 ( H ) · kR H ( z ± iε ) k · kR 0 ( z ± iε ) k . No w, since z 7→ σ ( z ± iε ) is holomorphic on S a ( K ) ∩ C ± , we may apply S tok es’ theorem to obtain: Z ± Im( z ) > 0 ∂ ¯ z ˜ f ( z ) σ ( z ± iε ) d x d y = ± 1 2 i Z Γ ± a ˜ f ( λ ) σ ( λ ± iε ) d λ, (4.4) where Γ ± a denotes the b oundary of S a ( K ) ∩ C ± . This b oundary integral can b e decomp osed into four parts, three of whic h v anish due to the supp ort prop erties of ˜ f . Therefore, ± 1 2 i Z Γ ± a ˜ f ( λ ) σ ( λ ± iε ) d λ = ± 1 2 i Z supp( f ) f ( λ ) σ ( λ ± iε ) d λ. (4.5) Substituting (4.5) into (4.3) yields the d esired identit y (4.2).  18 V. BR UNEAU, N. FRANTZ, AND F. NICOLEAU Remark 4.5. Pr op osition 4.3 al lows to give the fol lowing extension to non-selfadjoint op er ators of the r epr esentation formula (1.2) . U nder the assumptions of The or em 4.1, let us intr o duc e the p erturb ation determinant D V ( z ) := Det  ( H 0 + V − z )( H 0 − z ) − 1  = Det  I + V ( H 0 − z ) − 1  . (4.6) It is a non vanishing analytic function on e ach S a ( K ) , K ⊂ I c omp act (b e c ause it is in the r esolvent set of H and H 0 ) and it satisﬁes (se e e.g. [31, Chapter 8] ): D V ( z ) − 1 D ′ V ( z ) = T r  R H ( z ) − R 0 ( z )  = σ ( z ) . It fol lows that, up to a c onstant, the function ln D V ( z ) i s wel l deﬁne d on e ach S a ( K ) and satisﬁes: d dλ ln D V ( λ ± iε ) = σ ( λ ± iε ) , λ ∈ J, ε > 0 . Then, up to a c onstant, Pr op osition 4.3 sug g est the fol lowing extension of (1.2) (when the limit exists): ξ ( λ ; H , H 0 ) = 1 2 π i lim ε → 0 + (ln D V ( λ + iε ) − ln D V ( λ − iε )) . (4.7) Of c ourse, when H 0 + V is selfadjoint, it c orr esp onds to (1.2) b e c ause in this c ase, ln D V ( λ − iε ) i s the c omplex c onjugate of ln D V ( λ + iε ) . Remark 4.6. F r om the ab ove r e sults it fol lows that the SSF for the adjoint H ∗ c oincides with the c omplex c onjugate of the SSF for H : ξ ( λ ; H ∗ , H 0 ) = ξ ( λ ; H , H 0 ) . (4.8) 5. The SSF for non-self adjoint r ela tivel y trace class per tu rba tions In man y applications, the diﬀerence H − H 0 do es not b elong to th e trace class. Ho w ev er, the diﬀerence b et we en their resolv en ts, or b et we en p o wers of their resolv ents, ma y indeed b e trace class. The aim of this section is to extend the d eﬁnition of th e sp ectral shift fu n ction (SSF) to such settings. The cen tral idea is to consider the p air of op erators ( X, X 0 ) :=  ( H + c ) − m , ( H 0 + c ) − m  , where c ∈ R and m ∈ N ∗ are chosen so th at X − X 0 ∈ L 1 ( H ). In this case, Deﬁnition 4.2 can b e applied directly to ( X , X 0 ), thus deﬁn ing a sp ectral sh ift function for this p air. Th en, to r etriev e the S SF asso ciated with the original p air ( H, H 0 ), we p erform a c hange of v ariables: for λ > − c , the sp ectral parameter µ for X is take n as µ = ( λ + c ) − m . Accordingly , we deﬁn e ξ ( λ ; H , H 0 ) := ξ  ( λ + c ) − m ; ( H + c ) − m , ( H 0 + c ) − m  . (5.1) 5.1. Deﬁnition of the SSF for relat iv ely trace class p erturbations. W e b egin with a theorem that allo ws the deﬁ n ition of the sp ectral sh ift fun ction (SSF) in the case where th e p erturb ation is relativ ely trace class, i.e., wh en a su itable p o w er of the resolv en t diﬀerence b elongs to L 1 ( H ). Prop osition 5.1. L et ( H 0 , D ( H 0 )) b e a self-adjoint, semi-b ounde d op er ator, and let ( H , D ( H 0 )) b e a close d op er ator acting on a c omplex sep ar able Hilb ert sp ac e H . Assume ther e exist c ∈ R and m ∈ N ∗ such that: (1) V := H − H 0 is a b ounde d op e r ator (2) The op er ator H + c is inve rtible with σ  ( H + c ) − m  ⊂ (0 , + ∞ ) · e i ( − π 2 , π 2 ) (5.2) THE SPECTRAL SHIFT FUNCTION FOR NON- SELF-ADJOINT PER TURBA TIONS 19 (3) Hyp otheses 1, 2 and 3 hold on I = ( s 0 , s 1 ) , s 0 < s 1 for some m ∈ N ∗ Then the sp e ctr al shift function ξ c,m := ξ ( · ; ( H + c ) − m , ( H 0 + c ) − m ) i s wel l-deﬁne d on (( s 1 + c ) − m , ( s 0 + c ) − m ) up to a c onstant. Mor e over, the distribution ξ ( · ; H , H 0 ) deﬁne d by ξ ( λ ; H , H 0 ) := ξ c,m (( λ + c ) − m ) (5.3) for λ ∈ I = ( s 0 , s 1 ) , is indep e ndent of the choic e of ( c, m ) and satisﬁes the tr ac e formula (1.1) . Uniqueness is guar ante e d by cho osing ξ ( · ; H , H 0 ) = 0 on a sp e ctr al gap (if it exists). Pr o of. Since the sp ectrum of ( H + c ) − m lies in (0 , + ∞ ) · e i ( − π 2 , π 2 ) , there exists θ 0 ∈ ]0 , π 2 m [ suc h that σ ( H + c ) ⊂ ]0 , + ∞ [ e i [ − θ 0 ,θ 0 ] . Then, by Pr op osition 3.13, for any inte rv al J = ( r 0 , r 1 ) relativ ely compact in (( s 1 + c ) − m , ( s 0 + c ) − m ) ⊂ ]0 , + ∞ [, the Hyp otheses 1, 2 are satisﬁed b y ( H + c ) − m on J . Applyin g Theorem 4.1, the sp ectral shift function ξ m,c := ξ ( · ; ( H + c ) − m , ( H 0 + c ) − m ) is therefore well d eﬁned on J , up to an additiv e constan t. F or ev ery f ∈ D ( I ), w e ha v e: T r  f (( H + c ) − m ) − f (( H 0 + c ) − m )  = −h ξ ′ m,c , f i . F ur thermore, for an y g ∈ D ((( s 1 + c ) − m , ( s 0 + c ) − m )), Pr op osition 3.13 (applied to the function f ∈ D ( I ) d eﬁned by f ( µ ) = g ( µ − 1 m − c )) yields: f (( H + c ) − m ) = g ( H ) , and consequent ly: T r  g ( H ) − g ( H 0 )  = h ξ , g ′ i . Since g ( H ) and g ( H 0 ) are indep endent of the choice of ( m, c ), the distrib ution ξ is u niquely d eter- mined on I by the normalization ξ = 0 on a sp ectral gap (if it exists).  Thanks to this P r op osition, in th e con text of relativ ely trace-class p erturb ations, the s p ectral shift fu nction can still b e deﬁned, as stated in the follo wing d eﬁ nition. Deﬁnition 5.2. Under the assumptions of Pr op osition 5.1, we deﬁne the Sp e ctr al Shift F unction (SSF) asso ciate d with the p air ( H , H 0 ) by the r elation (5.3) . Remark 5.3. L et s m in := inf σ ( H ) ∩ σ ( H 0 ) ∩ R . If, in Pr op osition 5.1, s 0 < s m in , then the SSF c an b e extende d to 0 thr oughout the sp e ctr al gap ( −∞ , s m in ) . 5.2. Representation form ula in t he case of relativ ely trace class p erturbations. In this section, w e establish an analogue of Lemma 4.3 in th e framework w here th e p ertur bation is only relativ ely trace class, rather than trace class. This generaliza tion is essen tial for extending the trace form ula and the d eﬁnition of the s p ectral shift function to broader classes of op erators. Prop osition 5.4. Supp ose that the p air ( H 0 , H ) satisﬁes the assumptions of Pr op osition 5.1 for some I = ( s 0 , s 1 ) , c ∈ R and m ≥ 1 , and that ( H 0 + c ) − m  ( H + c ) − 1 − ( H 0 + c ) − 1  ∈ L 1 ( H ) . (5.4) Then, in the sense of distributions, on I , we have: ξ ′ ( · ; H , H 0 ) = 1 2 π i lim ε → 0 + ( σ m ( · + iε ) − σ m ( · − iε )) , (5.5) wher e σ m ( z ) := ( z + c ) m − 1 T r  ( H + c ) − m +1 R H ( z ) − ( H 0 + c ) − m +1 R 0 ( z )  , Im( z ) 6 = 0 . 