Counting nodal domains on surfaces of revolution

Coun ting no dal domains on su rfaces of rev olution P anos D. Karageorge 2 , 3 and Uzy Smilansky 1 , 2 1 Department of Physics of Complex Systems, The W eizmann Institute of Science, Rehov ot 76100, Israel. 2 School o f Mathematics, Universit y of B ristol, B r istol BS8 1TW, UK . 3 Department of Physics, Universit y of Crete, Hera klion 710 0 3, Greece. Abstract. W e consider eigenfunctions o f the Laplace-B eltrami oper ator on sp ecial surfaces of r evolution. F or this s eparable system, the no dal domains of the (real) eigenfunctions form a chec k er-b oar d pattern, and their num ber ν n is prop ortional to the pro duct of the angular and the “surface” quantum num bers. Arranging the wa v e functions b y incre a sing v alues of the Laplace -Beltrami sp ectrum, w e obtain the no dal sequence, whose statistical pro p er ties we study . In pa rticular we inv estigate the distribution of the normalized counts ν n n for sequences of eigenfunctions with K ≤ n ≤ K + ∆ K wher e K, ∆ K ∈ N . W e show that the distributio n a pproaches a limit as K , ∆ K → ∞ (the class ical limit), and study the leading corr ections in the semi-classica l limit. With this informa tion, we der ive the central result of this work: the no dal sequence of a mirro r -symmetric s urface is sufficient to uniquely determine its s ha p e (mo dulo scaling ). 1. In tro duction No dal domains of a real, con tin uous function are the maximally connected domains where the function do es not c hange its sign. The no dal domains of eigenfunctions of the Laplacian on compact domains hav e b een studied since Chladni first observ ed the no dal structure of vibration mo des o f thin plates (eigenfunctions o f the bi-ha r monic op erator) , in the early y ears of the 19 th cen tury . In the presen t man uscript w e are not in terested Counting no dal dom ains on surfac es of r evolution 2 in the geometric prop erties of no dal domains of La placian eigenfunctions, but rather in their coun t. F o llowing Couran t, w e order the eigenfunctions so that the corresp onding eigen v alues fo rm a non-decreasing sequence. Denoting b y ν n the n um b er of no da l domains of the n -th eigenfunc tion, w e form the normalized no dal seq uence ξ n := ν n n , n ∈ N . Courant’s theorem [16] guar a n tees that ξ n ≤ 1, and w e w ould lik e to study the distribution o f the v alues of ξ n in the unit in terv al (0 , 1]. In previous pap ers [1, 2] the distribution o f the ξ n for v arious planar domains and 2- ma nif o lds w ere studied, and it w as concluded that the features o f the distribution dep end crucially on the type of classical dynamics it supp orts. If the classical dynamics o n the manif o ld (geo desics) a re in tegrable (and quan t um mech anically separable - for suc h systems, actually , quantum separablilt y is equiv alen t t o classical separability [29]), the limit distribution exists, and displa ys certain features whic h are common to a ll suc h systems. O n the other hand, if the classical dynamics is ch aotic, the distribution of the normalized no dal sequence is w ell repro duced by using a random wa ve mo del for the eigenfunctions [5]. Bogomolny and Schmit computed the mean and the distribution by using ideas fro m p ercolation theory [4]. The first work w it h implications on the geometric con ten t of the no dal sequ ence w as of Smilansky and Sank aranaray anan [2], where it w as sho wn that the asp ect ratio o f a rectangular domain on the plane (with Diric hlet b oundar y conditions) can b e determined b y coun