WKB for semiclassical operators: How to fly over caustics (and more)

WKB for semiclassical op erators: Ho w to fly o v er caustics (and more) V ˜ u Ngo . c San ∗ Marc h 27, 2026 Abstract The metho d initiated b y W en tzel, Kramers, and Brillouin to find appro ximate solutions to the Schrödinger equation lies at the origin of the spectacular developmen t of microlocal and semiclassical analysis. When used naiv ely , the approac h appears to break do wn at caustics, but Maslo v sh o w ed ho w a simple generalization could ov ercome this difficult y . In this pap er, after a partial historical review, we tak e adv an- tage of more recen t adv ances in microlo cal analysis to presen t a unified treatmen t of this generalized Maslov-WKB metho d, using the microlo- cal sheaf-theoretic approac h of [ 63 ]. This framew ork provides a rigor- ous pro of of the Bohr–Sommerfeld–Einstein–Brillouin–Keller quan ti- zation conditions for the eigen v alues of general semiclassical op erators (pseudo differen tial and Berezin–T o eplitz) in one degree of freedom. W e also review some applications and extensions. 1 EBK and WKB In 1926, almost simultaneously , three pap ers app eared to prop ose an approx- imate solution to the newly published Schrödinger equation [ 60 ]: − ℏ 2 2 m d 2 Ψ dx 2 + V Ψ = E Ψ , (1.1) where m is the particle mass, V : x 7→ V ( x ) the p oten tial, Ψ : x 7→ Ψ( x ) the wa vefunction, and E the energy eigen v alue. The three authors are W en tzel [ 69 ] (received on June 18, 1926), Brillouin [ 4 , 5 ] (receiv ed on July 1, 1926 and October 13, 1926), Kramers [ 45 ] (receiv ed on Septem b er 9, 1926). ∗ Univ Rennes, CNRS, IRMAR — UMR 6625, F-35000 Rennes, F rance 1 This approximate solution, no w called the WKB Ansatz, has the form Ψ( x ) = a ℏ ( x ) exp  i ℏ φ ( x )  , (1.2) where the reduced Planck constant ℏ is treated as a small real parameter, and the function a ℏ ( x ) admits an asymptotic expansion in non-negative integral p o w ers of ℏ : a ℏ ( x ) = a 0 ( x ) + ℏ a 1 ( x ) + ℏ 2 a 2 ( x ) + · · · in some adequate top ology . The fact that the small parameter app ears in fron t of the deriv ative in ( 1.1 ) make the limit ℏ → 0 singular , in the sense that the “kinetic” term − ℏ 2 2 m d 2 Ψ dx 2 should not be considered smaller than the “p oten tial” term V Ψ . The WKB Ansatz ( 1.2 ) reflects this by the highly oscillatory b eha vior of the phase (the function φ is real-v alued.) Both historically and scientifically , it w ould hav e b een in teresting to b e there, bac k in 1926, to interview these three scientists and understand how the v ery same idea came up to their minds at the same time. In vestigating this now ada ys would require the comp etence of a historian, which I do not p ossess; the only thing I can say is that Brillouin’s pap ers men tion W entzel, and Kramers’ pap er, which, by the wa y , was written in Utrech t 1 , do es men- tion Brillouin and W entzel’s work, whic h makes the usual ordering of the three initials difficult to justify 2 . I ha ve no clue as to whether this w as recipro cal. On the other hand, this idea didn’t come from no where, not at all. Al- ready in 1899, there were several pap ers by Horn [ 41 , 42 ] treating the same differen tial equation with the same Ansatz, and he himself based his work on previous studies by Sturm and others. Historical pap ers usually also men- tion the work of Jeffreys [ 43 ], whic h in terestingly w as published just a few mon ths b efore the famous Schrödinger pap ers. So, it is actually difficult to claim that the WKB Ansatz was really no vel in 1926. On the con trary , one should not b e surprised that, as so on as Schrödinger published his equation, man y researchers tried to find a wa y to solve it and came up with this Ansatz. What is really in teresting in the WKB approac h is that these authors w ere the first — to my knowledge — to realize that the asymptotic expansion of Horn and others actually enlightens the relationship b et ween the old clas- sical mechanics of Hamilton and the new quantum mec hanics of Schrödinger. 1 where Duistermaat, another name in this story , sp en t most of his career. 2 The WKB ordering reflects the publications dates if one omits the CRAS note [ 4 ]. F rench researchers often use BKW b ecause it’s alphab etical — or for a less honorable reason. 2 In particular, the phase φ must satisfy the no w called Hamilton–Jacobi equa- tion: H  x, ∂ φ ∂ x  = E , (1.3) where H is Hamilton’s function 3 : H ( x, ξ ) = 1 2 ∥ ξ ∥ 2 + V ( x ) . Ev en more fascinating, they show ho w this Ansatz gives a very simple justi- fication of the EBK quantization rule of the old quan tum theory . This rule, a generalization of the Bohr-Sommerfeld quantization formula, claims that in order for the energy E to b e a correct eigenv alue of the Sc hrö dinger op erator, it should satisfy the following geometrical equation: 1 2 π ℏ I γ E ξ d x = n ∈ Z , (1.4) where γ E denotes a closed classical orbit in phase space with energy E , and ξ is the classical momentum (denoted b y p in the physics literature) conjugate to the p osition coordinate x (often denoted by q ). The in teger n is called the quantum num b er. Lik e WKB, EBK is another concatenation of three authors’ initials: Ein- stein, Brillouin (the same Brillouin), and Keller. But this time, con trary to WKB, the historical progression is clear. T o mak e it simple, one can sa y that Einstein [ 29 ] prop osed the quantization rule based on geometric consid- erations (and of course on earlier w orks, in particular by Bohr, Ehrenfest, Epstein, Sc hw arzsc hild, Sommerfeld), then Brillouin show ed the connection with Sc hrö dinger’s new quan tum mec hanics (this is the same paper as the one cited for WKB!), and finally Keller [ 44 ] explained that, in order to b e more accurate, the rules require a correction (no w also called the Maslo v correction): 1 2 π ℏ I γ E ξ dx = n + µ 4 , n ∈ Z , (1.5) where µ is called the (Keller-)Maslo v index; it is defined from top ological prop erties of the lo op γ E and accounts for phase shifts at caustics or turning p oin ts. This remark ably simple rule, (partly) justified by the WKB Ansatz, bridges classical mec hanics and quantum mec hanics b y imposing discrete energy levels based on classical tra jectories and corrected b y topological in- formation. 3 F rom no w on, I incorp orate the mass m in to the new semiclassical parameter ℏ m . 