20 V. BR UNEAU, N. FRANTZ, AND F. NICOLEAU Pr o of. Let u s denote X = ( H + c ) − m and X 0 = ( H 0 + c ) − m . Using the iden tit y ( H + c ) − m +1 R H ( z ) − ( H 0 + c ) − m +1 R 0 ( z ) = ( X − X 0 )( H + c ) R H ( z ) + X 0  ( H + c ) R H ( z ) − ( H 0 + c ) R 0 ( z )  , together with the resolv en t ident ity an d the strong mapping sp ectral theorem (see e.g., [25, Lemma 2, XI I I.4]), on e chec ks that σ m ( z ) is well-deﬁned for all z ∈ ρ ( H ) ∩ ρ ( H 0 ) (see also the relation (5.6) b elo w for furth er justiﬁcation). Moreo ver, f or λ ∈ R ∩ ρ ( H ) ∩ ρ ( H 0 ), we ha ve lim ε → 0 + σ m ( λ + iε ) = σ m ( λ ) = lim ε → 0 + σ m ( λ − iε ) , so that b oth side of (5.5) v anishes on R ∩ ρ ( H ) ∩ ρ ( H 0 ) ⊃ ( − ∞ , inf ( σ ( H ) ∪ σ ( H 0 ))) ⊃ ( −∞ , − c ]. By Deﬁnition 5.2, on I , we hav e ξ ′ ( λ ; H , H 0 ) = − m ( λ + c ) − m − 1 ξ ′ c,m (( λ + c ) − m ) . W e now app ly Prop osition 4.3 to the p air (( H + c ) − m , ( H 0 + c ) − m ), which yields, in the sense of distributions: ξ ′ c,m ( µ ) = 1 2 π i lim ε → 0 + (Σ( µ + iε ) − Σ( µ − iε )) , where Σ( Z ) = T r  ( X − Z ) − 1 − ( X 0 − Z ) − 1  , Im( Z ) 6 = 0 . F or Z = µ ± iε with µ > 0, ε > 0, there exists z = λ ∓ iδ ( ε ) w ith λ > − c and δ ( ε ) → 0 + suc h that Z = ( z + c ) − m . Using Lemma 3.12 for with L = H + c and w ith z + c instead of z , w e ha v e: ( H + c ) − m +1 R H ( z ) = X ( H + c ) R H ( z ) = X + ( z + c ) X R H ( z ) = X − mZ X ( X − Z ) − 1 − X B H ( z ) = X − mZ − mZ 2 ( X − Z ) − 1 − X B H ( z ) where B H ( z ) := m − 1 X k =1 G k (( z + c )( H + c ) − 1 ) G m (( z + c )( H + c ) − 1 ) − 1 , is holomorphic near I × { 0 } . The same iden tit y holds for H 0 . Ther efore, we obtain ( z + c ) m − 1  ( H + c ) − m +1 R H ( z ) − ( H 0 + c ) − m +1 R 0 ( z )  = ( z + c ) m − 1 ( X − X 0 ) − m ( z + c ) − m − 1  ( X − Z ) − 1 − ( X 0 − Z ) − 1  (5.6) − ( z + c ) m − 1  X B H ( z ) − X 0 B H 0 ( z )  . The ﬁ rst and third terms in the right -hand side of (5.6) are h olomorph ic n ear the r eal axis and hence do n ot contribute to the jump in the limit. Th e only con tribution to the diﬀerence of b oun dary v alues comes from the second term. T his yields − m ( λ + c ) − m − 1 lim ε → 0 +  Σ(( λ + c ) − m + iε ) − Σ(( λ + c ) − m − iε )  = lim ε → 0 + ( σ m ( λ + iε ) − σ m ( λ − iε )) , whic h pro v es (5.5). Finally , w e brieﬂy ju stify the trace class prop ert y of the th ird term in (5.6) . By the structure of B H ( z ) and the assumption (5.4), one can v erify that z 7→ X 0  B H ( z ) − B H 0 ( z )  is holomorphic with v alues in L 1 ( H ).  THE SPECTRAL SHIFT FUNCTION FOR NON- SELF-ADJOINT PER TURBA TIONS 21 6. Limiting absorp tion principle for non -self-adjo int per turb a tion In this section, we sho w h o w th e results of the previous sections can b e app lied in the general setting introd uced by J. F aupin and the second author in [11]. First, we recall some we ll-kno wn deﬁnitions related to the limiting absorption pr inciple for non -self-adjoint p erturbations of self- adjoin t op erators, and then w e p ro vide suﬃcient conditions for Hyp othesis 2 to hold. W e consider H 0 and H = H 0 + V as in S ection 2.1, and we supp ose that V is of the form V = C W C, where C ∈ B ( H ) is selfadjoint, relativ ely compact with resp ect to H 0 , and W ∈ B ( H ). 6.1. Sp ectra l singularities. A cen tral concept in the study of n on-self-adjoin t op erators in tro- duced in [11] is that of a sp e ctr al singu larity , whic h refers to p oin ts of the essent ial sp ectrum th at fail to b e regular in the follo wing sense. Deﬁnition 6.1 (Regular sp ectral p oint and sp ectral singularity) . L et λ ∈ Λ := σ ess ( H ) = σ ess ( H 0 ) . (i) We say that λ is an outgoing (r esp e ctiv e ly incoming ) r egular sp ectral p oint of H if λ i s not an ac cumulation p oint of eig envalues lying in the set λ ± i (0 , ∞ ) , and if the limit C R H ( λ ± i 0 + ) C W := lim ε → 0 + C R H ( λ ± iε ) C W (6.1) exists in the norm top olo gy of B ( H ) . If this limit do es not exist, then λ is said to b e an outgoing (r esp e ctively incoming ) sp e ctr al singularity of H . (ii) We say that λ is a r egular sp ectral p oint of H if it is b oth an outgoing and an inc oming r e g u lar sp e ctr al p oint. O therwise, λ is c al le d a sp ectral singularit y of H . (iii) We say that inﬁnity is an outgoing/inc oming r e gular sp e ctr al p oint of H if ther e exi sts m > 0 such that f or al l λ > m , λ is an outgoing/inc oming r e gular sp e ctr al p oint and if the map [ m, ∞ ) ∋ λ 7→ C R H ( λ ± i 0 + ) C W is b ounde d for the top olo g y of the norm of op er ator. If one of this c ondition do es not hold, we say that inﬁnity is an outgoing (r esp e ctive ly incoming ) sp e ctr al singularity of H . The notion of sp ectral singularit y is closely related to the concept of sp ectral pro jection for non-self-adjoin t op erators introduced in [29]. Ass u ming a limiting absorption principle for H 0 , a c haracterizatio n of sp ectral singularities h as b een pro vided in [11]. This notion also plays a fundamental role in the stud y of the dynamics of solutions to the S c hr¨ odinger equation go ve rn ed b y a non-self-adjoint Hamiltonian. F or instance, F aupin and F r ¨ ohlic h [10] sho w that the dissip ativ e w a ve op erators are complete if and only if H has no sp ectral singularities, while F aupin and Nicole au [14] demonstrate that the dissipativ e scattering matrix fails to b e inv ertible at sp ectral singularities. Finally we refer to [13] for a construction of wa ve op erators (named ”regularized w a ve op erators”) taking into accoun t sp ectral s in gularities in the n on-dissipativ e case. In the sequel, w e denote b y Λ reg the set of regular s p ectral p oint s of H . It is of particular interest to understand h o w rapidly the w eigh ted r esolv en t of H diverge s as the sp ectral parameter appr oac h es a s p ectral singularit y from the upp er or lo w er half-plane. This leads to the notion of the or der of a sp ectral singularity . Deﬁnition 6.2 (Order of a sp ectral singularity) . L et λ ∈ σ ess ( H ) b e an outgoing or inc oming sp e ctr al singularity of H . We say that λ i s a sp e ctr al singularity of ﬁnite order if ther e exi st an inte ger n ∈ N and ε > 0 such that sup z ∈ D ( λ,ε ) ∩ C ± | λ − z | n k C R H ( z ) C W k B ( H ) < ∞ . (6.2) 22 V. BR UNEAU, N. FRANTZ, AND F. NICOLEAU Otherwise, λ i s said to b e an outgoing or inc oming sp e ctr al singularity of inﬁnite order . If λ is an outgoing or inc oming sp e ctr al singularity of ﬁnite or der, we deﬁne its ord er as the smal lest inte ger n for which (6.2 ) holds. We say that inﬁnity is a outgoing or inc oming sp e ctr al singularity of ﬁnite or der if ther e exists an inte ger n , ε 0 > 0 , m > 0 and z 0 ∈ ρ ( H ) \ R such that sup Re( z ) >m |± Im( z ) | <ε 0 | z − z 0 | − n k C R H ( z ) C W k B ( H ) < ∞ (6.3) Otherwise, inﬁnity is said to b e an outgoing or inc oming sp e ctr al singularity of inﬁnite order . If inﬁnity is an outgoing or inc oming sp e ctr al singularity of ﬁnite or der, we deﬁne its ord er as the smal lest inte ger n for wich (6.3) holds. In other words, a sp ectral singularit y is of ﬁnite ord er if th e wei ghte d resolv en t of H can b e regularized by a p olynomial factor in a neighb orh o o d of the singularity within the upp er or lo wer half-plane and a singularit y at inﬁnit y can b e regularized b y a rational function in a neigh b orho o d of inﬁnity in the upp er or lo we r half-plane. As we will recall in Section 7, for the Sc hr¨ odinger op erator sp ectral singularities are r elated to real resonances (see Section 7.1). 