ting its no da l domains. In [6] the no dal sequence for eigenfunctions of the Laplace-Beltrami o p erator fo r “simple” surfaces of rev olution w as discuss ed. A t r ace form ula for the no dal coun t w as deriv ed, a nd w a s shown to dep end explicitly b o th on some mean g eometric prop erties o f the surface, a s we ll as the lengths of its geo desics. In spite of the formal similarit y b etw een the sp e ctr al a nd the no dal trace form ulae, the geometrical information is included in differen t w a ys. F urther studies [3 , 7, 32] ha ve sho wn that isosp ectral domains hav e differen t no dal sequence s, th us supp orting the conjecture t ha t the geometrical info rmation is stored in the no dal and sp ectral Counting no da l doma ins on surfac es of r evolution 3 sequence s in differen t w a ys. In the presen t w o rk w e go one step further and inquire whether one c an de duc e the s hap e (up to scaling) of a domain given the distribution of the normalize d numb e r of n o da l domains, or, paraphrasing the classical sp ectral inv ersion question p osed b y M. Ka c [19], c an o ne c ount the shap e of a drum? It m ust b e stressed that the only use of the sp ectral information is lexicographical - it is o rdered as a non-decreasing sequence. Otherwise, there is no reference t o the a ctual v a lues of the eigen v alues. In the presen t man uscript w e shall confine ourselv es to the inte grable (and separable) case, particularly t o a special class o f surfaces o f rev olution. W e pres en t new results which p ertain to the distribution P ( ξ , I K ) of the normalized no dal coun ts of eigenfunctions with indices n in the interv al I K = [ K , K + ∆ K ]. In [1] it was sho w ed that there exists a limit distribution P ( ξ ) when the s ize of the index in terv al I K b ecomes infinite ( cor r espo nding to the semi-classical limit, K → ∞ ). W e provide the leading order term of the difference b etw een P ( ξ , I K ) and t he limit distribution P ( ξ ): P ( ξ , I K ) = P ( ξ ) + 1 √ K P 1 ( ξ ) + O  1 K  . (1) W e show that the know ledge o f the function P ( ξ ) and P 1 ( ξ ) suffices for no dal doma in in ve rsion, provid ed that the surface is mirror-symmetric. In other w ords, giv en the normalized no dal sequence, w e can deduce uniquely the pro file function of t he surface of rev olution (prov ided it is smo oth and symmetric). Numerical sim ulations w ere carried out fo r ellipsoids of rev olution, whic h illustrate our theoretical findings. In what follo ws w e shall use the classical notation of asymptotic analysis; the standard ‘ O , o ’ o rder notat ion, the sym b ol ‘ ∼ ’ standing for an asymptotic relatio n, the sym b ol ‘ ≍ ’ denoting same order of magnitude, a nd the sym b o l ‘ ≫ ’ denoting greater order of magnitude. 2. Surfaces of rev olution W e consider surfaces of rev olution M in R 3 , whic h are generated b y the complete rotation of the line y = f ( x ) , x ∈ I := [ − 1 , 1], a b out the x axis. W e confine our Counting no da l doma ins on surfac es of r evolution 4 atten tio n to a sp ecial subset of functions whic h satisfy the following requiremen ts: i. f 2 is analytic in I , and v anishes at ± 1, where f ( x ) ∼ a ± (1 ∓ x ) 1 / 2 , with a ± > 0 . This requiremen t guarantees that M is compact, has no b oundary and is smo o th ev en at the p oin ts where it is in tersected by the axis of rotation. ii. The second deriv ativ e of f is strictly negativ e, so that f ( x ) has a single maxim um at some x = x max , where it reac hes the v alue f max . This requiremen t guar a n tees con ve xit y of M . Surfaces which satisfy the requiremen ts a b ov e will b e referred to as simple surfac es of r evolution , and are con ve x, mild defo r ma t ions of ellipsoids of revolution. The induced Riemannian metric on M is d s 2 = ( 1 + f ′ ( x ) 2 )d x 2 + f ( x ) 2 d θ 2 , (2) where the prime denotes differen t ia tion with resp ect to x , and θ ∈ [0 , 2 π ) is the azim uthal angle. In the pro ceeding subsections we shall review the prop erties o f geo desics (classical mec hanics) and the sp ectrum of the Laplace-Beltrami op erator (quan tum mec hanics) on M . 2.1. Th e ge o desics The geo desics on M are the classical t r a jectories of free motion. They can b e deriv ed from t he Euler-La grange v ariation principle with the Lagrang ia n L = 1 2  (1 + f ′ ( x ) 2 ) ˙ x 2 + f ( x ) 2 ˙ θ 2  , (3) where a dot ab o v e denotes time deriv ativ e. The angular momen tum along the axis of rotation f ( x ) 2 ˙ θ is conserv ed, and we shall denote its v alue by m . The momentum conjugate to x is p x = ( 1 + f ′ ( x ) 2 ) ˙ x , and the conserv ed energy is E = (1 + f ′ ( x ) 2 ) ˙ x 2 + m 2 f ( x ) 2 . (4) It is con ve nien t to introduce the a ction v a r ia ble n , n ( E , m ) := 1 2 π I p x d x = 1 π Z x + x − p E f ( x ) 2 − m 2 p 1 + f ′ ( x ) 2 f ( x ) d x . (5) Counting no da l doma ins on surfac es of r evolution 5 Here, x ± are the classical turning p oints , where E f ( x ) 2 − m 2 = 0 , with x − ≤ x max ≤ x + , whic h corresp ond to tw o meridians γ ± (pro jections of caustics onto M ) b etw een which all geo desics with m 6 = 0 wind around M . Real classical tra j ectories exist o nly if E > ( m/f max ) 2 . The conv exity of M guarantees that the action v ariables ( n, m ) alo ng with their conjugate angle v ariables constitute a global co ordinate system on phase space [25]. The classical Hamiltonian H ( n, m ) in the action- angle represen tation is obtained b y in ve rting ( 5 ) to express the energy in terms o f n and m . H ( n, m ) is a homogeneous function o f order 2, i.e. H ( λn, λm ) = λ 2 H ( n, m ) , λ > 0 [13]. It suffices, therefore, t o study the function n ( m ) := n (1 , m ), which defines a smo oth line Γ in the ( n, m )-plane (the pro jection o f the unit energy shell o n the action plane). The function n ( m ) is one of the main building blo ck s of the semi-classical theory whic h will b e used througho ut this w ork. W e shall list some of its prop erties which will b e used in the sequel: 1. The reflection symmetry , n ( − m ) = n ( m ), follows from the definition (5). Th us, w e restrict our atten tion to m ≥ 0 when referring to Γ. 2. n ( | m | ) is defined in the in terv al I µ = ( 0 , m max ], where m max = f max . In this interv al, n ( m ) is analytic and decreases monotonically since d n ( m ) d m = − m π Z x + x − 1 p f ( x ) 2 − m 2 p 1 + f ′ ( x ) 2 f ( x ) d x ≤ 0 . (6) 3. n ( m ) assumes its maxim um v alue at m = 0 , n (0) = 1 π Z 1 − 1 p 1 + f ′ ( x ) 2 d x = L π , (7) where L is t he length of the rotating line. W e sho w in App endix A that n ( m ) is not analytic at m = 0, and in that vicinit y n ( m ) ∼ L π − | m | . (8) A t the o ther endp oin t , n ( m ) v anishes, n ( m ) ∼ r 2 ω ( m max − m ) , ω = | 2 m max f ′′ ( x max ) | . (9) 4. The phase space v o lume is 1 (2 π ) 2 Z R 2 + Θ  E − H ( n, m )  d v = 2 E Z m max 0 n ( m )d m = 2 A E . (10) Counting no da l doma ins on surfac es of r evolution 6 A is the area enclosed b etw een t he line Γ and the n and m axes. It is related to the area o f M b y || M|| = 8 π A . 