3 2 F rom WKB to Microlo cal Analysis This is not the end of the story . In fact, I would rather say that the WKB Ansatz w as the starting p oint of a new and fascinating mathematical devel- opmen t of in timidating breadth: microlo cal analysis. Indeed, mathematicians had decades of headaches trying to prop erly jus- tify the WKB approach. The first problem comes from turning p oin ts, or caustics. These are p oints where the appro ximation ceases to b e uniform and in fact blows up. F or the one-dimensional Schrödinger equation, these turn- ing p oin ts corresp ond, on the Hamiltonian side, to a p osition x where the momen tum ξ of the particle v anishes. Thus, the energy level set has there a vertical tangent, and hence the phase function φ has an infinite deriv a- tiv e, whic h forbids the Hamilton–Jacobi equation to b e solv ed by smo oth functions. A t this p oin t, it is difficult not to notice that the EBK quantization rule, b y con trast, has no singular b eha vior whatso ev er at turning p oin ts. So is EBK b etter than WKB, or more general? This is really tempting to sa y so, b ecause in fact not only do es EBK ha v e no singularit y , but it can also b e generalized to an y kind of Hamiltonian function, not sp ecifically the one obtained from Sc hrö dinger op erators, which indicates that it could be used to solv e eigenv alue problems arising from more general partial differen tial equations. The extension of the WKB method to a large class of op era- tors was done thirty years later b y Lax [ 46 ], but he was still constrained by the caustic issue. It seems that Maslov [ 53 ] w as the first to substan tially extend the WKB Ansatz to a form that would swallo w caustics using par- tial F ourier transforms. Then, Hörmander [ 39 , 40 ], also inspired by Sato’s microfunctions [ 59 ], wrapp ed up the whole thing in to a formidable theory no w called (homogeneous) microlo cal analysis. Unfortunately , Hörmander neither wan ted to motiv ate his theory b y quantum mec hanics nor to apply it to this field. He was interested in the regularity theory of distributional solutions to general partial differen tial equations and did not use an y small parameter ℏ . Duistermaat [ 23 ] was the first to incorp orate the semi-classical parameter ℏ back again in order to use microlo cal analysis tow ards recov er- ing Maslov’s results. This was the birth of semiclassical analysis. In doing so, Duistermaat relied on the theory of F ourier integral op erators, recently dev elop ed b y Hörmander and himself [ 24 ] (follo wing a thread of ideas includ- ing in particular the work of Egoro v [ 28 ]), in order to generalize the notion of WKB solutions, now called semi-classical lagrangian distributions. F or- mally (that is, ignoring the problem of con vergence of integrals), a lagrangian distribution is simply a linear sup erp osition of standard WKB Ansätze, as 4 follo ws: Ψ( x ) = 1 (2 π ℏ ) n/ 2 Z R n e i ℏ φ ( x,θ ) a ℏ ( x, θ ) d θ . (2.1) 3 The lagrangian manifold viewp oin t Consider the phase space R 2 n equipp ed with the canonical symplectic form ω = d ξ ∧ d x = P n j =1 d ξ j ∧ d x j . A lagrangian 4 (sub)manifold is a subman- ifold of R 2 n of dimension n on which the symplectic form v anishes. F or instance, the “horizontal fib er” { ξ = const } is a lagrangian submanifold (of some imp ortance, as we shall see later). Of course, the same notion extends to any symplectic manifold. What do lagrangian manifolds hav e to do with the Schrödinger equation? W ell, Alan W einstein had this now famous sen- tence: “ev erything is a lagrangian manifold.” [ 68 ]. I’m not going to fully justify this, but let me explain wh y an y WKB solution — ev en in the gen- eralized Maslov sense ( 2.1 ) — of the Schrödinger equation is asso ciated to a lagrangian manifold. Recall that the phase φ is a solution to the Hamilton-Jacobi equation H  x, ∂ φ ∂ x  = E . (3.1) So w e hav e a natural subset of the energy lev el set, defined by the set Λ φ of all pairs ( x, ∂ φ ∂ x ) , which is the graph of the differen tial of φ . I claim that this graph is a lagrangian submanifold. Indeed, if you restrict the symplectic form d ξ ∧ d x to this graph, y ou obtain: d ξ ∧ d x ↾ Λ φ = d( ξ d x ) ↾ Λ φ = d( d φ ) = 0 . (3.2) (The Liouville 1-form α = ξ d x has this remark able “tautological” property that, when restricted to a section of the cotangent bundle — i.e. , a 1-form —, it produces bac k the 1-form itself, here d φ .) In the case of the generalized WKB Ansatz, as in Equation ( 2.1 ), and under generic conditions on φ , we can see by the stationary phase lemma that the in tegral lo calizes on the critical p oin ts of the map θ 7→ φ ( x, θ ) , 4 I follow here W einstein’s use [ 67 ] of the fully adjectiv al form “lagrangian”— that is, without capital “L”; I also second W einstein in ending the word with “ian” instead of “ean”; after all, Joseph Louis de Lagrange was b orn Giusepp e Luigi Lagrangia. 5 that is, on the set C φ =  ( x, θ )     ∂ φ ∂ θ ( x, θ ) = 0  . A t these lo calized points, the asymptotic expansion of Ψ( x ) b ecomes a usual WKB Ansatz. Hence, one can show that if it is a solution to the Schrödinger equation, we must hav e the Hamilton-Jacobi equation H  x, ∂ φ ∂ x ( x, θ )  = E , where ( x, θ ) ∈ C φ . And again, one can chec k that the image of C φ under the map ( x, θ ) 7→  x, ∂ φ ∂ x ( x, θ )  is a lagrangian submanifold. Remark ably , one can pro ve conv ersely that a (non-empty) class of lagrangian distributions whic h are lo cally of the form ( 2.1 ) can b e asso ciated to any lagrangian submanifold; this is one the main ac hievemen ts of Maslo v, extended to a great generalit y by Hörmander (but for homogeneous lagrangian submanifolds) and then Duistermaat; for a thorough explanation, I recommend Duistermaat’s pap er [ 23 ]. It follo ws that, in order to find a go od phase for a (generalized) WKB solution to the Sc hrö dinger equation (or to an y go o d semiclassical equation, see b elo w), it is simply enough to find a arbitrary lagrangian submanifold contained in the required energy lev el set H ( x, ξ ) = E . One of the (many) magical aspects of lagrangian manifolds is that suc h a lagrangian is automatically invariant by the Hamiltonian flow of H . Indeed, if v is tangent to Λ , then ω ( X H , v ) = − d H · v = 0 , the first equalit y defining the Hamiltonian v ector field X H and the second one expressing that v is tangen t to the level set of H . Therefore, X H is symplectically orthogonal to Λ . But since Λ is lagrangian, its symplectic orthogonal is itself, and hence X H is tangent to Λ ! 5 W e now see that solving the Sc hrö dinger equation b y means of the WKB Ansatz, at first order O ( ℏ ) , amoun ts to finding an invariant lagr angian sub- manifold of the phase space. (By inv ariant w e shall alw a ys mean in v arian t under the Hamiltonian flo w of the classical Hamiltonian H .) 5 More details concerning the geometry of the WKB approximation can b e found in the nice b o ok [ 3 ]. 6 The reader ma y w onder why we introduce general lagrangian manifolds here, while in this pap er we are mostly interested in the one-dimensional case: the phase space is R 2 , or a t wo-dimensional surface, and a lagrangian manifold is just a curv e in that space. W ell, one reason is that the first part of the WKB pap ers, whic h deals with appro ximate solutions to the Sc hrö dinger equation, actually w orks in an y dimension and hence defines general lagrangian submanifolds, whic h pro vides a nice geometric interpre- tation of the w av e function Ψ . How ever, the second part, whic h aims at justifying the Bohr–Sommerfeld quan tization condition, is m uch more deli- cate and only works in dimension 1, or for separable or completely integrable systems. But, and this is our second reason, it turns out that our main to ol for a rigorous justification of the Bohr-Sommerfeld condition, ev en in di- mension 1, will b e F ourier Integral Op erators, which themselves are actually WKB functions defined in a higher dimensional space. 