6.2. Resolv ent estimates. Here suﬃ cient conditions on sp ectral sin gularities and eigen v alues are giv en so that H satisﬁes the resolv ent estimate of Hyp othesis 2. Prop osition 6.3. A ssume that H satisﬁes Hyp othesis 1 on an op en interval I and that its closur e I c ontains a ﬁnite numb e r of sp e ctr al singularities of ﬁnite or der. Then Hyp othesis 2 holds for H on I . Pr o of. If I is b oun d from ab o v e, it is suﬃcien t to pro ve the r esu lt for I b ounded b ecause b elo w the essen tial sp ectrum Hyp othesis 2 is alw a ys tru e (see Remark 2.1). By com bin ing Hyp othesis 1 with (6.2), near eac h s p ectral singularity there exists c 1 > 0 and n 1 ∈ N suc h that k C R H ( z ) C W k B ( H ) ≤ c 1 | Im( z ) | − n 1 , (6.4) and n ear regular p oin ts w e can take n 1 = 0. Then ha ving a ﬁnite num b er of sp ectral singularities, b y compactness of I , (6.4) holds on S a ( I ) for s ome a > 0 and with the maxim um of all the ﬁnite order instead of n 1 . If w e denote ν 1 , . . . , ν n the ord er of the n sp ectral singularities of H in I , w e deduce Hyp othesis 2 on I with n I = max( ν k ) k ∈{ 1 ,...,n } + 1 b y using the r esolv en t ident it y R H ( z ) = R 0 ( z ) − R 0 ( z ) V R 0 ( z ) + R 0 ( z ) C W C R H ( z ) C W C R 0 ( z ) , (6.5) and the resolv en t estimate f or the self-adjoin t op erator H 0 : kR 0 ( z ) k B ( H ) ≤ | Im( z ) | − 1 . If I has no upp er b ound, it is suﬃcien t to prov e the resu lt on an in terv al [ s 1 , + ∞ ) ha ving only a sp ectral singularity at inﬁ nit y (on I ∩ ( −∞ , s 1 ] we apply th e previous argumen t). Hyp othesis 1 and with (6.3 ) giv e k C R H ( z ) C W k ≤ c | Re( z ) | n , on S a ([ s 1 , + ∞ )) for some n ∈ N , c > 0 and a > 0. W e conclud e by using again (6.5).  THE SPECTRAL SHIFT FUNCTION FOR NON- SELF-ADJOINT PER TURBA TIONS 23 7. S SF for Schr ¨ odinger O pera tors with Complex-V alued Potential In this section w e consider the self-adjoint op erator H 0 = − ∆ on L 2 ( R 3 ) , with domain D ( H 0 ) = H 2 ( R 3 ) := { u ∈ L 2 ( R 3 ); ∆ u ∈ L 2 ( R 3 ) } and H is the Sc hr¨ odinger op erator H := H 0 + V where V is a m ultiplication op erator by a f unction V ∈ L ∞ ( R 3 ; C ). The relativ e compactness is guaran teed b y the assumption: | V ( x ) | ≤ M h x i − δ , ∀ x ∈ R 3 , δ > 0 , (7.1) for some constan t M > 0 and the essentia l sp ectrum is giv en b y Λ := σ ess ( H ) = σ ess ( H 0 ) = [0 , + ∞ ). W e f o cus on the thr ee-dimensional setting bu t this section could easily b e extended to any o dd dimension. W e sh o w ho w to apply the pr evious results to this non-self-adjoin t Schr¨ odinger op erator; in particular, we recall the link b et wee n resonances and sp ectral sin gularities, and then we giv e regularit y p r op erties of the sp ectral shift f u nction ou tsid e of sp ectral singularities. Finally , we show that the b eha vior at h igh energies is very close to the self-adjoin t case. The existence of th e sp ectral shift fu nction will b e giv en und er the s hort-range condition δ > 3 W e hav e V = C W C by consid ering C th e m ultiplication op erator by C ( x ) := h x i − δ/ 2 and W := h·i δ V . If V is a compactly su pp orted p oten tial, w e can also deﬁne C , (resp. W ) as th e m ultiplication op erator by ρ , (resp. V ) with ρ a smo oth compactly sup p orted function suc h that ρ ≡ 1 on supp( V ). 7.1. Sp ectra l singularities and resonances. F or Sc hr¨ odinger op erators, the notion of sp ectral singularit y is closely related to th at of resonances. W e presen t here tw o related notions, dep end in g on the d eca y of the p oten tial V . 7.1.1. Short-r ange c ondition. W e deﬁne the w eigh ted sp ace L 2 δ := n f : R 3 → C | x 7→ h x i δ f ( x ) ∈ L 2 ( R 3 ) o , In [11], F aupin and F ran tz established th e follo w ing Prop osition: Prop osition 7.1. [11, Pr op osition 3.10] Supp ose that V i s a c omplex-value d p otential satisfying the short-r ange c ondition (7.1 ) with δ > 1 . Then for al l λ > 0 , the fol lowing c onditions ar e e quivalent: (i) λ is an outgoing/inc oming sp e ctr al singularity of H in the sense of De ﬁnition 6.1, (ii) Ther e exists Ψ ∈ L 2 − δ 2 , with Ψ 6 = 0 , such that ( − ∆ + V ( x ) − λ )Ψ = 0 . If the function Ψ from (ii) lies in L 2 ( R 3 ), then λ is an eigen v alue of H . Otherwise, λ is referr ed to as a real resonance, asso ciated with a resonant state Ψ / ∈ L 2 ( R 3 ). Suc h a state satisﬁes the outgoing/inco min g S ommerfeld radiation condition: Ψ( x ) = | x | 1 / 2 e ± iλ 1 / 2 | x |  a  x | x |  + o (1)  , | x | → ∞ , where a ∈ L 2 ( S 2 ) and a 6 = 0. Moreo ve r, if the op erator H is diss ipativ e, (in th e sense that Im( h u, H u i ) ≤ 0 f or all u in the d omain of H ), H cannot hav e outgoing sp ectral singularities in (0 , ∞ ) (see [30, Corollary 3.2]). But, in general, sp ectral singularities ma y still o ccur. As sh o wn in [30, Remark 5.4], for any λ > 0, one can construct a sm o oth, compactly s upp orted p oten tial V suc h that λ is an incoming sp ectral singularit y of H in the dissipativ e setting. 24 V. BR UNEAU, N. FRANTZ, AND F. NICOLEAU 7.1.2. Comp actly supp orte d p otential. If V ∈ L ∞ c ( R d , C ), with d ≥ 3 o dd, then the map { z ∈ C | Im( z ) > 0 } ∋ z 7→ ( H − z 2 ) − 1 : L 2 ( R d , C ) → L 2 ( R d , C ) is meromorph ic and ad m its a meromorphic extension to the whole complex plane as a map C ∋ z 7→ F ( z ) : L 2 c ( R d ) → L 2 lo c ( R d ) , (7.2) where L 2 c ( R d ) := { u ∈ L 2 ( R d ) | supp( u ) is compact } , L 2 lo c ( R d ) := { u : R d → C | u ∈ L 2 ( K ) for all compact K ⊂ R d } . The p oles of the meromorphic extension in (7.2) are called r esonanc es of H . One can v erify that an y real r esonance ± λ 0 of H , with λ 0 ≥ 0, corresp onds to an outgoing/incoming sp ectral singularit y λ 2 0 in the sens e of Deﬁnition 6.1 . Moreov er, H has only ﬁnitely many s p ectral singularities, and the order of a sp ectral singularit y coincides with the multiplicit y of th e corr e- sp ond in g resonance p ole; s ee [9, Th eorem 3.8]. W e refer the reader to [9 ] and the references therein for an o verview of the resonance theory for Sc hr¨ odinger op er ators, and to [11, Section 3.3.1] for a more detailed comparison b etw een the notions of resonances and sp ectral singularities considered in this p ap er. Finally note that in b oth cases, ∞ is an outgoing and an incoming regular sp ectral p oin t. This is a consequence of p oin t (3) of Prop osition 7.2. (See e.g the pro of of Prop osition 7.14 ab out the high energy asymptotic of the deriv ativ e of the SSF for more details). 7.2. Preliminary results. Here w e recall some u seful results on the free resolv en t, R 0 ( z ), th at will b e u sed to derive qualitativ e results on the sp ectral shift fu nction. W e deﬁne √ z on C \ R + b y requiring that Im( √ z ) > 0. Hence, for all λ ∈ (0 , ∞ ), w e h a v e lim ε → 0 + √ λ ± iε = ± √ λ. F or z ∈ C \ R + , set T 0 ( z ) := C R 0 ( z ) C W , and recall that, for suc h z , the integral ke rn el of T 0 ( z ) is th e function K 0 ( z )( x, y ) := 1 4 π C ( x ) e i √ z | x − y | | x − y | C ( y ) W ( y ) , ( x, y ) ∈ R 3 × R 3 . The w ell-kno wn next prop osition colle cts some useful p rop erties of the op erator T 0 ( z ), (see [32] for instance). Prop osition 7.2. Assu me that V is a short-r ange p otential (i.e. satisﬁes (7.1) ) with δ > 3 . Then, for al l z ∈ C \ R + , the inte gr al kernel K 0 ( z ) b elongs to L 2 ( R 3 × R 3 ) . In p articular, T 0 ( z ) is a Hilb ert–Schmidt op er ator. Mor e over: (1) F or al l λ ∈ Λ , the limits T 0 ( λ ± i 0 + ) := lim ε → 0 + T 0 ( λ ± iε ) exist in the Hilb ert–Schmidt top olo gy. (2) The maps Λ ∋ λ 7→ T 0 ( λ ± i 0 + ) ar e c ontinuous in the Hilb ert–Schmidt top olo gy. (3) F or any δ > 0 , ther e exist c onstants c > 0 and C > 0 such that, for al l z ∈ C \ R + with | z | > c , one has k T 0 ( z ) k B ( H ) ≤ C | z | − 1 2 + δ . (7.3) THE SPECTRAL SHIFT FUNCTION FOR NON- SELF-ADJOINT PER TURBA TIONS 25 The k -deriv ativ e with resp ect to z of T 0 is given by T ( k ) 0 ( z ) := ( − 1) k C R 0 ( z ) k +1 C W . F or k = 1, its in tegral k ernel is giv en for all ( x, y ) ∈ R 3 × R 3 b y K (1) 0 ( z )( x, y ) = i 8 π √ z C ( x ) e i √ z | x − y | C ( y ) W ( y ) , ± I m ( z ) > 0 . The follo w ing Prop osition collects some useful prop erties of the op erators T ( k ) 0 ( z ), (see [22], Lemma 2.2): Prop osition 7.3. A ssume that V satisﬁes (7.1) ) with δ > 3 . Then: (1) F or al l k ∈ N \ { 0 } and al l z ∈ C \ R + , the op er ator T ( k ) 0 ( z ) i s of tr ac e class. (2) If δ > 2 k + 1 , then for al l λ ∈ (0 , + ∞ ) , the limits T ( k ) 0 ( λ ± i 0 + ) := lim ε → 0 + T ( k ) 0 ( λ ± iε ) exist in the Hilb ert–Schmidt top olo gy. (3) Mor e over, the map (0 , + ∞ ) ∋ λ 7− → T 0 ( λ ± i 0 + ) b elongs to C k ((0 , + ∞ ) , L 2 ( H )) , and one has T ( k ) 0 ( λ ± i 0 + ) = ∂ ( k ) λ T 0 ( λ ± i 0 + ) . W e no w derive explicit trace formulas for T (1) 0 ( z ) and T (1) 0 ( z ) T 0 ( z ), which will p la y a key role in the computation of the sp ectral shif t function. Lemma 7.4. Assume that V is a short-r ange p otential with exp onent δ > 3 , and let λ ∈ (0 , + ∞ ) . Then: (1) One has lim ε → 0 + T r  T (1) 0 ( λ + iε ) − T (1) 0 ( λ − iε )  = i 4 π √ λ Z R 3 V ( x ) d x. (7.4) (2) Mor e over, T r  T (1) 0 ( λ ± i 0 + ) T 0 ( λ ± i 0 + )  = ± i 32 π 2 √ λ Z R 6 V ( x ) V ( y ) e ± 2 i √ λ | x − y | | x − y | d x d y . (7.5) Pr o of. (1) F or all ε > 0, the op erator T (1) 0 ( λ ± iε ) is of trace class, and its trace is giv en b y T r  T (1) 0 ( λ ± iε )  = Z R 3 K (1) 0 ( λ ± iε )( x, x ) d x = i 8 π √ λ ± iε Z R 3 V ( x ) d x. Subtracting the t w o expressions and taking the limit as ε → 0 + yields (7.4). (2) Let z ∈ C \ R + . The integral ke rn el of the pro du ct T (1) 0 ( z ) T 0 ( z ), d enoted K (0 , 1) 0 ( z ), is giv en b y K (0 , 1) 0 ( z )( x, y ) = i 32 π 2 √ z Z R 3 C ( x ) e i √ z | x − t | V ( t ) e i √ z | t − y | | t − y | C ( y ) W ( y ) d t . Since T (1) 0 ( z ) T 0 ( z ) is of trace class, w e ha v e T r  T (1) 0 ( z ) T 0 ( z )  = i 32 π 2 √ z Z R 6 V ( x ) V ( y ) e 2 i √ z | x − y | | x − y | d x d y . 26 V. BR UNEAU, N. FRANTZ, AND F. NICOLEAU Finally , taking z = λ ± iε and letting ε → 0 + yields (7.5).  Before stating the next lemma, let us recall that, by th e r esolv en t iden tit y , for eve ry z ∈ ρ ( H ) ∩ ρ ( H 0 ) the op erator Id + T 0 ( z ) is inv ertible and w e ha v e  Id + T 0 ( z )  − 1 = Id − C R H ( z ) C W. (7.6) W e conclude this section with the follo wing u seful identit y : Lemma 7.5. Supp ose that V i s a short-r ange p otential with de c ay e xp onent δ > 3 . Then, for every z ∈ ρ ( H ) ∩ ρ ( H 0 ) , the diﬀer enc e R H ( z ) − R 0 ( z ) is a tr ac e- class op er ator. Mor e over, the fol lowing identities hold: T r  R H ( z ) − R 0 ( z )  = T r  T (1) 0 ( z ) [ I + T 0 ( z )] − 1  , (7.7) = T r  T (1) 0 ( z )  − T r  T (1) 0 ( z ) [ I + T 0 ( z )] − 1 T 0 ( z )  , (7.8) = T r  T (1) 0 ( z )  − T r  T (1) 0 ( z ) T 0 ( z )  + T r  T (1) 0 ( z ) C R H ( z ) C W T 0 ( z )  . (7.9) Pr o of. Let z ∈ ρ ( H ) ∩ ρ ( H 0 ). Since δ > 3, the resolv en t identit y implies that R H ( z ) − R 0 ( z ) is a trace-cla ss op erator (see, for instance, [25, Th eorem XI.21]). App lyin g again the resolven t iden tit y successiv ely (twice and three times, resp ectiv ely), we obtain R H ( z ) − R 0 ( z ) = − R 0 ( z ) V R 0 ( z ) + R 0 ( z ) V R H ( z ) V R 0 ( z ) = − R 0 ( z ) V R 0 ( z ) + R 0 ( z ) V R 0 ( z ) V R 0 ( z ) − R 0 ( z ) V R H ( z ) V R 0 ( z ) V R 0 ( z ) . Next, by a clev er use of the cyclicit y of the trace, we obtain T r  R H ( z ) − R 0 ( z )  = T r  T (1) 0 ( z ) [Id − C R H ( z ) C W ]  = T r  T (1) 0 ( z ) [Id + T 0 ( z )] − 1  , b y (7.6), whic h pro v es (7.7). Th e identi ties (7.8) and (7.9) follo w immediately .  7.3. Existence of the SSF. Th e f ollo win g theorem pro vides suﬃcient conditions for the existence of the sp ectral shift function ξ ( · ; − ∆ + V ( x ) , − ∆). Prop osition 7.6. L et V b e a short-r ange p otential with de c ay exp onent δ > 3 , and assume that the op er ator H admits only ﬁnitely many eigenvalues and ﬁnitely many sp e ctr al singularities, e ach of ﬁnite or der. Then ther e exists c > 0 such that the sp e ctr al shift function ξ ( · ; − ∆ , − ∆ + V ) := ξ  · ; ( H + c ) − 1 , ( H 0 + c ) − 1  is wel l deﬁne d and do es not dep e nd on the p articular choic e of c > 0 . Pr o of. As H has a ﬁnite num b er of eigen v alues and a ﬁn ite num b er of sp ectral s ingularities of ﬁnite order, it follo ws from Prop osition 6.3 that Hyp otheses 1 are 2 is satisﬁed on R . F ur thermore, since V is b ound ed and H has only ﬁnitely man y eigen v alues, th er e exists c > 0 such that − c ∈ ρ ( H ), and a > 0 su c h th at σ ( H ) ⊂ { z ∈ C | Re( z ) > − c and | Im( z ) | < a } . Moreo ver, as δ > 3, thanks to Lemma 7.5, ( H 0 + c ) − 1 − ( H + c ) − 1 is a trace class op erator. Th us, condition (2) in Prop osition 5.1 is also satisﬁed. W e conclude th at the sp ectral sh if t function ξ ( · , − ∆ , − ∆ + V ) := ξ ( · , ( H + c ) − 1 , ( H 0 + c ) − 1 ) THE SPECTRAL SHIFT FUNCTION FOR NON- SELF-ADJOINT PER TURBA TIONS 27 exists and do es not dep end on the particular c hoice of c .  As a consequen ce of the trace-class pr op ert y of ( H − c ) − 1 − ( H 0 − c ) − 1 and P rop osition 5.4 with m = 1, the deriv ativ e ξ ′ ( H , H 0 , · ) adm its the follo win g represent ation: Prop osition 7.7. L et V b e a short-r ange p otential with de c ay exp onent δ > 3 , and assume that the op er ator H has only ﬁnitely many eigenvalues and ﬁnitely many sp e ctr al singularities, e ach of ﬁnite or der. Then, in the sense of distributions, one has ξ ′ ( H , H 0 ; λ ) = 1 2 π i lim ε → 0 + T r  R H ( λ + iε ) − R 0 ( λ + iε )  − T r  R H ( λ − iε ) − R 0 ( λ − iε )  . Remark 7.8. The assumption that the op er ator has only ﬁnitely many eigenvalues may app e ar r estrictive. N e vertheless, ther e exist suﬃcient c onditions on the p otential V ensuring that − ∆ + V admits only ﬁnitely many eigenvalues. F or instanc e, i n o dd dimensions, F r ank, L aptev, and Safr onov [12 ] showe d that if V de c ays exp onential ly at inﬁnity, then the op er ator − ∆ + V has ﬁnitely many eigenvalues. 7.4. Regularity of the Sp ectral Shift F unction. In this section, w e show that the sp ectral shift function extends to a regular function in a n eigh b orh o o d of any regular sp ectral p oin t. Since the sp ectral singularities are isolated, this will allo w us to d eﬁne the SSF lo cally aw ay from these singularities. W e h a v e the follo wing p rop osition: Prop osition 7.9. Assume that V is a short-r ange p otential with de c ay exp onent δ > 2 k + 1 , wher e k ∈ N ∗ . A ssume also that H has only ﬁnitely many eigenvalues and ﬁnitely many sp e ctr al singularities, e ach of ﬁnite or der. L et λ ∈ (0 , + ∞ ) b e a r e gular sp e ctr al p oint of H . Then the sp e ctr al shift function ξ ( · ; H , H 0 ) is of c lass C k +1 in a neighb orho o d of λ . Pr o of. It follo ws f r om p oint (2) of Prop osition 7.2 together with [11, Prop osition 4.7] th at there exists a neigh b orho o d ω λ of λ free of sp ectral singularities. Using (7.8), w e obtain, in the sense of distributions and for all µ ∈ ω λ , ξ ′ ( µ ; H , H 0 ) = 1 2 π i lim ε → 0 +  T r  T (1) 0 ( µ + iε )  − T r  T (1) 0 ( µ − iε )  (7.10) − T r  T (1) 0 ( µ + iε ) [ I + T 0 ( µ + iε )] − 1 T 0 ( µ + iε )  + T r  T (1) 0 ( µ − iε ) [ I + T 0 ( µ − iε )] − 1 T 0 ( µ − iε )   . (7.11) Since eve ry µ ∈ ω λ is a regular sp ectral p oint of H , [11, Prop osition 4.7] ensu res that the op erators I + T 0 ( µ ± i 0 + ) are in vertible, and that lim ε → 0 + [ I + T 0 ( µ ± iε )] − 1 = [ I + T 0 ( µ ± i 0 + )] − 1 exist in B ( H ). Moreo v er, th e mapp