5. The computat ion of the higher deriv ativ es of n ( m ) cannot pro ceed simply b y taking the deriv atives of ( 6) - the resulting integrals div erge. T o o v ercome this difficult y the in tegral defining n ( m ) needs regularization [24]. This is done in App endix A. 6. Some authors (e.g., [13]) prefer to use the Clairaut in tegral I instead of the angular momen tum. They are r elat ed by I = m √ 2 E . (11) The twis t c o ndition is introduced in [13] to distinguish the class of simple surfaces o f rev olution, for whic h the dynamics are particularly simple. In the presen t notation, the t wist condition is expressed b y the requiremen t ∂ 2 n ( E , m ) ∂ m 2 6 = 0 , 0 < | m | ≤ m max (12) i.e. n ( m ) is either con vex or conca ve on (0 , m max ). Throughout this paper w e shall use t he ellipsoid of rev o lut io n to illustrate graphically our findings. The ellipsoids are generated b y f ( x ) 2 = ε 2 (1 − x 2 ), with ε > 0 b eing the eccen tricity of the generating semi-ellipse. The action v aria ble n ( m ) reduces to n ( m ) = 2 x 2 ± π Z 1 0 √ 1 − t 2 1 − x 2 ± t 2 q 1 − (1 − ε 2 ) x 2 ± t 2 d t , (13) where x ± = ± p 1 − m 2 /ε 2 are the classical turning p oints. This in tegral can b e ev aluated in terms of elliptic functions giving, n ( m ) = 2 π 1 ε √ b h b E(1 − ε 2 b ) − (1 − ε 2 ) m 2 K(1 − ε 2 b ) − ε 4 Π(1 − ε 2 m 2 , 1 − ε 2 b ) i , (14) where b = ε 4 + (1 − ε 2 ) m 2 , and K( k ) := Z π / 2 0 (1 − k sin 2 θ ) − 1 / 2 d θ , E( k ) := Z π / 2 0 (1 − k sin 2 θ ) 1 / 2 d θ Π( k , l ) := Z π / 2 0 (1 − k sin 2 θ ) − 1 (1 − l sin 2 θ ) − 1 / 2 d θ Counting no da l doma ins on surfac es of r evolution 7 are the complete elliptic in t egrals of first, second and third kind r esp ective ly . Figure 1. sho ws the functions Γ := n ( m ) for a few ellipsoids of rev olutio n 0 1 2 0 1 m n H m L Figure 1. The curv e s Γ for ellipsoids of revolution with ec c ent ricities ε = 0 . 5 (prolate), ε = 1 (a sphere ) and ε = 2 (oblate) are shown as da shed line, solid line and sparsely dashed line res p. 2.2. Th e L aplac e-Be ltr ami op er ator The Laplace-Beltrami op erator o n M reads ∆ = − 1 f ( x ) σ ( x ) ∂ ∂ x f ( x ) σ ( x ) ∂ ∂ x − 1 f ( x ) 2 ∂ 2 ∂ θ 2 , (15) where σ ( x ) := p 1 + f ′ ( x ) 2 . The domain of ∆ are Ψ ∈ W 2 2 ( I × S 1 ), require d to b e 2 π - p erio dic in θ . Under t hese conditions, the op erator is self-adjoint, and its sp ectrum is discrete and non-negat ive. ∆ is separable, and the eigenfunctions can b e written in the pro duct form Ψ( x, θ ) = exp( imθ ) ψ m ( x ), where m ∈ Z . It is c on venie n t to in tr o duce a new v ariable t , through d t = σ ( x ) f ( x ) d x , (16) Counting no da l doma ins on surfac es of r evolution 8 whic h maps the in terv al [ − 1 , 1] to R ∪ {∞} . F or an y m and eigen v alue E n,m , the sp ectral equation for (15) reduces to the Sturm-Liouville ODE  − d 2 d t 2 +  m 2 − E n,m f ( x ( t )) 2   ψ n,m ( t ) = 0 . (17) The sp ectrum o f the Laplace-Beltrami op erator is doubly degenerate for a ll m 6 = 0 , with E n, − m = E n,m . The semi-classical sp ectrum is constructed b y using the Einstein- Brillouin-Keller approx imation [1 3], E scl n,m = H ( n + 1 2 , m ) , n ∈ N 0 , m ∈ Z , (18) where H ( n, m ) is the class ical Hamiltonian defin ed in terms of the action v ariables. The semi-classical appro ximatio n for t he sp ectral sequence with m = 0 assumes a v ery simple form. Since n + 1 2 = √ E π Z 1 − 1 p 1 + f ′ ( x ) 2 d x = √ E π L , (19) the semi-classical quan tization condition reads: E scl n, 0 = h π ( n + 1 2 ) L i 2 , n ∈ N 0 . (20) Because of the degeneracy of t he sp ectrum, w e hav e to choose a particular represen tation of the w av