4 Semiclassical op erators and microlo cal solutions Microlo cal analysis w as first dev elop ed for (semiclassical) pseudo differential op erators, whic h is a large class of linear op erators con taining all differential op erators with smo oth co efficien ts (and hence the Sc hrö dinger op erator) 6 . It was then realized that the whole theory can also be applied to a very differen t class of operators, the so-called Berezin-T oeplitz op erators (see for instance [ 50 ]). These are not differential op erators. Instead, for each fixed v alue of ℏ , they act on a finite-dimensional Hilb ert space, the dimension of whic h gro ws to infinit y as ℏ → 0 . In this pap er, following an idea of [ 51 , App endix A], we will call semiclassic al op er ators operators that are either semiclassical pseudo differential op erators or Berezin-T oeplitz operators. In b oth cases, w e shall denote b y H the natural Hilbert space on whic h they act (this space dep ends on ℏ in the Berezin-T o eplitz case), and by M the phase space (so H = L 2 ( R n ) and M = R 2 n in the pseudodifferential case). In the Berezin-T o eplitz case, the admissible v alues of ℏ are quantized: t ypically , ℏ = 1 /k with k ∈ N ∗ . How ev er, I wrote this text with a bias to- w ard pseudo differen tial op erators; hence, for notational simplicit y , we shall sa y “ ℏ ∈ ]0 , 1] ” when we actually mean “ ℏ admissible and inside ]0 , 1] ”. W e 6 T echnically , one requires a mo derate growth of the co efficien ts; for a Schrödinger op- erator, this means that V m ust b e smo oth and b eha ve at most p olynomially at infinity . In current microlo cal terminology , the symbols of our op erators must b elong to an appro- priate class , which we shall implicitly assume throughout this article. See for instance [ 70 ] for details. 7 shall also denote by ( x, ξ ) a p oint in phase space, ev en though for a gen- eral symplectic manifold there might not b e such global co ordinates. In the Berezin-T o eplitz quan tization sc heme, the WKB phase e i ℏ φ ( x ) m ust b e re- placed b y a holomorphic section of a prequantum line bundle, whic h is flat on some lagrangian manifold Λ , see [ 7 ]. (This also co vers the “generalized WKB Ansatz” ( 2.1 ).) Let P b e a semiclassical op erator with symbol H ( x, ξ ) . W e sa y that ψ ∈ H is a WKB solution of order N to the equation ( P − E ) ψ = 0 if ψ is a lagrangian distribution whose associated lagrangian submanifold is con tained in the energy level set H = E , and suc h that ψ is a quasimode of order N , that is ( P − E ) ψ = O ( ℏ N ) . (4.1) F or instance, when P = ℏ i ∂ ∂ x , whic h is arguably the simplest semiclassical op erator, then ψ ( x ) = C e i ℏ E x , x ∈ R , is a plane wave and a lagrangian distribution associated with the lagrangian manifold { ( x, ξ ); ξ = E } , and is a WKB solution to ( 4.1 ) at any order. F or a lagrangian distribution, the quasimo de condition ( 4.1 ) with k = 1 is automatic. Ho wev er, for larger k , one has to solv e transp ort equations b ey ond the Hamilton-Jacobi equation in order to obtain higher-order solu- tions. In this pap er, I do not describ e the transp ort equations, b ecause they will b e hidden in the more general theory of semiclassical F ourier integral op erators. In whic h sense, or in which topology , do w e wan t Equation ( 4.1 ) to hold? It will be con v enient to use the notion of a micr olo c al solution , whic h asserts that Equation ( 4.1 ) holds microlo cally near a p oin t ( x, ξ ) in phase space if and only if it holds in the H norm after microlo cal pro jection in a neigh b orho od of that p oin t. And, b y microlo cal pro jection, w e mean a semiclassical op erator whose symbol is 1 in a small neigh b orho od of ( x, ξ ) and 0 outside some compact set. An element u ℏ ∈ H is said to b e micr olo c al ly supp orte d 7 in some set Σ ⊂ M if it is microlo cally O ( ℏ ∞ ) outside of Σ . The microlo cal support is also called the semiclassical w av efron t set. See for instance [ 51 , Definitions A.1 and A.4] Using semiclassical partitions of unit y , one can sho w that the set of mi- crolo cal solutions has a nice sheaf structure. This means that if you ha ve t wo solutions defined on tw o different small open sets with a non-empty in- tersection, and whic h coincide on the intersection, then they can b e glued to pro duce a solution on the union of the op en sets. As w e shall see, one can 7 or even “microsupp orted” 8 alw ays choose the second solution suc h that the coincidence on this intersec- tion holds. In other w ords, an y lo c al se ction of the sheaf ov er the first op en set extends to a section ov er the union of b oth op en sets. Ho wev er, even if we ha ve a family of op en sets cov ering a compact in v arian t lagrangian subman- ifold, if that manifold is not simply connected, a glob al section of the sheaf migh t not alw ays exist. If it do es exist (which in general will impose some conditions on E ), then the solution actually extends to a true global solu- tion in L 2 of Equation ( 4.1 ). Therefore, this pro ves (at least for self-adjoint op erators) that the energy E is ℏ k -close to the sp ectrum of the operator P . These ideas w ere implemented b y Colin de V erdière in the pioneer article [ 9 ]. Let us now see ho w it works more precisely in the one-dimensional case. 5 The Bohr-Sommerfeld co cycle Let P b e a self-adjoint semiclassical op erator with discrete sp ectrum in a sp ectral window [ E 1 , E 2 ] (for instance, the Schrödinger op erator with a con- fining p oten tial, or any Berezin-T o eplitz op erator on a compact Kähler man- ifold). W e shall assume from no w on that P has only one degree of freedom: the phase space has dimension 2, and that the in terv al [ E 1 , E 2 ] is contained in the set of regular v alues of the principal sym b ol H of P . Moreo ver, w e assume that H − 1 ([ E 1 , E 2 ]) is compact. W e wish here not only to use the WKB Ansatz to obtain approximate solutions to ( P − E ) ψ = 0 , but actually to sho w that al l solutions are indeed obtained in this wa y , and in turn to obtain a v ery precise description of the spectrum of P in [ E 1 , E 2 ] . F or Schrödinger op erators, sp ecific pro ofs can b e found in the literature, see for instance the interesting lecture b y V oros [ 62 ]. But for a statement of this generality , a full microlo cal approach is more appropriate. T o my kno wledge, the pap ers [ 57 , 12 , 34 ] were among the first ones to emplo y deep microlo cal analysis to deriv e the Bohr-Sommerfeld rules. Here, m y plan is to b e “ev en more microlo cal”, in the sense that we start to describ e the problem lo cally near a p oint in phase space, and try to pass from local to global. The strategy follows [ 63 , 65 ], but I shall give more details here. Giv en an arbitrary function ℏ 7→ E ( ℏ ) , let us denote b y D = D ( P,E ) the sheaf of microlocal solutions of P − E ( ℏ ) . In other words, the set D (Ω) of sections of D o ver an op en subset Ω ⊂ M is the space of microlo cal solutions ψ ℏ to the equation ( P − E ( ℏ )) ψ ℏ = O ( ℏ ∞ ) on Ω . First note that, by standard semiclassical ellipticit y , if E ( ℏ ) → E 0 , then D is supported in the energy lev el set H = E 0 . Since the phase space has 9 dimension 2, this level set is generically just a curve and hence a lagrangian submanifold b y itself. The k ey result, which says that D is locally giv en b y WKB solutions near an y regular p oin t of H , is the following. Prop osition 5.1 ([ 63 ]) Ne ar any r e gular p oint m of H , the sp ac e of mi- cr olo c al solutions to ( P − E ) ψ = 0 is one-dimensional, mor e pr e cisely: • Ther e exists a fixe d 8 neighb orho o d Ω of m , and for any E ∈ H (Ω) , a WKB solution ( i.e. a lagr angian distribution) Ψ ℏ = Ψ ℏ ,E , dep ending smo othly on E , with ∥ Ψ ℏ ∥ = 1 , such that ( P − E )Ψ ℏ = O ( ℏ ∞ ) on Ω . • If ψ = ψ ℏ satisfies ( P − E ) ψ ℏ = O ( ℏ ∞ ) ne ar m and ∥ ψ ℏ ∥ = 1 , then ther e exists a WKB solution Ψ ℏ and a c onstant phase C = C ( E , ℏ ) ∈ U (1) , such that ψ ℏ − C ( ℏ )Ψ ℏ = O ( ℏ ∞ ) on Ω . In b oth items the r emainder O ( ℏ ∞ ) is lo c al ly uniform in E . When w e say that a WKB solution dep ends smo othly on E we mean that, in ( 2.1 ), b oth the phase φ and the amplitude a ℏ dep end smoothly on E , uniformly with resp ect to ℏ . This statement can b e seen as a semiclassical v arian t of [ 24 ]; it relies on the theory of F ourier in tegral op erators (FIO). FIOs are p erhaps the most p o w erful tool in microlo cal analysis. One of their celebrated uses is to quantize canonical transformations of phase space as unitary op erators. If U is suc h an FIO quan tizing the canonical transformation ( i.e. symplectic diffeomorphism) χ , then for any semiclassical op erator P with symbol H ( x, ξ ) , the conjugated op erator U − 1 P U is again a semiclassical op erator, and its symbol is H ◦ χ + O ( ℏ ) (this state- men t is called the Egoro v Theorem). See for instance [ 7 , Prop osition 4.3] or [ 70 , Theorem 11.5]. Remark ably , we hav e here another illustration of W einstein’s “symplectic creed” [ 68 ], b ecause FIOs are also lagrangian distributions! (More precisely , the graph of χ in the pro duct space R 2 n × R 2 n is automatically a lagrangian 8 i.e. indep endent of ℏ 10 submanifold 9 , and b y definition an FIO is an op erator whose Sch w artz kernel is a lagrangian distribution asso ciated with this graph.) Pro of of Prop osition 5.1 . Since m is regular: d H ( m )  = 0 , there exist local canonical co ordinates ( x, ξ ) near m such that H ( x, ξ ) − H ( m ) = ξ when ( x, ξ ) ∈ R 2 sta y in some neigh b orho od ˜ Ω of the origin (This the statement of Darb oux-Carathéodory’s theorem.) Let U b e an FIO asso ciated with this c hange of v ariables. Let E 0 = H ( m ) . W e obtain, microlo cally near m , U − 1 ( P − E 0 ) U = ℏ i ∂ ∂ x + O ( ℏ ) . By a standard pro cedure ( this is wher e the WKB tr ansp ort e quations ar e hidden! See [ 64 , Section 3.2] or the foundational paper [ 24 , Proposition 6.1.4] — in the homogeneous case.) one can m ultiply U b y a suitable unitary semiclassical op erator V such that, with ˜ U := U V , ˜ U − 1 ( P − E 0 ) ˜ U = ℏ i ∂ ∂ x + O ( ℏ ∞ ) . W orking in co ordinates, one can show that Prop osition 5.1 holds for the op erator ℏ i ∂ ∂ x − E ; indeed, its solutions are plane w av es ψ ( x ) = C e i ℏ E x . This pro ves the prop osition, with Ψ ℏ = C ˜ U e i ℏ ( E − E 0 ) x . Since E ∈ H (Ω) , there exists ( x 0 , ξ 0 ) ∈ ˜ Ω such that ξ 0 = E − E 0 ; hence the microsupp ort of e i ℏ ( E − E 0 ) x in tersects ˜ Ω , and we can normalize it b y a uniform constant. The fact that, for any fixed E , suc h a Ψ ℏ is indeed a generalized WKB Ansatz asso ciated with the lagrangian manifold { H = E } follo ws from the fact that the Sch wartz kernel of U is a lagrangian distribu- tion. By a stationary phase argument, one shows that the application of the semiclassical op erator V do es not mo dify this (this is called the semiclassical FIO calculus, see for instance [ 31 , section 9.7] or [ 7 , Prop osition 2.7]). □ Remark 5.2 In Prop osition 5.1 , the energy parameter E is purely decora- tiv e. Indeed, if P dep ends on an additional parameter λ , then the parametric v ersion of the prop osition holds. Therefore, one can set P E := P − E and describ e solutions to P E Ψ ℏ = O ( ℏ ∞ ) by viewing E as a parameter. This remarks b ecomes far-reac hing for sp ectral problems where the “sp ectral” pa- rameter do es not appear in a linear fashion. △ 9 this holds if the product R 2 n × R 2 n , sometimes denoted b y R 2 n × R 2 n , is equipp ed with the direct sum symplectic form ω ⊕ ( − ω ) 11 This prop osition allows us to understand in a simple wa y the sheaf D of microlo cal solutions to the equation ( P − E ) ψ = O ( ℏ ∞ ) , when the energy E is a regular v alue of H : this sheaf is then simply a flat bund le , meaning that for eac h point m = ( x 0 , ξ 0 ) of the level set H ( x, ξ ) = E , there exists a neigh b orho od U of m in which an y tw o microlo cal solutions must differ b y a multiplicativ e constant. Th us, at least theoretically , it is easy to determine when the sheaf has a glob al section ( i.e. a microlo cal solution in a full neighborho o d of the energy lev el set H = E ): this happ ens when the flat bundle is trivial, or equiv alen tly when its first Čec h cohomology group ˇ H 1 ( D ) v anishes (mo dulo O ( ℏ ∞ ) ). Ho w to compute this cohomology? When E is a regular v alue of H , we kno w that the level set H = E is a finite union of closed smo oth (em b edded) curv es C k = C k ( E ) , k = 1 , . . . , d . Let us consider a fixed j and denote b y D k the restriction of D to a small tubular neigh b orho od of C k . The curve C k is diffeomorphic to a circle, and has a natural orien tation giv en by the Hamiltonian v ector field X H , whic h does not v anish on it b y hypothesis. Let Ω j , j = 1 , . . . , N k b e an op en co ver of C k b y small balls, on eac h of whic h Prop osition 5.1 hold, and whose centers z j ∈ C k are ordered along the orien tation of C k . F or each j , the prop osition gives a WKB function Ψ j ℏ generating D (Ω j ) . By restricting to Ω j ∩ Ω j +1 (indices are written mo dulo N k ), the uniqueness part of the prop osition yields a constant phase C j = C j ( E , ℏ ) ∈ U (1) such that Ψ ( j +1) ℏ = C j Ψ ( j ) ℏ + O ( ℏ ∞ ) on Ω j ∩ Ω j +1 . (5.1) Definition 5.3 The assignment Ω j ∩ Ω j +1 7→ C j is c al le d the Bohr-Sommerfeld co cycle of the curve C k . (It is indeed a co cycle in the sense of Čech cohomology .) Prop osition 5.4 Ther e exists a micr olo c al solution of ( P − E ) on a neigh- b orho o d of C k if and only if the p air ( E , ℏ ) satisfies the e quation C 1 ( E , ℏ ) · · · C N k ( E , ℏ ) = 1 + O ( ℏ ∞ ) . (5.2) Pro of . The solution exists if and only if the Bohr-Sommerfeld co cycle is a coboundary , i.e. there exist constan ts d j = d j ( E , ℏ ) ∈ C ∗ , j = 1 , . . . N k , suc h that C j = d − 1 j +1 d j + O ( ℏ ∞ ) . (5.3) Indeed, in this case, we see from ( 5.1 ) that the lo cal sections d j Ψ ( j ) ℏ actually coincide (mo dulo O ( ℏ ∞ ) ) on the intersections Ω j ∩ Ω j +1 . Therefore, they 12 can b e glued together by a microlo cal partition of unity to form a solution on a full neigh b orho od of C k . Con versely , if such a global section Ψ ℏ exists, by Prop osition 5.1 we hav e constan ts d j suc h that Ψ ℏ = d j Ψ ( j ) ℏ + O ( ℏ ∞ ) on each Ω j . In view of ( 5.1 ), it follows that ( 5.3 ) holds. It remains to show why the cob oundary equation ( 5.3 ) is equiv alen t to ( 5.2 ). Since indices j are tak en mo dulo N k , it is clear that ( 5.3 ) im- plies ( 5.2 ). Conv ersely , if ( 5.2 ) holds, we let d 1 = 1 , d j = ( C 1 · · · C j − 1 ) − 1 for j = 2 , . . . , C k , which gives d − 1 1 d N k = ( C 1 · · · C N k − 1 ) − 1 = C N k + O ( ℏ ∞ ) , whic h pro v es ( 5.3 ) for j = N k (other j ’s are automatic.) □ Equation ( 5.2 ) will b e called the Bohr-Sommerfeld condition of the curve C k . Since the curv es for differen t k are disjoint, we obtain: Prop osition 5.5 Ther e exists a non-trivial se ction of D , i.e. a micr olo c al solution on a neighb orho o d of H = E , if and only if at least one of the Bohr- Sommerfeld c onditions asso ciate d with the curves C 1 , . . . , C d is satisfie d. A t this p oin t, it is not clear how to relate this Bohr-Sommerfeld condition to the EBK prediction. This is giv en b y the follo wing prop osition. Prop osition 5.6 The semiclassic al action A k ( E , ℏ ) := log ( C 1 · · · C N k ) ad- mits an asymptotic exp ansion of the form A k ( E , ℏ ) = 1 ℏ A k, 0 ( E ) + A k, 1 ( E ) + ℏ A k, 2 + ℏ 2 A k, 3 + · · · (5.4) with smo oth c o efficients A k,j ; and A k, 0 = Z C k α , (5.5) wher e α is the Liouvil le 1-form. Pro of . W e use the same quantum Darb oux-Carathéo dory normal form as in the pro of of Prop osition 5.1 , this time rep eated on each Ω j . Thus, we ma y ob- tain the Bohr-Sommerfeld co cycle ( C j ) by choosing Ψ ( j ) ℏ := ˜ U j e i ℏ ( E − H ( m )) x . (If w e viewed E as a parameter, see Remark 5.2 , we could simply tak e Ψ ( j ) ℏ := ˜ U j 1 .) F rom ( 5.1 ) w e obtain, microlocally on Ω j ∩ Ω j +1 , C j = e − i ℏ ( E − H ( m )) x U − 1 j U j +1 e i ℏ ( E − H ( m )) x + O ( ℏ ∞ ) . 