ings In v : µ ∈ ω λ 7− → [ I + T 0 ( µ ± i 0 + )] − 1 are con tinuous. Th e C k -regularit y of Inv on ω λ follo w s from the diﬀeren tiabilit y of the in ve rs ion map and from p oin t (2) of Prop osition 7.3. Combining this fact w ith p oint s (1) and (2) of Prop osition 7.2 and p oint (2) of Prop osition 7.3, we deduce that T r  T (1) 0 ( µ ± i 0 + ) [ I + T 0 ( µ ± i 0 + )] − 1 T 0 ( µ ± i 0 + )  := lim ε → 0 + T r  T (1) 0 ( µ ± iε ) [ I + T 0 ( µ ± iε )] − 1 T 0 ( µ ± iε )  exists, and that the mappings µ ∈ ω λ 7− → T r  T (1) 0 ( µ ± i 0 + ) [ I + T 0 ( µ ± i 0 + )] − 1 T 0 ( µ ± i 0 + )  28 V. BR UNEAU, N. FRANTZ, AND F. NICOLEAU are of class C k . Finally , by (7.4), the mapping µ ∈ ω λ 7− → T r  T 0 ( µ + i 0 + ) − T 0 ( µ − i 0 + )  is wel l deﬁned and of class C k . This completes the p ro of.  7.5. Asymptotics nea r a Sp ectral Singularity . In this section, w e deriv e the asymptotic b e- ha vior of the sp ectral shift function (SS F) in a neigh b orho o d of a sp ectral singularity , in the case of a compactly sup p orted and b ound ed p oten tial. Assume that V ∈ L ∞ c ( R 3 , C ), the space of essential ly b oun ded fun ctions with compact supp ort. Then there exists a compactly sup p orted fu nction ρ su c h th at ρ ≡ 1 on su p p( V ) , ρ V = V . As men tioned in Su bsection 7.1.2, the outgoing and incoming sp ectral s ingularities λ 0 > 0 of H corresp ond to the real p oles ± √ λ 0 of the meromorphic conti nuation to C of the map C + ∋ z 7− → ( H − z 2 ) − 1 : L 2 c ( R 3 ) → L 2 lo c ( R 3 ) . Equiv alentl y (see [9, Theorem 3.8]), they corresp ond to r eal p oles of the m eromorphic contin u ation to C of the map C + ∋ z 7− → ρ ( H − z 2 ) − 1 ρ : L 2 ( R 3 ) → L 2 ( R 3 ) . W e denote by F ( z ) this meromorph ic con tinuat ion. T o ﬁx ideas, supp ose that λ 0 > 0 is an outgoing sp ectral sin gularit y of H of ord er ν 0 . It then follo w s that √ λ 0 is a p ole of F ( z ). Acco rd ing to [9], there exist ﬁ nite-rank op erators A − j , 1 ≤ j ≤ ν 0 , and an op erator-v alued function z 7→ A 0 ( z ), holomorphic in a complex neighbourh o o d of √ λ 0 , such that F ( z ) = ν 0 X j =1 A − j ( z 2 − λ 0 ) j + A 0 ( z ) , (7.12) in a complex neigh b ourho o d of √ λ 0 . W e recall th e follo wing stand ard expansion, wh ich is a direct consequence of the theory of distribu- tions asso ciated with analytic functions (see, e.g., [15, Chap. I I I, § 3.5]). Lemma 7.10 (Multiplication of a p rincipal part by a smo oth function) . L et j ∈ N ∗ , λ 0 > 0 , and let g b e holomorphic in V λ 0 ∩ C ± and admit a smo oth e xtension up to the r e al axis, denote d by g ± , wher e V λ 0 denotes a c omplex neighb ourho o d of λ 0 . Then, in the sense of distributions in a neighb ourho o d of λ 0 on the r e al line, one has lim ε → 0 + g ( λ ± iε ) ( λ ± iε − λ 0 ) j = j − 1 X k =0 g ( k ) ± ( λ 0 ) k ! ( λ − λ 0 ± i 0) − ( j − k ) + h ± ( λ ) , (7.13) wher e h ± is smo oth ne ar λ 0 . Pr o of. W e only treat the case with the + sign, the other one b eing identica l. W e expand g in a T aylo r series around λ 0 up to order j − 1: g ( λ + iε ) = j − 1 X k =0 g ( k ) ± ( λ 0 ) k ! ( λ + iε − λ 0 ) k + R j ( λ + iε ) , THE SPECTRAL SHIFT FUNCTION FOR NON- SELF-ADJOINT PER TURBA TIONS 29 where the remaind er satisﬁes R j ( λ + iε ) = O  ( λ + iε − λ 0 ) j  in a n eigh b orho o d of V λ 0 . Dividing b y ( λ + iε − λ 0 ) j yields g ( λ + iε ) ( λ + iε − λ 0 ) j = j − 1 X k =0 g ( k ) ± ( λ 0 ) k ! ( λ + iε − λ 0 ) − ( j − k ) + r j ( λ, ε ) , where r j ( λ, ε ) is a sm o oth function and con v erges, as ε → 0 + , to a smo oth fun ction h + ( λ ) near λ 0 . Since eac h family ( λ + iε − λ 0 ) − m con v erges to ( λ − λ 0 + i 0) − m in D ′ ( R ), we ma y pass to the limit term by term , which giv es (7.13).  Remark 7.11 (Real and imaginary parts) . E ach distribution ( λ − λ 0 + i 0) − m app e aring in (7.13) admits the fol lowing de c omp osition i nto its r e al and imaginary p arts: ( λ − λ 0 + i 0) − m = p . v . 1 ( λ − λ 0 ) m − iπ ( − 1) m − 1 ( m − 1)! δ ( m − 1) ( λ − λ 0 ) , m ≥ 1 . (7.14) Henc e, the r e al p art of ( λ − λ 0 + i 0) − m c orr esp onds to the princip al value p . v . 1 ( λ − λ 0 ) m , while its imaginary p art is supp orte d at λ = λ 0 and involves derivatives of the D ir ac delta distribution. No w, according to Prop osition 7.7, and since v is compactly sup p orted, the follo wing rep resen tation holds in th e sens e of d istr ibutions: ξ ′ ( H , H 0 ; λ ) = 1 2 π i lim ε → 0 + T r  R H ( λ + iε ) − R 0 ( λ + iε )  − T r  R H ( λ − iε ) + R 0 ( λ − iε )  . F rom (7.9) and recalling that C = ρ , W = V , it follo w s that T r  R H ( λ ± iε ) − R 0 ( λ ± iε )  = T r  T (1) 0 ( λ ± iε )  − T r  T (1) 0 ( λ ± iε ) T 0 ( λ ± iε )  + T r  T (1) 0 ( λ ± iε ) ρ R H ( λ ± iε ) ρ V T 0 ( λ ± iε )  := 3 X n =1 a ± n ( ε ) . W e ﬁrst consider the case with the plus sign and in particular the third term a + 3 ( ǫ ). W e deﬁn e g j ( µ ) := T r  T (1) 0 ( µ ) A − j V T 0 ( µ )  . Since V is compactly sup p orted, g j satisﬁes the assu m ptions of Lemma 7.10. Let λ lie in a r eal neigh b our ho o d of λ 0 , and let ε > 0 small enough. S u bstituting z = √ λ + iε ∈ V λ 0 ∩ C + in to (7.12), where V λ 0 denotes a complex neighbour ho o d of λ 0 , we obtain F ( z ) = ρ R H ( λ + iε ) ρ, and a straigh tforw ard computation yields, in the sense of distrib utions lim ε → 0 + a + 3 ( ε ) = ν 0 X j =1 j − 1 X k =0 g ( k ) j, + ( λ 0 ) k ! 1 ( λ − λ 0 + i 0) j − k + H + ( λ ) = ν 0 X l =1 α l ( λ 0 ) ( λ − λ 0 + i 0) l + H + ( λ ) where g j, + denotes the smo oth extension of g j up to the real axis, H + denotes a smo oth fu nction near λ 0 and α l = ν 0 − l X k =0 1 k ! g ( k ) k + l, + . 30 V. BR UNEAU, N. FRANTZ, AND F. NICOLEAU The situation in the case w ith the min us sign is diﬀerent , sin ce λ − iε can b e wr itten as z 2 with z lying in a neigh b ourho o d of − √ λ 0 in tersected with C + . As − √ λ 0 is not a p ole, the trun cated resolv en t extends holomorphically to this region, and we obtain lim ε → 0 + a − 3 ( ε ) = H − ( λ ) , where H − denotes a smo oth fun ction near λ 0 . W e no w compare the ﬁrs t tw o terms w ith opp osite b oundary v alues on the r eal axis. F or eac h ε > 0, w e deﬁ ne a ± 1 ( ε ) := T r  T (1) 0 ( λ ± iε )  , a ± 2 ( ε ) := − T r  T (1) 0 ( λ ± iε ) T 0 ( λ ± iε )  . By (7.4 ), we immediately obtain, in the sense of distribu tions, lim ε → 0 +  a + 1 ( ε ) − a − 1 ( ε )  = i 4 π √ λ Z R 3 V ( x ) dx. In the same wa y , using (7.5) , we get in the sense of distrib utions, lim ε → 0 +  a + 2 ( ε ) − a − 2 ( ε )  = − i 32 π 2 √ λ Z R 6 V ( x ) V ( y ) e 2 i √ λ | x − y | + e − 2 i √ λ | x − y | | x − y | dx dy = − i 16 π 2 √ λ Z R 6 V ( x ) V ( y ) cos  2 √ λ | x − y |  | x − y | dx dy . Using the previous notations, we now d eriv e an explicit expression for th e singular p art of th e deriv ativ e of th e sp ectral shift fu nction near an outgoing sp ectral singularity . Theorem 7.12 (Explicit d istributional form ula for ξ ′ ( H , H 0 ; λ )) . Assume that v is c omp actly sup- p orte d and that λ 0 > 0 is an outgoing sp e ctr al singularity of H of ﬁnite or der ν 0 . Then, in the sense of distributions on a r e al neighb ourho o d of λ 0 , ξ ′ ( H , H 0 ; λ ) = ν 0 X l =1 α l ( λ 0 ) ( λ − λ 0 + i 0) l + H ( λ ) , wher e H ( λ ) is smo oth ne ar λ 0 , and wher e α l := ν 0 − l X k =0 1 k ! g ( k ) k + l, + with g j, + the smo oth extension up to the r e al axis of g j ( µ ) := T r  T (1) 0 ( µ ) A − j V T 0 ( µ )  , µ ∈ V λ 0 ∩ C + , with V λ 0 a c omplex neighb ourho o d of λ 0 . This form ula shows that th e s in gular b ehavio r of ξ ′ ( H , H 0 ; λ ) near λ 0 is en tirely determined by the ﬁnite-rank residues A − j of the w eigh ted resolv en t. In particular, the singular part of ξ ′ ( H , H 0 ; λ ) has the same distributional structure as the b ound ary v alues of th e r esolv en t R H ( λ + i 0 + ). T his phenomenon is quite natural, as it originates fr om th e loss of u niform con trol of th e r esolv en t in the upp er h alf-plane near the outgoing sp ectral sin gularit y . Remark 7.13. In the c ase wher e λ 0 > 0 i s an incoming sp e c tr al singularity of H , an analo gous formula holds with the b oundary value ( λ − λ 0 − i 0) − ( j − k ) and the c o eﬃcients g j, − . If b oth inc oming and outgoing sp e ctr al si ng u larities o c cu r at λ 0 , the distributional derivative ξ ′ ( H , H 0 ; λ ) r e c eives c ontributions fr om b oth sides. THE SPECTRAL SHIFT FUNCTION FOR NON- SELF-ADJOINT PER TURBA TIONS 31 7.6. High Energy Asymptotic. In this section, we are in terested in the asymptotic b eha vior of the d eriv ativ e of the sp ectral sh if t fu n ction as the sp ectral parameter tends to + ∞ . W e compu te the leading term in the asymptotic expansion and sh o w that it coincides with the self-adjoin t case. The metho d w e use is the same as in the self-adjoin t setting (see [3]). The next p rop osition establishes the h igh energy asymptotic of the sp ectral s h ift function asso- ciated to H and H 0 under a short range assumption on V . Prop osition 7.14. Supp ose that V is a b ounde d short r ange p otential with δ > 3 . Supp ose that H has a ﬁnite numb er of eig envalues and a ﬁnite numb er of sp e ctr al singularities of ﬁnite or der. Then ξ ′ ( λ, H , H 0 ) = 1 8 π 2 √ λ Z R 3 V ( x ) d x + o  1 √ λ  λ → ∞ . (7.15) Mor e over if V b elongs to C 1 ( R 3 , C ) and satisﬁes |∇ V ( x ) | ≤ C α h x i − δ for some δ > 3 , then ξ ′ ( λ, H , H 0 ) = 1 8 π 2 √ λ Z R 3 V ( x ) d x + O  1 λ  λ → ∞ . (7.16) Pr o of. It follo ws from p oint (3) of Pr op osition 7.2 that there exist constan ts λ 0 > 1 and ε 0 > 0 such that for all λ ≥ λ 0 and all 0 ≤ ε ≤ ε 0 , one has   T 0 ( λ ± iε )   B ( L 2 ) < 1 in the u n iform (op erator-norm) top ology . Hence Id + T 0 ( λ ± iε ) is in vertible, and its in v erse admits the Neumann series expansion [Id + T 0 ( λ ± iε )] − 1 = ∞ X k =0 ( − 1) k T 0 ( λ ± iε ) k , (7.17) with the series con v erging uniformly in ε with r esp ect to the op er ator norm up to ε → 0 + . Com- bined with [11, Prop osition 4.7] th is implies that if λ > λ 0 , then λ is a regular sp ectral p oin t. With Prop osition 7.9, ξ ′ ( H , H 0 , . ) is con tinuous on ( λ 0 , + ∞ ) and this allo w s us to compute the asymptotic of ξ ′ ( H , H 0 , . ) w hen λ → + ∞ . Next with (7.7) and (7.17) we hav e that for all λ ∈ ( λ 0 , + ∞ ), ξ ′ ( λ, H , H 0 ) = (2 π i ) − 1  lim ε → 0 + T r  T (1) 0 ( λ + iε ) − T (1) 0 ( λ − iε )  (7.18) − lim ε → 0 + T r  T (1) 0 ( λ + iε ) T 0 ( λ + iε ) − T (1) 0 ( λ − iε ) T 0 ( λ − iε )  (7.19) + ∞ X k =2 ( − 1) k lim ε → 0 + T r  T (1) 0 ( λ + iε ) T 0 ( λ + iε ) k − T (1) 0 ( λ − iε ) T 0 ( λ − iε ) k   (7.20) T o compute the limit of (7.18) it su ﬃces to u se (7.4) and we ha ve (2 π i ) − 1 lim ε → 0 + T r  T (1) 0 ( µ + iε ) − T (1) 0 ( µ − iε )  = 1 8 π 2 √ λ Z R 3 V ( x )d x . 32 V. BR UNEAU, N. FRANTZ, AND F. NICOLEAU T o analyze the asymptotic b eha vior of the terms corresp onding to (7.19), one computes directly b y using (7.5) lim ε → 0 + h T r  T (1) 0 ( λ + iε ) T 0 ( λ + iε ) − T (1) 0 ( λ − iε ) T 0 ( λ − iε ) i = T r  T (1) 0 ( λ + i 0 + ) T 0 ( λ + i 0 + ) − T (1) 0 ( λ − i 0 + ) T 0 ( λ − i 0 + )  = i 32 π 2 √ λ Z R 6 V ( x ) V ( y ) cos  2 √ λ | x − y |  | x − y | d x d y . (7.21) By the Riemann–Leb esgue lemma, th is term d eca ys lik e o ( λ − 1 / 2 ) as λ → ∞ . Moreo ver, if V ∈ C 1 ( R 3 , C ) and its gradien t satisﬁes |∇ V ( x ) | = O  | x | − δ  for some δ > 3 , then the remainder is O (1 /λ ). It remains to estimate (7.20). App lying (3) of Prop osition 7.2 w e see that   T 0 ( λ ± iε ) k   B ( H ) = O  λ − k 2  . Then for k ≥ 2, one has :   T (1) 0 ( λ ± iε ) T 0 ( λ ± iǫ ) k   L 1 ≤   T (1) 0 ( λ ± iε )   L 2   T 0 ( λ ± iǫ ) k − 1   B ( H )   T 0 ( λ ± iε )   L 2 (7.22) = O  λ − k 2  , uniformly with resp ect to ε > 0. T h us (7.20) = O ( λ − 1 ) uniform ly with resp ect to ε .  Remark 7.15. In [26 ] , D. R ob ert investigate d the high-ener gy asymptotics of the sc attering phase for the fr e e L aplacian H 0 p erturb e d by a smo oth, r e al-value d, de c aying p otential V on R n . U nder the de c ay c ondition | ∂ α x V ( x ) | ≤ C α h x i − ρ −| α | , ρ > n, the sp e ctr al shift function is deﬁne d in terms of the sc attering matrix S ( λ ) by det S ( λ ) = exp  − 2 π i ξ ( λ ; H , H 0 )  . (7.23) In the dissip ative c ase (i.e., when Im V ≤ 0 ), F aupin and Nic ole au [14] have shown that the sc attering matric es S ( λ ) ar e wel l-deﬁne d and admit an explicit r epr esentation formula. It is imp ortant to note that, although the r elation (7.23) has not b e en pr oven in the non-selfadjoint setting, the r esult of our the or em is c onsistent with the fact that, in the dissip ative c ase, S ( λ ) is a c ontr action. 8. Exp licit s imple exampl es In order to illustrate and commen t our deﬁn ition of the sp ectral sh ift function (SSF), let u s giv e some exp licit calculations for very simple examples. On these to y mo d els we hav e phen omena kno wn in the self-adjoin t conte xt (in teger j u mp at real eigen v alues, fractional ju mp at sp ectral singulari- ties/resonances, smo othness outside these singularities, ...), but also some n ew phenomena. A ﬁrst no v elt y is the non-inte grabilit y of the SSF in the presence of non-real eigen v alues (for p erturb ations of trace class). A second is th e f act that th e SSF is no longer real. On our to y mo d els, it remains real if the p ertur bation does n ot in teract w ith the con tin uous sp ectrum and the sign of its imaginary part is related to that of the p erturbation. THE SPECTRAL SHIFT FUNCTION FOR NON- SELF-ADJOINT PER TURBA TIONS 33 8.1. In ﬁnit e dimension: diagonalizable op e ra tors. When the Hilb ert space is H = C , con- sider H 0 the multiplicat ion op erator by λ 0 ∈ R and H the multiplica tion by λ 0 + v , v ∈ R . F or f ∈ D ( R ), f ( H 0 ) is th e multiplicatio n by f ( λ 0 ). F or H , it dep ends if v is real or n ot. When v ∈ R , f ( H ) is the m ultiplication by f ( λ 0 + v ), wh ile when v ∈ C \ R , f ( H ) = 0. Th us when v ∈ R the SSF corresp ond s to the standard S SF. F rom the F un damen tal Theorem of Calculus, we ha ve T r ( f ( H ) − f ( H 0 )) = f ( λ 0 + v ) − f ( λ 0 ) = Z λ 0 + v λ 0 f ′ ( λ ) dλ and, up to an add itiv e constan t, almost ev erywh ere, we ha ve ξ ( λ ; H , H 0 ) = ( 1 [ λ 0 ,λ 0 + v ] ( λ ) if v ≥ 0 − 1 [ λ 0 + v, λ 0 ] ( λ ) if v ≤ 0 . The SSF has t wo jump s at the eigen v alues of H 0 and of H . When v ∈ C \ R , T r ( f ( H ) − f ( H 0 )) = − f ( λ 0 ) = Z + ∞ λ 0 f ′ ( λ ) dλ and, up to an add itiv e constan t, w e ha v e ξ ( λ ; H , H 0 ) = 1 [ λ 0 , + ∞ ) ( λ ) , whic h has a unique jump at the real eigenv alue λ 0 whic h b ecomes a non-real eigen v alue under the p ertur b ation. Unlik e the selfadjoin t case, ξ ( · ; H , H 0 ) is not compactly su pp orted (nor integ rable). This is related to the fact that the p erturb ed eigen v alue λ 0 + v is no longer real. Ob viously , we hav e the same p henomena in H = C 4 for d iagonal matrices (or diagonalizable matrices). F or example, if w e consider th e diagonal matrices H 0 :=     λ 1 0 0 0 0 λ 2 0 0 0 0 λ 3 0 0 0 0 λ 4     H := H 0 + V , V :=     v 1 0 0 0 0 v 2 0 0 0 0 v 3 0 0 0 0 0     with λ 1 < λ 2 < λ 3 < λ 4 and v 1 > 0, v 2 < 0, v 3 ∈ C \ R th en T r( f ( H ) − f ( H 0 )) = f ( λ 1 + v 1 ) − f ( λ 1 ) + f ( λ 2 + v 2 ) − f ( λ 2 ) − f ( λ 3 ) , and, up to an add itiv e constan t, w e ha v e ξ ( λ ; H , H 0 ) = 1 [ λ 1 ,λ 1 + v 1 ] ( λ ) − 1 [ λ 2 + v 2 ,λ 2 ] ( λ ) + 1 [ λ 3 , + ∞ ) ( λ ) . This fun ction has jumps at the real eigen v alues of H 0 and H , excepted at the un p erturb ed eigen v alue λ 4 . T he fact that the SSF is equal to 1 up to inﬁnity is related to the app earance of the non-real eigen v alue λ 3 + v 3 . 