e functions. W e do this by asso ciating cos( mθ ) with m ≥ 0 and sin( mθ ) for m < 0 , i.e. Ψ n,m ( x, θ ) = ψ n,m ( x )      sin( mθ ) if m < 0 cos( mθ ) if m ≥ 0 . (21) The no dal pattern o f Ψ is that of a ch ec k erb oard, t ypical for separable systems (as a matter of fact, the no da l pattern remains a c hec kerboard for an y linear com bination of the basis functions. It is only rotated aro und the symmetry axis of the surface). F o r m = 0, the n um b er of no dal domains is ν n, 0 = n + 1, a nd for all other m , ν n,m = 2( n + 1) | m | . In summary ν n,m = ( n + 1)(2 | m | + δ m, 0 ) . (22) T o end this section w e illustrate its con ten t by an application to t he simplest surface - t he sphere, considered here as a surface of rev o lutio n with f ( x ) 2 = 1 − x 2 . Counting no da l doma ins on surfac es of r evolution 9 The action v aria ble ( 5 ) can b e computed explicitly n ( E , m ) = 2 √ E π Z √ 1 − m 2 /E 0 p 1 − m 2 /E − x 2 1 − x 2 d x = √ E − | m | . (23) Th us, H ( n, m ) = ( n + | m | ) 2 , and the EBK quantization for the sp ectrum is E n,m = ( l + 1 2 ) 2 ∼ l ( l + 1) where l := n + | m | . The (2 l + 1)-fold degeneracy follows immediately b y counting t he num b er o f integer pairs ( n, m ) which satisfy l = n + | m | . T urning to the quan tum description, the v ariable t defined in (16) can b e explicitly computed, x = ta nh t . W riting E = l ( l + 1) , l ∈ N 0 , transfor ms (17) to  − d 2 d t 2 +  m 2 − l ( l + 1) cosh 2 t  ψ l,m ( t ) = 0 , (24) whic h is equiv alen t to the Legendre equation. Finally , the n um b er o f no dal domains of spherical harmonics is kno wn, and coincides with (22) whe n the identific ation l = n + | m | is ma de. 3. Coun ting no dal domains W e shall start this section b y reviewing some of the general definitions and r esults obtained in [1], where the limit distribution of the normalized noda l coun ts w as first studied. W e shall then deriv e t he next to leading term, and sho w that it pro vides further information on the geometry of M . The no dal structure o f the wa v e functions was review ed in the preceding section, and an explicit expression for the dep endence of the n umber of no dal do mains ν n,m on the quan tum n umbers ( n, m ) is giv en in (22). The ob ject whic h is in v estigated in this w ork is t he no dal se quenc e which is defined as follows : A rrange the sp ectrum as a no n- decreasing sequence. This amoun ts to assigning to eac h pair o f quan tum n umbers ( n, m ) a coun ting index N ( n, m ), whic h giv es the n umber of eigenv alues of the Laplace-Beltrami op erator (counted with multiplicit y) whic h are strictly smalle r than the eigenv alue E n,m , i.e. N ( n, m ) := # { E ∈ Sp ec(∆) : E < E n,m } . Ob viously N ( n, m ) = N ( E n,m ), where N ( E ) is the sp ectral counting function. Counting no da l doma ins on surfac es of r evolution 10 T o a ccoun t for the sp ectral degeneracy , w e mo dify the definition a b o ve f or | m | 6 = 0, so t ha t N ( n, | m | ) = N ( n, −| m | ) + 1. The no dal seq uence is t he sequence of no dal coun ts ordered b y N : { ν N } ∞ N =1 . By this conv en tion, the systematic degeneracy of the sp ectrum is tak en care of. In g eneral, ho w ev er, acciden tal degeneracies cannot b e excluded. The ordering am biguity may app ear, e.g., when the degeneracy class in volv es states with differen t (no n-negativ e) m v alues, suc h as e.g., for the sphere. In this case a p ossible w ay to remo ve this problem is to consider the sphere a s a limiting case of an ellipsoid with (a p o sitiv e) eccen tricit y approaching