13 Since C j is constant, it can b e ev aluated on any p oin t in Ω j ∩ Ω j +1 . Since the Sc hw artz k ernel of U j and U j +1 are lagrangian distributions, i.e. tak e the form ( 2.1 ), with an amplitude a ℏ admitting an asymptotic expansion in integral p o wers of ℏ , we obtain that, for eac h fixed E , C j m ust ha ve an asymptotic expansion of the form C j ∼ e i ℏ S 0 ( a 0 + ℏ a 1 + ℏ 2 a 2 + · · · ) , (5.6) and all co efficients dep end smo othly on E . Since | C j | = 1 + O ( ℏ ∞ ) we m ust ha ve | a 0 | = 1 . Hence we may also write C j ∼ e i ℏ P j ≥ 0 ℏ j S j , (as Brillouin [ 5 ] suggested) which gives ( 5.4 ). In order to compute A k, 0 , recall that Ψ ( j ) ℏ is a WKB solution associated to the curve C k ⊂ { H = E } . Hence, near an y p oin t m in the lagrangian manifold C k , w e hav e a phase function φ j as in ( 2.1 ); using ( 3.2 ), this phase, view ed as a function on C k , satisfies d φ j = α , where α = ξ d x is the Liouville 1-form. Ev aluating ( 5.1 ) on a p oin t m of Ω j ∩ Ω j +1 , we deduce that the co efficien t S 0 = S ( j ) 0 of ( 5.6 ) takes the form S ( j ) 0 = φ j +1 ( m ) − φ j ( m ) but since d( φ j +1 − φ j ) = 0 , the function φ j +1 − φ j is constant, and the set of all of these constants, i.e. ( S ( j ) 0 ) j =1 ,...,N k is exactly the Čech co cycle asso ciated with the closed 1-form α on C k . (See for instance [ 3 , Appendix C].) Thus, P N k j =1 S ( j ) 0 = R C k α , which giv es ( 5.5 ). □ A ctually , it is also possible to describ e the second term A k, 1 ; in the pseudo differen tial case, if P has no subprincipal symbol, A k, 1 = µ π 2 where µ is the (Keller-Leray-)Maslo v index of C k whic h, in R 2 , is alwa ys equal to 2 . In the Berezin-T o eplitz case, see [ 7 ]. In the sp ecial case of the Sc hrö dinger op erator, an iterative scheme to obtain A k,j for all j is kno wn [ 14 , 2 ]. 6 Quasi-mo des and eigenv alues Let us no w upgrade the microlo cal result of Prop osition 5.1 to a concrete statemen t ab out honest eigen v alues and eigenfunctions, in order to fully justify the EBK rule. 14 First of all, as exp ected, the generalized WKB construction will give us go od quasimodes, i.e. solutions (Ψ ℏ , E ( ℏ )) to ∥ ( P − E ( ℏ ))Ψ ℏ ∥ H = O ( ℏ ∞ ) . But w e hav e muc h more: exact eigenfunctions must b e O ( ℏ ∞ ) -close to these quasimo des, and hence the whole sp ectrum in [ E 1 , E 2 ] is O ( ℏ ∞ ) -close (in ev ery possible sense, including m ultiplicit y) to the set of E ’s for whic h the microlo cal WKB construction w as v alid. Most treatments of the Bohr-Sommerfeld rule in the literature make the simplifying assumption that energy lev el sets are connected 10 ; this is quite understandable, since the presen tation b ecomes m uch simpler, and adding connected comp onents is “well-kno wn to the experts”. F rom the p oin t of view o f symplectic geometry , this mak es no difference indeed, since eac h comp onen t can b e treated separately 11 . The microlo cal treatmen t is equally similar. Ho wev er, proving that WKB solutions exhaust the spectrum (which is the goal of this section) requires an additional argument, which we presen t here. One w ay to achiev e this (and again, this is known to exp erts, I don’t claim m uch originalit y) is to use the fact that quasimodes asso ciated to differen t comp onen ts are mutually orthogonal mo dulo O ( ℏ ∞ ) [ 7 , Prop osition 2.6]. One of the first mathematical accounts of the Bohr-Sommerfeld rule is due to Helffer and Rob ert [ 34 ]; interestingly , motiv ated by quantum tunnel- ing, they did consider m ultiple components, but with a symmetry assump- tion implying that all p erio ds of the Hamiltonian flo w on H − 1 ( E ) are equal, allo wing them to apply an idea of Colin de V erdière [ 11 ]. Theorem 6.1 (Bohr-Sommerfeld eigen v alues) L et P b e a self-adjoint semiclassic al op er ator with one de gr e e of fr e e dom and sp e ctrum σ ( P ) . Con- sider a sp e ctr al window I = [ E 1 , E 2 ] which c ontains only r e gular values of the princip al symb ol H of P . Assume mor e over that H − 1 ( I + [ − ϵ, ϵ ]) is c omp act in the phase sp ac e M , for some ϵ > 0 . Then σ ( P ) ∩ I is discr ete, and is describ e d as fol lows. 10 A notable exception is the recen t paper [ 21 ]. 11 A ctually , there is a top ological subtlety , whic h is never discussed to my kno wledge, concerning a distinction b etw een the global fibration b y H and the semiglobal one — i.e. in a neigh b orhoo d of a connected comp onen t: if H − 1 ( E ) is assumed to b e compact, then it is a union of circles, and near any circle C , the action-angle theorem asserts in particular that neigh b oring fibers are also compact and connected. . . which means: in a satur ate d neighb orho o d of C ; but this do es not imply that neighboring global fib ers of H : M → R are compact. 15 1. Ther e exists an inte ger d ≥ 0 such that for e ach E ∈ I , the ener gy level set H − 1 ( E ) is the disjoint union of d smo oth curves C 1 ( E ) , . . . , C d ( E ) . 2. Ther e exist smo oth functions E 7→ A k,j ( E ) , for k = 1 , . . . , d and j ∈ N , and smo oth functions I × ]0 , 1] ∋ ( E , ℏ ) 7→ A k ( E ; ℏ ) admitting the asymptotic exp ansion A k ( E ; ℏ ) ∼ 1 ℏ ∞ X j =0 ℏ j A k,j ( E ) in the smo oth top olo gy, such that σ ( P ) ∩ I c oincides mo dulo O ( ℏ ∞ ) , and including multiplicities, with the disjoint union F d k =1 σ k ( ℏ ) , which we denote by σ ( P ) ∩ I = d G k =1 σ k ( ℏ ) + O ( ℏ ∞ ) (6.1) with σ k ( ℏ ) := A k ( · ; ℏ ) − 1 (2 π Z ) . (6.2) pr e cisely, for e ach eigenvalue E = E ( ℏ ) of P , for any the sp e ctr al multiplicity of the interval B = [ E − ℏ N , E + ℏ N ] is, for ℏ smal l enough, e qual to the numb er of non-empty sets σ k ∩ B . 3. A k, 0 ( E ) = R C k ( E ) α , wher e α is the Liouvil le 1-form, and the map E → A k, 0 ( E ) is a diffe omorphism fr om I to its image. 4. In p articular, if d = 1 , then the sp e ctrum in I is simple for ℏ smal l enough, and gaps ar e of or der O ( ℏ ) . 5. If P has no subprincip al symb ol, then A k, 1 is the Kel ler-Maslov index of C k . Remark 6.2 W e will b e sloppy here on t wo tec hnical points: 1. The precise definition of “t wo subsets of R dep ending on ℏ which co- incide mo dulo O ( ℏ ∞ ) including m ultiplicit y”: see [ 30 , Definition 1.5]. This notion is natural but some care has to b e taken at the “b oundary” of the sets. See Step 4 of the proof below. 2. The precise definition of the Keller-Maslo v index, b oth in the pseudo d- ifferen tial [ 23 ] (see also [ 31 , Section 5.13]) and the Berezin-T o eplitz [ 7 ] cases. △ Pro of . 16 Step 1. Discrete sp ectrum. This step is only necessary in the pseu- do differen tial case, where the phase space M is not compact, and it is no w standard. Let a smooth function f b e equal to 1 on I and compactly sup- p orted in I + [ − ϵ, ϵ ] . Then 1 I f = 1 I and hence b y functional calculus 1 I ( P ) = 1 I ( P ) f ( P ) . The sym b ol f ( H ( x, ξ )) and all its deriv ativ es v anish outside the compact set H − 1 ( I + [ − ϵ, ϵ ]) , so by pseudo differen tial functional calculus, f ( P ) is compact (see [ 22 , Theorem 8.7, Theorem 9.4], and also [ 31 , Section 13.6]). Th us 1 I ( P ) is compact as well, which implies that its range is finite dimensional. Step 2. Quasimo des. Let us fix k ∈ { 1 , . . . , d } . W e define the map A k to b e the semiclassical action of Prop osition 5.6 . Th us, the map E → A k, 0 ( E ) is the action integral on the energy set H = E ; a classical computation shows that its deriv ativ e is the map E 7→ τ k ( E ) where τ k ( E ) is the p eriod of the Hamiltonian flo w of H on C k ( E ) . Since E is a regular v alue, the Hamiltonian vector field cannot v anish on the lev el set, and hence τ k ( E )  = 0 . This sho ws that A k, 0 is a diffeomorphism, and hence that the map ℏ A k ( · ; ℏ ) is in v ertible for ℏ small enough. This also implies that the in verse G k ( ℏ ) := [ ℏ A k ( · ; ℏ )] − 1 admits an asymptotic expansion in p o w ers of ℏ . Hence, for con v enience, we may define this in v erse modulo O ( ℏ ∞ ) for all ℏ ∈ ]0 , 1] by a Borel summation, which w e call G k ( ℏ ) again. It now follows from ( 6.2 ) that E ( ℏ ) ∈ σ k ( ℏ ) ⇐ ⇒ E ( ℏ ) = G k ( ℏ )(2 π ℏ n ) for some n ∈ Z (6.3) (the integer n b eing of course sub jected to the fact that 2 π n must b elong to the range of A k ( · , ℏ ) .) F or such a family { E ( ℏ ); ℏ ∈ J } , where J ⊂ ]0 , 1] accum ulates at zero, w e ma y apply Prop osition 5.4 and obtain a microlo cal solution to ( P − E )Ψ ℏ = O ( ℏ ∞ ) in a neigh b orho od of C k . By a microlocal cutoff outside of C k and v anishing on the other C k ′ ’s, k ′  = k , we obtain a normalized quasimo de Ψ ( k ) ℏ :    ( P − E )Ψ ( k ) ℏ    H = O ( ℏ ∞ ) . By the spectral theorem, there m ust be an element λ ( ℏ ) ∈ σ ( P ) such that | λ ( ℏ ) − E ( ℏ ) | = O ( ℏ ∞ ) . This sho ws σ k ⊂ σ ( P ) + O ( ℏ ∞ ) . 17 Step 3. Microsupport. Let ψ = ψ ℏ ∈ H be an eigenfunction of P for the eigen v alue E = E ( ℏ ) ∈ I . Assume that E ( ℏ ) → E 0 , at least for some subsequence of v alues of ℏ . By semiclassical ellipticit y , the microsupp ort of ψ is contained in H − 1 ( E 0 ) , and is not empty if w e normalize ∥ ψ ∥ = 1 . This microsupp ort is inv ariant by the Hamiltonian flo w of H (this follows for instance from Proposition 5.1 ). Hence it is a finite union of curves C k ( E 0 ) . Let W k b e a saturated neighborho o d of C k ( E 0 ) : for E close to E 0 , the whole curv e C E is contained in W k . No w, the microlo cal restriction of ψ on W k is a section of the sheaf D for all E near E 0 , and hence b y Prop osition 5.4 the corresp onding Bohr- Sommerfeld conditions m ust b e satisfied: th us E ∈ σ k ( ℏ ) . This sho ws σ ( P ) ⊂ σ k + O ( ℏ ∞ ) . Step 4. Multiplicities. F or simplicity , w e did not state explicitly the result about multiplicities within the theorem, see Remark 6.2 . Here is a more precise statemen t, whic h w e could not find in the literature (although it is not very far from the metho d employ ed in [ 35 ]), except v ery recently in the Berezin-T o eplitz case [ 21 ]. While the idea is simple (the m ultiplicity of E should b e exactly the num b er of k for which E ∈ σ k ), the correct statement is more inv olv ed due to b oundary effects (eigen v alues ma y surreptitiously en ter or leav e an y given set when ℏ v aries arbitrarily little, and this may not b e detected b ecause of the O ( ℏ ∞ ) error). Let us pro v e the follo wing: F or each eigen v alue E = E ( ℏ ) ∈ I of P , let N ≥ 2 , let B = B ( ℏ ) b e the ball around E ( ℏ ) , of radius ℏ N . Let N ′ > N and let ˜ B b e a slightly smaller ball, or radius ℏ N − ℏ N ′ . Then, when ℏ is small enough, 1. the sp ectral multiplicit y of B is at least equal to the num ber of k ’s such that σ k in tersects ˜ B ; 2. the sp ectral multiplicit y of ˜ B is at most equal to the num b er of k ’s such that σ k in tersects B . Here, we used the terminology “sp ectral multiplicit y of B” to denote the rank of the sp ectral pro jector of P onto B . First, remark that in each σ k , the discussion is simple: if t wo energies E 1 ( ℏ ) and E 2 ( ℏ ) are separated b y some C h N , N ≥ 1 , then they m ust correspond to differen t integers n 1 and n 2 in ( 6.3 ): hence | n 1 ( ℏ ) − n 2 ( ℏ ) | ≥ 1 , and hence | E 1 ( ℏ ) − E 2 ( ℏ ) | ≥ c ℏ 18 for some c > 0 . Let B = B ( ℏ ) b e a ball around some energy E ( ℏ ) ∈ I , of radius ℏ N with N ≥ 2 , and let ˜ B b e a slightly smaller ball as in the statement, so that ˜ B + O ( ℏ ∞ ) ⊂ B . By the ab o ve argumen t, for ℏ small enough, each set σ k + O ( ℏ ∞ ) can contain at most one element of ˜ B . If this happ ens for t wo differen t in tegers k  = ℓ , then the corresp onding WKB quasimo des are microsupp orted on different curves and hence their scalar pro duct is O ( ℏ ∞ ) . Therefore they cannot b e collinear for ℏ small enough, whic h means, using the v ariational c haracterisation of the sp ectrum, that the sp ectral multiplicit y of B is at least equal to the num b er of k ’s such that σ k in tersects ˜ B . Con versely , as in the previous step, an y eigenfunction in the range of 1 ˜ B ( P ) must b e supp orted on some union of C k ’s and hence is microlo cally equal to a linear combination of the corresp onding WKB quasimo des. Since w e hav e an orthonormal basis of eigenfunctions, and since, again, the WKB quasimo des are m utually almost orthogonal, we deduce that the spectral m ultiplicity cannot exceed this num b er of k ’s. □ W e see from the pro of that one actually has a precise description of eigenfunctions and quasimodes. Theorem 6.3 (Bohr-Sommerfeld eigenfunctions) With the same hy- p othesis as The or em 6.1 , 1. A ny quasimo de of P (and henc e, any eigenfunction) is e qual, mo dulo O ( ℏ ∞ ) , to a line ar c ombination of the WKB solutions. 2. Each of these WKB solutions, with appr oximate eigenvalue E = E ( ℏ ) , must c orr esp ond to a c omp onent C k such that σ k interse cts an O ( ℏ ∞ ) b al l ar ound E . 