8.2. In ﬁnite dimension: an undiagonalizable case. Let us consider, in H = C 2 , a case when the non-selfadjoint p ertur bation is n ot diagonalizable. F or example let H 0 :=  λ 0 0 λ  H := H 0 + V , V :=  0 v 0 0  , λ ∈ R , v ∈ R ∗ . W e ha v e f ( H 0 ) = f ( λ ) I 2 and f ( H ) = f ( λ ) I 2 + f ′ ( λ )  0 v 0 0  b ecause ( H − z ) − 1 = ( λ − z ) − 1 I 2 − ( λ − z ) − 2  0 v 0 0  . 34 V. BR UNEAU, N. FRANTZ, AND F. NICOLEAU Then T r( f ( H ) − f ( H 0 )) = 0 and in th is case where the real sp ectrum is u nc hanged u n der the p ertur b ation, the SSF is a constan t that can b e chosen to b e zero. In the previous examples of op er ators ha ving discrete sp ectra, the S SF is simp ly a step function whic h coun ts the eigen v alues of H 0 whic h are p erturb ed and the p ertur b ed real eigen v alues of H γ . The n otion of S SF is mainly introdu ced to deal with op erators ha ving a n on-empt y con tinuous sp ectrum but the explicit calculus of th e SSF can quic kly b ecome complicated. Let us discuss a simple example of a rank 1 p erturb ation of a selfadjoint op erator ha ving contin u ous sp ectrum. 8.3. Rank one p erturbation: weak interaction wit h the con tinuous sp ectrum. In th e space H = L 2 ( R ), we consider H 0 the op erator of m ultiplication b y th e fu nction h 0 ( x ) := x 1 [0 , 1] ( x ) and for γ ∈ C , the p erturb ation V γ = γ Π 0 , wh er e Π 0 is the orthogonal pro jection on to a normalized function u 0 ∈ L 2 ( R ): Π 0 := h· , u 0 i u 0 , k u 0 k = 1. The sp ectrum of H 0 is [0 , 1] and 0 is an eigen v alue of inﬁ nite m ultiplicit y (eac h fu n ction sup p orted outside [0 , 1] is an eigenfunction). Since V γ is a compact p erturbation of H 0 , th en H γ = H 0 + V γ is closed and its essential sp ectrum is [0 , 1]. Let us assume that u 0 is supp orted in R \ [0 , 1], then H 0 u 0 = 0 and H γ u 0 := ( H 0 + V γ ) u 0 = γ u 0 , that is u 0 is an eigenfun ction corresp onding to the eigenv alue γ for H γ . In this case, we ha v e ( H γ − z ) − 1 = ( H 0 − z ) − 1 +  1 z + 1 γ − z  Π 0 , where ( H 0 − z ) − 1 is the op erator of multiplic ation by ( x − z ) − 1 1 [0 , 1] − z − 1 1 R \ [0 , 1] . Th en for any f ∈ D ( R ), T r( f ( H γ ) − f ( H 0 )) = ( f ( γ ) − f (0) if γ ∈ R − f (0) if γ ∈ C \ R , and, up to a constan t, the SSF is giv en, almost everywhere, by: ξ ( λ ; H γ , H 0 ) =      1 [0 ,γ ] ( λ ) if γ > 0 − 1 [ γ , 0] ( λ ) if γ < 0 1 [0 , + ∞ [ ( λ ) if γ ∈ C \ R . In this case, ev en though there is con tin uous sp ectrum, the sp ectra of H γ and H 0 diﬀer by only one eigen v alue, and the SSF is still a step function with ju mps at the real eigen v alues in ﬂ uenced b y the p ertur b ation. Let us mention that the form ula (4.7) give s the same expression by c ho osing arg D V ( λ ) = 0 wh en λ → − ∞ . Let us chec k it for γ = iβ , β > 0 (the case γ ∈ R is treated in [2]). W e ha ve , D V γ ( z ) := Det  I + V γ ( H 0 − z ) − 1  = 1 + γ h ( H 0 − z ) − 1 u 0 , u 0 i = 1 − γ z whose the p ole z = 0 and the zero z = γ are the eigen v alues of H . When λ → −∞ , D V γ ( λ ) tends to 1 and w e c ho ose its argumen t equal to 0. T hen the logarithm of D V γ ( λ ± iε ) is well deﬁned and smo oth with resp ect to λ ∈ R for ε > 0 su ﬃcien tly s mall. W e h a v e ln D V γ ( λ + iε ) − ln D V γ ( λ − iε ) = a ε ( λ ) + i b ε ( λ ) , with a ε ( λ ) = ln     D V γ ( λ + iε ) D V γ ( λ − iε )     , b ε ( λ ) = arg D V γ ( λ + iε ) − arg D V γ ( λ − iε ) . Clearly , lim ε → 0 + a ε ( λ ) = 0 b ecause   D V γ ( λ ± iε )   2 =     1 − iβ λ ± iε     2 = λ 2 + ( β ∓ ε ) 2 λ 2 + ε 2 and lim ε → 0 + λ 2 + ( β + ε ) 2 λ 2 + ( β − ε ) 2 = 1 . THE SPECTRAL SHIFT FUNCTION FOR NON- SELF-ADJOINT PER TURBA TIONS 35 F or the imaginary p art b ε ( λ ), w e h a v e to follo w the argum en t of 1 − iβ λ ± iε when λ go es from − ∞ to + ∞ on the real axis. The case ” − ” is s im p ler b ecause 1 − iβ λ − iε = 1 + β ε λ 2 + ε 2 − iβ λ λ 2 + ε 2 alw a ys has p ositiv e real part and then its argumen t remains in ] − π 2 , π 2 [. In particular wh en this quan tit y is real the argum ent v an ish es (there is no turning p oin t). In the case ”+” we ha v e 1 − iβ λ + iε = 1 − β ε λ 2 + ε 2 − iβ λ λ 2 + ε 2 whose real p art v anish es twice (for λ = ± p β ε − ε 2 ) and the imaginary part once (for λ = 0). Th en its argument go es f rom 0 (when λ tends to −∞ ) to 2 π (when λ tends to + ∞ ). When λ = 0 and ε is small, these quanti ties ha ve op p osite sign and the phase shift is π . F or λ 6 = 0, 1 − iβ λ + iε and 1 − iβ λ − iε ha v e the s ame limit 1 − iβ λ but the turn in g p oint 0 pro du ces a ph ase shift of 2 π , and we ha v e: lim ε → 0 + b ε ( λ ) =  0 if λ < 0 2 π if λ > 0 . W e obtain 1 2 iπ lim ε → 0 + ln D V γ ( λ + iε ) − ln D V γ ( λ − iε ) = 1 2  1 [0 , + ∞ [ ( λ ) + 1 ]0 , + ∞ [ ( λ )  , whic h is consistent with the ab o v e expr ession of ξ ( λ ; H γ , H 0 ) (it coincides with 1 [0 , + ∞ [ ( λ ) excepted at the p oin t λ = 0). 8.4. Rank one p erturbat ion: stronger interaction with the con tinuous sp ectrum. In the previous example, the calculatio n of the SSF was simpliﬁed by the fact that the sp ectra of H γ and H 0 diﬀer b y only t w o eigen v alues and the SSF is still a step function. A more in teresting example for whic h the p erturb ation interac ts with the con tinuous sp ectrum is the case where supp u 0 ∩ [0 , 1] 6 = ∅ . Let us consider the previous op erators H 0 and H γ = H 0 + V γ with u 0 = 1 [0 , 1] . In this case, the computation of f ( H γ ) is m ore tric ky , so we p r efer to compute the SSF ξ ( λ ; H γ , H 0 ) by using the form ula (4.7), for γ = iβ , β > 0. Th anks to (4.8) for β < 0, it suﬃces to tak e the complex conjugate. As in the selfadjoint case (see [2] for γ ∈ R ), for any u ∈ L 2 ( R ), V γ R 0 ( z ) u = γ h ( H 0 − z ) − 1 u , u 0 i u 0 = γ Z [0 , 1] u ( x ) x − z dx ! u 0 , and for γ = iβ , β > 0, z = x + iy , D V γ ( z ) = 1 + γ hR 0 ( z ) u 0 , u 0 i = 1 − β  arctan 1 − x y + arctan x y  + i β 2 ln  (1 − x ) 2 + y 2 x 2 + y 2  . The eigen v alues of H γ are 0 (as for H 0 , the asso ciated eigenfunctions are s upp orted ou tsid e [0 , 1]) and the zeroes of D V γ in C \ [0 , 1], that is z = x + iy satisfying 1 = β  arctan 1 − x y + arctan x y  ; ln  (1 − x ) 2 + y 2 x 2 + y 2  = 0 , y 6 = 0 or x / ∈ [0 , 1] . It admits a unique solution if and only if β > 1 π : z β := 1 2 (1 + i coth 1 2 β ). Thus for β ≤ 1 π the sp ectrum of H γ is [0 , 1] and f or β > 1 π , it is [0 , 1] ∪ { z β } where z β is an eigenv alue of m ultiplicit y one asso ciated w ith the eigenfunction w β ( x ) := 1 z β − x 1 [0 , 1] ( x ). F or β = 1 π , w e see that z β = 1 2 is an 36 V. BR UNEAU, N. FRANTZ, AND F. NICOLEAU outgoing sp ectral singularit y of order 1. Indeed, V iβ = C W C with C = Π 0 , W = iβ , and thanks to (7.6), C R H iβ ( z ) C W = Id − (Id + T 0 ( z )) − 1 =  1 − 1 D V iβ ( z )  Π 0 , where we u sed that T 0 ( z ) := C R H 0 ( z ) C W = iβ hR 0 ( z ) u 0 , u 0 i Π 0 . Then f rom the ab o v e expression of D V iβ ( z ) for z = 1 2 + δ + iε , ε > 0, w e deduce that | z − 1 2 | n k C R H iβ ( z ) C W k is uniformly b ounded with resp ect to δ + iε small enough, for n ≥ 1 (but not for n = 0). No w in order to apply (4.7 ), let us study the mo dulus and the argum ent of D V γ ( z ) for z = λ ± iε , ε > 0, giv en by: D V γ ( λ ± iε ) = 1 ∓ β  arctan 1 − λ ε + arctan λ ε  + i β 2 ln  (1 − λ ) 2 + ε 2 λ 2 + ε 2  . 