3. F or e ach k , to any solution E ( ℏ ) of the Bohr-Sommerfeld c ondition ( 5.2 ) on the curve C k , one c an asso ciate (for ℏ smal l enough) a unique eigen- value of P , for which the c orr esp onding WKB A nsatz is a quasimo de. 7 Caustics? Comparing with most texts on WKB (quasi)mo des, lik e [ 32 ], it is p erhaps surprising that no discussion of turning p oin ts or introduction of the Airy function has been necessary here. A ctually , in our treatmen t, caustics ha ve b een sup erbly ignored. What has b ecome of them? 19 As w e saw in the preceding sections, WKB solutions live intrinsically on the curv es C k , whic h are smo oth lagrangian submanifolds of the phase space M . In the usual pseudo differen tial setting, a problem o ccurs due to the fact that we ultimately need eigenfunctions that dep end on the p osition v ariable only , x ; and in general the curves C k cannot b e globally parameterized by x . Definition 7.1 L et Λ ⊂ R 2 n b e a lagr angian manifold. A p oint ( x, ξ ) ∈ Λ is c al le d caustic if the map ( x, ξ ) → x r estricte d to Λ is not a lo c al diffe o- morphism. Th us, if n = 1 , any closed smooth curv e C in R 2 m ust hav e at least tw o caustics, corresp onding to the extrema of x on the curve. The Maslov index of C is the intersection num b er of the curv e with the vertical fibration: i.e. , the n umber of times the tangent to the curve crosses the v ertical direction, coun ted algebraically: +1 if the oriented curv e stays lo cally on the left of the v ertical tangen t. F or a circle in R 2 orien ted coun ter-clo c kwise, the Maslov index is 2. By homotopy in v ariance, the Maslov index of an y smooth lo op in R 2 is ± 2 . Remark 7.2 In the Berezin-T o eplitz category , the caustic issue does not exist, b ecause elemen ts of the Hilb ert space H are directly defined on the phase space M . As a matter of fact, w e could ha v e a voided caustics al- together b y transforming pseudo differential op erators into Berezin-T o eplitz op erators (in the non-compact phase space C n ) using the FBI transform. This is not our metho d here, but it can b e done efficiently , see [ 55 , 26 ]. △ Symplectically , due to the Darb oux-Carathéo dory theorem, an y regular p oin t of H − 1 ( E ) plays the same role, be it in R 2 n or in a compact Käh- ler manifold, b e it caustic or not. Since our metho d is to quan tize the Darb oux-Carathéodory c harts, the microlo cal description completely flies o ver the caustics. But, looking carefully , they are still there: they influ- ence the dimension of the θ v ariable in the generalized WKB Ansatz ( 2.1 ) used for the k ernel of the F ourier In tegral Op erator quantizing the Darb oux- Carathéo dory c hart. By stationary phase, this dimension sho ws up in the asymptotic expansion of the transition constant b etw een tw o WKB solutions, whic h is our Bohr-Sommerfeld co cycle ( 5.6 ), see the pro of of Prop osition 5.6 . Then, in the final result, Theorem 6.1 , the only remaining footprint of the caustic analysis is the Maslo v index, causing the famous 1 2 -shift in the quan- tum num b ers. 20 8 Applications The description of Theorem 6.1 is so precise that almost an y asymptotic statemen t ab out eigenv alues of P in I = [ E 1 , E 2 ] should directly follow from it. Here are some examples, certainly kno wn to exp erts, but which w e could not find explicitly in the literature. The hypothesis are the same as those of the theorem. Corollary 8.1 (Densit y of the sp ectrum) Given any E 0 ∈ I , ther e ex- ists a family of eigenvalues E ( ℏ ) ∈ σ ( P ) such that E ( ℏ ) → E 0 , and mor e pr e cisely E ( ℏ ) − E 0 = O ( ℏ ) . Pro of . Let us apply F ormula ( 6.2 ). Let c ( ℏ ) = ℏ A k ( E 0 ; ℏ ) . W e ha v e c ( ℏ ) = c 0 + O ( ℏ ) c 0 = A k, 0 ( E 0 ) , and G k ( ℏ )( c ( ℏ )) = E 0 + O ( ℏ ∞ ) . Let n ( ℏ ) = ⌊ c ( ℏ ) 2 π ℏ ⌋ ∈ Z , so that | 2 π n ( ℏ ) ℏ − c ( ℏ ) | ≤ 2 π ℏ . Since G k ( ℏ ) is locally Lipsc hitz, uniformly in ℏ , we obtain G k ( ℏ )(2 π n ( ℏ ) ℏ ) − G k ( ℏ )( c ( ℏ )) = O ( ℏ ) and by ( 6.2 ) w e obtain E ( ℏ ) ∈ σ ( P ) with G k ( ℏ )(2 π n ( ℏ ) ℏ ) = E ( ℏ ) + O ( ℏ ∞ ) , whic h giv es the result. □ In the ab o ve result, we notice that the quan tum num b er n ( ℏ ) which lab els the eigen v alues is allo w ed to depend on ℏ , and the distance | E ( ℏ ) − E 0 | cannot in general b e b etter than O ( ℏ ) . A differen t question is to try to in vestigate the behaviour, as ℏ v aries, of an eigen v alue with a given, fixed, n . This is what concerns the follo wing statemen t. Corollary 8.2 ( ℏ -b eha viour of individual eigen v alues) Al l eigenvalues of P in I «dep end smo othly on ℏ mo dulo O ( ℏ ∞ ) » in the fol lowing sense: for any ε > 0 and any N ≥ 2 , ther e exists ℏ 1 > 0 such that the fol lowing holds. L et ℏ 0 ∈ ]0 , ℏ 1 ] , and c onsider an eigenvalue E ( ℏ 0 ) ∈ σ ( P ) ∩ [ E 1 , E 2 ] . Ther e exist a family of eigenvalues { E ( ℏ ) , ℏ ∈ ]0 , 1] } , and a smo oth map ℏ 7→ λ ( ℏ ) such that, as long as λ ( ℏ ) ∈ I , | E ( ℏ ) − λ ( ℏ ) | ≤ ε ℏ N . In the pseudo differ ential c ase, this family always le aves the interval I for ℏ smal l enough. 21 Pro of . W e use the bijection giv en b y ( 6.1 ). This defines, for ℏ = ℏ 0 , an integer k ∈ { 1 , . . . , d } , labelling a connected comp onen t of the energy lev el set, and another in teger n ∈ Z (the quantum n umber) lab elling the appro ximate eigen v alue A k ( · ; ℏ 0 ) − 1 (2 π n ) . W e let λ ( ℏ ) := G k ( ℏ )(2 π ℏ n ) as in ( 6.3 ), and the result follo ws. In other w ords, w e just reform ulate ( 6.1 ) as σ ( P ) ∩ I = G k =1 ,...,d n ∈ Z , 2 π ℏ n ∈ Dom ( G k ( ℏ )) G k ( ℏ )(2 π ℏ n ) + O ( ℏ ∞ ) whic h expresses the spectrum as a union of ℏ -smo oth branc hes indexed by ( k , n ) . Note that G k ( ℏ ) = G k, 0 + O ( ℏ ) , and G k, 0 is the recipro cal of the action v ariable E 7→ A k, 0 ( E ) . In the pseudodifferential case, the action is the area b elo w C k and hence cannot v anish unless E is a critical v alue. This means that G k, 0 (0) cannot belong to I , whic h forces the eigen v alue to exit this in terv al. □ Note that the drifting of eigenv alues, forcing them to exit any interv al of regular v alues, w as describ ed for general quantum integrable systems in [ 17 ]. F rom this corollary , all eigenv alues belong to smooth branc hes. In gen- eral, the branc hes asso ciated to differen t connected comp onents ma y inter- sect. Whether the exact eigen v alues follow the branches as ℏ mov es con tinu- ously (in the pseudo differential setting), or instead “hop” to another branch, creating avoiding cr ossing seems to be an op en question for general pseu- do differen tial op erators. Of course, for 1D Sc hrö dinger op erators, b ecause of the simplicity of the sp ectrum, we know that all crossings m ust be a v oided. Corollary 8.3 (P o or man’s tunnel effect) Assume that two c onne cte d c omp onents C k , C ℓ ar e “quantum symmetric” (for instanc e they ar e exchange d by an affine symple ctic map and P is invariant under the c orr esp onding metaple ctic tr ansformation, or P is a Schr ö dinger op er ator with symmetric p otential wel ls). Then we have doublets of eigenvalues E k ( ℏ ) and E ℓ ( ℏ ) such that 1. E k ( ℏ ) = E ℓ ( ℏ ) + O ( ℏ ∞ ) ; 2. the sp e ctr al multiplicity of any b al l of r adius ℏ N , N ≥ 2 ar ound E k ( ℏ ) (or E ℓ ( ℏ ) ) is at le ast 2. It is exactly 2 if d = 2 . Pro of . The symmetry ensures that the Bohr-Sommerfeld cocycles are the same for both connected comp onen ts. The result then follows directly from Theorem 6.1 . □ 22 See [ 25 ] for a thorough discussion of tunnelling for 1D pseudo differen tial op erators in relation to the Bohr-Sommerfeld rules. Let us no w count eigenv alues. The usual W eyl la w (see for instance [ 22 , Theorem 10.1]) giv es the n um b er of eigen v alues (with m ultiplicities) in I : N ( P , I ; ℏ ) = 1 2 π ℏ V ol ( H − 1 ( I )) + O (1) where V ol is the symplectic volume. So V ol ( H − 1 ( I )) = R H − 1 ( I ) | ω | which by Stok es is (recall that d α = ω )      Z H ( x,ξ )= E 2 α − Z H ( x,ξ )= E 2 α      = d X k =1      I C k ( E 2 ) α − I C k ( E 1 ) α      = d X k =1 | A k, 0 ( E 2 ) − A k, 0 ( E 1 ) | . Using Theorem 6.1 w e can not only go one step further in the approxima- tion, but also, surprisingly (maybe), obtain an exact expression, without remainder. Corollary 8.4 (Exact W eyl la w) Assume P has no subprincip al symb ol. L et ˜ E j ( ℏ ) ∈ I , wher e ℏ ∈ ]0 , 1] or ℏ b elongs to some subset ac cumulating at 0, b e such that 1. ˜ E j ( ℏ ) → E j as h → 0 ; 2. dist ( ˜ E j , σ ( P )) ≥ ϵ ℏ for some ϵ > 0 . Then, for ℏ smal l enough, the numb er of eigenvalues of P in ˜ I := [ ˜ E 1 ( ℏ ) , ˜ E 2 ( ℏ )] is exactly N ( P , ˜ I ; ℏ ) = d X k =1 ⌊ 1 2 π A k, 0 ( ˜ E 2 ) + 1 2 ⌋ − ⌊ 1 2 π A k, 0 ( ˜ E 1 ) + 1 2 ⌋ . (8.1) Mor e over, for the fixe d interval I , this gives N ( P , I ; ℏ ) = 1 2 π ℏ V ol ( H − 1 ( I )) + 1 2 π  τ k ( E 2 ) − τ k ( E 1 )  + δ ( ℏ ) + O ( ℏ ) , wher e τ k ( E ) is the p erio d of the Hamiltonian flow of H on C k ( E ) , and | δ ( ℏ ) | < 1 . 23 Pro of . W e use the decomposition ( 6.1 ) and count the n um b er of elemen ts N k ( ℏ ) of eac h set σ k ( ℏ ) . By ( 6.2 ), N k ( ℏ ) = #( A k ( · ; ℏ ) − 1 (2 π Z )) ; in other w ords, it is the num b er of integers n ∈ Z suc h that A k ( E ; ℏ ) = 2 π n for some E ∈ I . Since A k, 0 is a lo cal diffeomorphism, A k is, for ℏ small enough, a monotonous bijection from I into its image, so N k ( ℏ ) is the num b er of in tegers in the in terv al [ β 1 , β 2 ] , where β j := β j ( ℏ ) = A k ( ˜ E j ( ℏ ); ℏ ) / 2 π , j = 1 , 2 . By assumption, β j ’s stay a wa y from integers by a distance at least ˜ ϵ , for some ˜ ϵ > 0 dep ending on ϵ . So they cannot b e integers, and N k ( ℏ ) = ⌊ β 2 ( ℏ ) ⌋ − ⌊ β 1 ( ℏ ) ⌋ . W e hav e β j ( ℏ ) = 1 2 π ℏ A k, 0 ( ˜ E j ) + 1 2 π A k, 1 ( ˜ E j ) + O ( ℏ ) and if P has no subprincipal sym b ol, w e hav e A k, 1 ( E ) = π . What’s more, when ℏ is small enough so that the remainder O ( ℏ ) is less than ˜ ϵ 2 , the sum 1 2 π ℏ A k, 0 ( E j ) + 1 2 π A k, 1 ( E j ) cannot be an in teger either. This gives N k ( ℏ ) = ⌊ 1 2 π ℏ A k, 0 ( ˜ E 2 ) + 1 2 ⌋ − ⌊ 1 2 π ℏ A k, 0 ( ˜ E 1 ) + 1 2 ⌋ . Summing ov er k , we obtain the correct n umber modulo O ( ℏ ∞ ) . But since w e are after an in teger n umber, w e can disregard the O ( ℏ ∞ ) remainder for ℏ small enough, whic h pro v es ( 8.1 ). Finally , we can write N k ( ℏ ) = 1 2 π ℏ A k, 0 ( ˜ E 2 ) − 1 2 π ℏ A k, 0 ( ˜ E 1 ) + δ where | δ | < 1 . Cho osing ˜ E j suc h that E j − ˜ E j = O ( ℏ ) , and T aylor expanding at order 2, w e obtain: N k ( ℏ ) = 1 2 π ℏ ( A k, 0 ( E 2 ) − A k, 0 ( E 1 )) + 1 2 π ( τ k ( E 2 ) − τ k ( E 1 )) + δ + O ( ℏ ) (8.2) whic h yields the result. □ Of course, in the statement of this corollary , we ma y replace the interv al I by any closed subinterv al. The assumption on the v anishing subprincipal sym b ol w as not crucial; w e obtain a general formula b y using the correct v alue for A k, 1 , which is the sum of the Maslov index and the integral ov er C k of the subprincipal symbol, see [ 63 ]. It also follows from Theorem 6.1 that the moving endp oin ts ˜ E j ( ℏ ) in the statemen t of this corollary alwa ys exist. A ctually , in the pseudo differen tial case, one can even choose ˜ E j ( ℏ ) = E j , provided we select a subsequence of ℏ ’s; this is due to the drifting phenomenon of Corollary 8.2 . 24 In vers e problems. Theorem 6.1 is also key in solving in v erse problems. Let me just men tion the Sc hrö dinger case, where w e wish to reco ver the p oten tial V from the sp ectrum [ 15 ], and the general pseudo differen tial or Berezin-T o eplitz case, where we recov er the principal symbol modulo sym- plectomorphisms [ 66 , 49 ]. In b oth case the Bohr-Sommerfeld rules are cru- cial. 9 Bey ond EBK What’s nice ab out the sheaf approac h that w e hav e presented here is the ease of generalization. Let us end this pap er by exploring some extensions to the WKB/EKB metho ds. Smaller errors. While an error term of size O ( ℏ ∞ ) is extremely accurate, it is still insufficient for some imp ortan t applications like quantum tunnel effects. A b etter accuracy , namely O ( e − ϵ/ ℏ ) , can b e reac hed under analyticity assumptions. One needs for this Sjöstrand’s microlo cal theory [ 61 ], which w as recently pro ven to extend to Berezin-T o eplitz operators [ 56 , 18 , 8 , 20 ]. The application to exp onentially precise Bohr-Sommerfeld rules was recently obtained b y Duraffour [ 26 ], based on ideas from [ 36 ], in the pseudo differen tial case, and Deleporte-Le Floch [ 21 ] in the Berezin-T o eplitz case. In tegrable systems. As was already noted b y Brillouin [ 5 ], the equations for the semiclassical action e i ℏ S ℏ ( x ) can b e solv ed b y quadrature in the 1D case, or if the v ariables are separated. In fact, one can go even further. A quan tum completely integrable systems, with a phase space of dimen- sion 2 n , is a set of n pairwise commuting self-adjoin t semiclassical op erators ( P 1 , . . . , P n ) . The corresp onding classical symbols ( H 1 , . . . , H n ) form a Liou- ville in tegrable system. Since the Darb oux-Carathéo dory theorem extends to such integrable systems, it can b e sho wn that the whole microlo cal strat- egy holds, see [ 63 , 64 ]. This establishes b oth WKB solutions whic h are joint quasimo des of the system, and Bohr-Sommerfeld rules for the join t spec- trum. T o m y kno wledge, the first mathematical treatmen t of these join t Bohr-Sommerfeld rules can b e found in [ 12 , 6 , 1 ]. A review of these results is presented in [ 65 ]. Singularities. While the Darb oux-Carathéo dory theorem applies only to r e gular p oints, the microlo cal sheaf strategy can, surprisingly , b e extended to energy level sets con taining singular p oints . F or instance, a local extrem um 25 of the p otential V . The easiest case concerns elliptic singularities, for which the Hamiltonian is a microlocal p erturbation of the Harmonic oscillator; see [ 33 , 10 , 48 , 19 ]. The most sp ectacular application concerns hyperb olic p oin ts (for instance, a lo cal maximum of V ). It has b een work ed out by Colin de V erdière-Parisse [ 16 ] for the pseudo differential case, and Le Flo c h [ 47 ] in the Berezin-T o eplitz case. The main idea is to extend Prop osition 5.1 to a neigh b orho od of the critical p oin t. One can show that, this time, the space of microlo cal solutions has dimension not one but two . Of course, the level sets are not circles an ymore, but immersed curv es with transv ersal intersections. The n umber of branches (4) at each crossing fits nicely with the dimension 2 of the microlo cal solutions, and pro duces new singular Bohr-Sommerfeld conditions b y expressing the v anishing of some determinan t related to the top ology of the critical fib er [ 16 ]. The analogous question for degenerate singularities, for instance a quartic oscillator ξ 2 + x 4 + O ( x 5 ) , is still quite op en; see [ 13 , 52 ]. Non-selfadjoin t operators. Giv en the hea vy use of real symplectic ge- ometry that the microlo cal metho d requires, it seems hazardous to extend it to non-selfadjoint op erators, i.e. semiclassical op erators whose symbols are not necessarily real-v alued. Nevertheless, it turned out that this can b e achiev ed, and symplectic normal forms in the complexified phase space remain v ery effective. F or small non-selfadjoint p erturbations of self-adjoint pseudo differen tial op erators, Bohr-Sommerfeld rules with O ( ℏ ∞ ) remainder ha ve b een obtained b y Rouby [ 55 ], follo wing the pioneer works of Melin- Sjöstrand [ 54 ] and Hitrik [ 37 ]. The extension to more general 1D op erators is currently an activ e area of research, see for instance [ 38 , 21 , 27 , 58 ]. A ckno wledgemen ts. I w ould like to warmly thank Antide Duraffour and the anonymous referee for their positive feedback and their v aluable remarks and corrections. References [1] C. Anné and A.-M. Charb onnel. 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