8.4.1. Imaginary p art of the SSF. The im aginary part of ξ ( λ ; H γ , H 0 ) is: Im ξ ( λ ; H γ , H 0 ) = − 1 2 π lim ε → 0 + ln     D V γ ( λ + iε ) D V γ ( λ − iε )     . When λ ∈ R \ [0 , 1], (1 − λ ) and λ h a v e opp osite signs then lim ε → 0 + D V γ ( λ ± iε ) = 1 + iβ ln  λ − 1 λ  , is indep endent of the sign in front of ε and Im ξ ( λ ; H γ , H 0 ) = − 1 2 π ln 1 = 0 , for λ ∈ R \ [0 , 1] . (8.1) When λ ∈ ]0 , 1[, we hav e lim ε → 0 +  arctan 1 − λ ε + arctan λ ε  = π . Then, Im ξ ( λ ; H γ , H 0 ) = − 1 4 π ln  (1 − β π ) 2 + β 2 f ( λ ) 2 (1 + β π ) 2 + β 2 f ( λ ) 2  = G β ( f ( λ )) , (8.2) where G β is th e ev en function d eﬁned on R by: G β ( X ) := 1 4 π ln  1 + 4 β π (1 − β π ) 2 + β 2 X 2  , and f is the decreasing f u nction from ]0 , 1[ onto R , giv en by: f ( λ ) := ln( λ − 1 − 1) , λ ∈ ]0 , 1[ . (8.3) It is a symmetric function with resp ect to 1 / 2 (i.e. f (1 − λ ) = − f ( λ )). It f ollo ws that the imaginary part of the SS F is given by: Im ξ ( λ ; H γ , H 0 ) = ( 0 if λ / ∈ [0 , 1] G β ( f ( λ )) if λ ∈ ]0 , 1[ (8.4) In particular it h as the sign of the imaginary part of V γ and it is contin u ous excepted for β = 1 π (the sp ectral singularit y z β = 1 2 is a singularit y of Im ξ ( · ; H γ , H 0 )). THE SPECTRAL SHIFT FUNCTION FOR NON- SELF-ADJOINT PER TURBA TIONS 37 8.4.2. R e al p art of the SSF. The real part of the SSF is Re ξ ( λ ; H γ , H 0 ) = 1 2 π lim ε → 0 +  arg D V γ ( λ + iε ) − arg D V γ ( λ − iε )  , where the argumen t is c hosen with th e condition lim λ →−∞ arg D V γ ( λ ) = arg 1 = 0 . • When β ∈ ]0 , 1 π [, the real part of D V γ ( λ ± iε ) is alw a ys p ositiv e, then arg D V γ ( λ ± iε ) ∈ ] − π 2 , π 2 [ and there is n o turnin g p oint . It follo ws th at lim ε → 0 + D V γ ( λ + iε ) = lim ε → 0 + D V γ ( λ − iε ) = ⇒ lim ε → 0 +  arg D V γ ( λ + iε ) − arg D V γ ( λ − iε )  = 0 . It is the case for λ / ∈ [0 , 1] b ecause lim ε → 0 + D V γ ( λ ± iε ) = 1 + i β 2 ln  (1 − λ ) 2 λ 2  = 1 + iβ ln (1 − λ − 1 ) . F or λ ∈ ]0 , 1[, by usin g again that the real part of D V γ ( λ ± iε ) is alwa ys p ositive and th at for a > 0, arg( a + ib ) = arctan b a , for f ( λ ) = 1 2 ln (1 − λ ) 2 λ 2 = ln( λ − 1 − 1), we h a v e: lim ε → 0 +  arg D V γ ( λ + iε ) − arg D V γ ( λ − iε )  = arctan β f ( λ ) 1 − β π − arctan β f ( λ ) 1 + β π = 2 π F β ( f ( λ )) where F β is th e o dd f unction deﬁn ed on R by F β ( X ) := 1 2 π  arctan β X 1 − β π − arctan β X 1 + β π  , β 6 = 1 π . W e dedu ce that for β ∈ ]0 , 1 π [, the SSF is the follo wing con tin uous fu n ction sup p orted on [0 , 1]: ξ ( λ ; H γ , H 0 ) = ( 0 if λ / ∈ [0 , 1] ( F β + iG β )( f ( λ )) if λ ∈ ]0 , 1[ . (8.5) • No w, let us consider the case β > 1 π for wh ic h the sp ectrum of H γ is [0 , 1] ∪ { z β } w ith z β := 1 2 (1 + i coth 1 2 β ). In th is case, the real part of D V γ ( λ − iε ) r emains p ositiv e, while D V γ ( λ + iε ) turns around 0. It will yield a p hase shift of 2 π for λ ∈ ]0 , 1[. First, as ab ov e, for λ < 0, b oth D V γ ( λ ± iε ) tend to 1 + iβ ln(1 − λ − 1 ) as ε ց 0 and lim ε → 0 +  arg D V γ ( λ + iε ) − arg D V γ ( λ − iε )  = 0 . F or λ ∈ ]0 , 1[, we hav e lim ε → 0 + D V γ ( λ ± iε ) = 1 ∓ β π + iβ f ( λ ); f ( λ ) = ln( λ − 1 − 1) . Then as b efore lim ε → 0 + arg D V γ ( λ − iε ) = arctan β f ( λ ) 1 + β π . Con trariwise since (1 − β π ) < 0, for arg D V γ ( λ + iε ), w e hav e to take into accoun t a phase shift of π and lim ε → 0 + arg D V γ ( λ + iε ) = π + arctan β f ( λ ) 1 − β π . 38 V. BR UNEAU, N. FRANTZ, AND F. NICOLEAU F or λ > 1, D V γ ( λ + iε ) and D V γ ( λ − iε ) ha v e the same limit (as for β < 1 π ) but th e phase shift b ecomes 2 π . S o ﬁnally , for β > 1 π w e obtain: ξ ( λ ; H γ , H 0 ) =        0 if λ < 0 ( 1 2 + F β + iG β )( f ( λ )) if λ ∈ ]0 , 1[ 1 if λ > 1 . (8.6) • T o conclude this example, let us discuss the case β = 1 π for which z β = 1 2 is a sp ectral singularit y . As in the previous cases, for λ < 0, we obtain ξ ( λ ; H γ , H 0 ) = 0. F or λ ∈ ]0 , 1[, the real part of D V γ ( λ + iε ) tends to 0 + when ε ց 0 an d the im aginary part c hange of sign at λ = 1 2 . Then, we ha v e lim ε → 0 + arg D V γ ( λ + iε ) = π 2 1 ]0 , 1 2 [ ( λ ) − π 2 1 ] 1 2 , 1[ ( λ ) , while lim ε → 0 + arg D V γ ( λ − iε ) = arctan β f ( λ ) 1 + β π = arctan f ( λ ) 2 π . F or λ > 1, the real part of D V γ ( λ + iε ) and of D V γ ( λ − iε ) are b oth p ositive and since they ha v e the same limit there is n o p hase shift. C onsequen tly , w hen β = 1 π , the SS F is the follo w ing function, d iscontin u ous at the sp ectral singularity z β = 1 2 : ξ ( λ ; H γ , H 0 ) =    0 if λ / ∈ [0 , 1] F π − 1 ( f ( λ )) if λ ∈ ]0 , 1[ \{ 1 2 } , (8.7) with F π − 1 ( X ) := lim β → (1 /π ) − F β ( X ) =        − 1 2 π  π 2 + arctan X 2 π  if X < 0 1 2 π  π 2 − arctan X 2 π  if X > 0 . (8.8) In ord er to visualize the d iﬀeren ce b et we en the th r ee formulas (8.5), (8.6) and (8.7 ), let us lo ok at the graphical r epresen tation of the real and imaginary parts of the sp ectral s hift function ξ ( · ; H γ , H 0 ) in the particular cases β = 0 . 2 < 1 π , β = 1 π and β = 0 . 4 > 1 π . − 0 . 5 0 0 . 5 1 1 . 5 − 0 . 5 0 0 . 5 1 λ Re SSF − 0 . 5 0 0 . 5 1 1 . 5 0 1 2 3 4 λ Im S SF β = 0 , 2 β = 0 , 4 β = 1 /π When 0 < β < 1 π the p erturbation V iβ do es not create a new eigen v alue or sp ectral singularit y , and ξ ( · ; H iβ , H 0 ) is con tinuous on R . F or β > 1 π , the j ump of 1 when λ crosses the con tin uous sp ectrum THE SPECTRAL SHIFT FUNCTION FOR NON- SELF-ADJOINT PER TURBA TIONS 39 [0 , 1] can b e explained by the app earance of th e non-real eigen v alue z β . Finally , for β = 1 π , the real part of ξ ( · ; H iβ , H 0 ) has a jump of h igh 1 2 at the sp ectral singularit y λ = 1 2 . Moreo v er at this p oin t, the imaginary part of the SSF blo ws up. A cknowledgments This work w as partially condu cted within the F rance 2030 f ramew ork p rogramme, C entre Henri Leb esgue ANR-11-LABX-0020 -01. V.B is p artially su pp orted b y the ANR-24-CE40- 2939-01 gran t. N.F. is sup p orted b y the R ´ egion P a ys de la Loire for the Conn ect T alent Pro ject HiF rAn 2022 07750 led by C lotilde F ermanian K ammerer and the ANR-25-CE40-729 6 (La Gabare). V.B and F.N. also thank the F rench GDR Dyn q u a for his supp ort. Referen ces [1] A. Alexander, J. F aup in, S. 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American Mathematical So ciety , Providence, R I, 2010. (V. Brun eau) Institut de Math´ ematiques de Bordeaux, UMR CNRS 5251, Univ ersit ´ e d e Bordeaux 351 cours d e la Lib ´ eration, 33405 T alence cedex, F rance Email adr ess: vb runeau@m ath.u-b ordeaux.fr (N. F ran tz) Univ Angers, CNRS, LAREMA, F-49000 Angers, F rance Email addr ess: nicolas.frantz @u n iv-angers.fr (F.Nicol eau) Lab oratoire de Math ´ ematiques Jean Lera y , UMR CNRS 6629. Nant es Universit ´ e F- 44000 Nan tes Email adr ess : francois.nicoleau@univ-nant es.fr

The Spectral Shift Function for Non-Self-Adjoint Perturbations

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