zero. The m degeneracy is remov ed for any ellipsoid, a nd the order o f the eigenv alues is monotonic in | m | for a rbitrary small eccen tricit y . Similar constructions can b e used fo r other acciden tal degeneracies. Couran t’s theorem [16] ensures that, f or an y ordering o f the eigenfunctions in the degeneracy classes ν N ( n,m ) ≤ N ( n, m ) . (25) It is natural therefore, to define the norm alize d no dal se quenc e ξ N := ν N N , 0 < ξ N ≤ 1 . (26) W e study the distribution of t he v alues of the normalized no dal sequence for a finite index set, N ∈ { K , ..., K + ∆ K } , P ( ξ , I K ) := 1 ∆ K X ( n,m ) ∈ N 0 × Z χ I K ( N ( n, m )) δ ( ξ − ξ N ( n,m ) ) , (27) and χ I K the c haracteristic function of the in terv al I K , χ I K ( x ) =      1 if K ≤ x ≤ K + ∆ K 0 o therwise . (28) Since P ( ξ , I K ) is not a f unction but a distribution ov er the interv al (0 , 1), one m ust tak e care in its manipulations. Most of the limits and estimates ar e considered in the w eak sense (e.g. [15]). In some sections though, related distributions are view ed a s functions (or to b e more precise, the functions whose v alues these distributions tak e on some subin terv al of (0 , 1)). In other cases they will b e manipulated as functions af ter an appro priate regularization. Counting no da l doma ins on surfac es of r evolution 11 In [1], it was assumed for con ven ience that I K gro ws linearly in K , ∆ K = g K , g b eing a p ositiv e constan t (suc h that g K ∈ N ) . The existence of the limiting distribution P ( ξ ) of P ( ξ , I K ) in the K → ∞ limit (the classical limit) w a s prov ed, and it s univ ersal features were presen t ed. The existence can b e prov en b y sho wing 1 ∆ K X j ∈ I K ϕ ( ξ j ) − → Z 1 0 P ( ξ ) ϕ ( ξ )d ξ , (29) uniformly in g , for a n y smo oth and compactly supp orted t est function ϕ on (0 , 1). Here, w e shall rep eat the deriv ation in more detail, and also compute the difference b etw een P ( ξ , I K ) and P ( ξ ) to leading order in 1 √ K (again, for the linear case ∆ K = g K - the sup erlinear case ∆ K ≫ K , f ollo ws trivially). W e rewrite (27) as a sum of an isotropic comp onen t, to whic h only isotropic states (with m = 0) contribute, and an anisotropic comp onen t (with m 6 = 0), P ( ξ , I K ) = P m =0 ( ξ , I K ) + P m 6 =0 ( ξ , I K ) (30) P m =0 ( ξ , I K ) = 1 g K X n χ I K ( N ( n, 0)) δ  ξ − n + 1 N ( n, 0)  P m 6 =0 ( ξ , I K ) = 1 g K X n,m 6 =0 χ I K ( N ( n, m )) δ  ξ − 2( n + 1) | m | N ( n, m )  . Since we are in terested in t he semi-classical limit, w e are allow ed to mak e the follow ing appro ximate steps, whic h incur errors of order higher than O ( 1 √ K ). i. The sp ectrum is approx imated by E n,m ∼ E scl n,m = H ( n + 1 2 , m ), as m 2 + n 2 → ∞ , whic h in tro duces a relativ e error b ounded b y O ( 1 E n,m ) [12 ]. In particular, as w as show n in (20), E scl n, 0 = [ π ( n + 1 2 ) L ] 2 . ii. T o the same order, N ( n, m ) can b e replaced b y the first term in the W eyl series [13], N ( n, m ) = 2 A H ( n + 1 2 , m )  1 + O ( 1 E 3 4 n,m )  , (31) where A w as defined in (1 0). In tro ducing these appro ximations in (30), we find t hat P m =0 ( ξ , I K ) is O ( 1 K ) (in the weak sense) and therefore w e defer its computation to a lat er stage. The sums o ver ( n, m ) in (30) are computed using the P oisson summation form ula, decomposing Counting no da l doma ins on surfac es of r evolution 12 P m 6 =0 ( ξ , I K ) in a smo oth and an oscillatory part, P m 6 =0 ( ξ , I K ) = ¯ P ( ξ , I K ) + Q ( ξ , I K ) := ¯ P ( ξ , I K ) + X ( N ,M ) ∈ Z 2 \{ 0 } Q N ,M ( ξ , I K ) , (32) where the F ourier co efficien t s are Q N ,M ( ξ , I K ) ∼ 2 g K Z ∞ 1 2 d m Z ∞ − 1 2 e 2 π i ( M m + N n ) χ I K  2 A H ( n + 1 2 , m )  δ  ξ − 2( n + 1) m 2 A H ( n + 1 2 , m )  d n , (33) and ob viously ¯ P ( ξ , I K ) = Q 0 , 0 ( ξ , I K ). The leading term in the sum ab ov e is the smo oth t erm ¯ P ( ξ , I K ) whic h w e calculate first. The oscillatory terms are of low er o r der in 1 √ K and will b e computed in a separate subsection. Pro ceeding with the smo oth part, we shift the integration v ariable n 7→ n + 1 2 and write t he r esult as ¯ P ( ξ , I K ) ∼ 2 g K Z ∞ 1 2 d m Z ∞ 0 χ I K  2 A H ( n, m )  δ  ξ − ( n + 1 2 ) m A H ( n, m )  d n . (34) W e c hange the in tegr a tion v ariables ( n, m ) 7→ ( E , s ), where E = H ( n, m ), and s is defined through the relations d E = ω n d n + ω m d m ; ω n = ∂ H ( n, m ) ∂ n , ω m = ∂ H ( n, m ) ∂ m , d s = − ω m d n + ω n d m ω 2 n + ω 2 m . (35) Note that with this definition the Jacobia n is unit y , and d n d m = d E d s . Th us, (34) is reduced to ¯ P ( ξ , I K ) ∼ 2 g K Z K (1+ g ) 2 A K 2 A d E Z Γ Θ  m ( s ) − 1 2 √ E  δ  ξ − n ( s ) m ( s ) A − m ( s ) 2 √ E A  d s , (36) where Θ( x ) is the Hea vyside step function. The pair of functions { n ( s ) , m ( s ) } constitute a par a metric represen tation of the line Γ, along which H  n ( s ) , m ( s )  = 1 . This allow s the scaling b y √ E which app ears in (36). The expression ab o ve can b e further reduced b y the following observ ation. On the line Γ we hav e ω n d n + ω m d m = 0, whic h induces a symplectic structure (where n and m pla y the ro les of canonically conjugate v ariables and s is the “ time”) d m ( s ) d s = ω n ; d n ( s ) d s = − ω m . (37) Counting no da l doma ins on surfac es of r evolution 13 Another imp o rtan t identit y follows from the fact t ha t H ( n, m ) is homogeneous of order 2 in ( n, m ). W e can write n = n ( H, m ) = 1 √ H n (1 , m √ H ), from whic h w e deduce that on the line { H ( n, m ) = 1 } , d m ( s ) d s = ω n = 2 n ( m ) − mn ′ ( m ) , (38) where n ′ ( m ) := d n ( m ) d m . Th us, (3 6) t ak es the f o rm ¯ P ( ξ , I K ) ∼ 1 g K Z K (1+ g ) 2 A K 2 A d E Z m max 1 2 √ E    n ( m ) − mn ′ ( m )    δ  ξ − n ( m ) m A − m 2 √ E A  d m . (39) A similar transformation can also b e applied in the computation of the oscillatory in tegrals Q N ,M ( ξ , I K ). 3.1. Th e limit dis tribution The limit distribution is the leading term in the expansion of the in tegra l (3 6) in p ow ers O ( 1 √ K ). T aking the limit K → ∞ , the E -in tegral can b e directly p erformed, resulting with P ( ξ ) = 1 A Z Γ δ  ξ − m ( s ) n ( s ) A  d s . (40) This expression allows a simple geometrical in terpretat io n: the pro duct n ( s ) m ( s ) is the area of the rectangle whose v ertices a re the points ( m ( s ) , n ( s )) on Γ, its pro jections ( m ( s ) , 0) and (0 , n ( s )) on the t w o axes and the origin (See F igure 2). P ( ξ ) is the probabilit y distribution of the areas of these rectangles ( scaled b y A , and therefore smaller than 1). The areas of rectangles whic h are based on p oints whic h are either near the m or the n axes approach 0 . Since Γ is either conv ex or conca ve , there exists a unique p o in t where ξ reaches the maximal v alue of scaled areas, ξ max . Thu s, P ( ξ ) ≡ 0 for ξ ≥ ξ max . Changing in tegr a tion v ariables using (38), w e get P ( ξ ) = 1 2 A Z m max 0 δ  ξ − m n ( m ) A  | n ( m ) − mn ′ ( m ) | d m . (41) Counting no da l doma ins on surfac es of r evolution 14 m n H m L Figure 2. The geometric interpretation of P ( ξ ) (Computed for an ellipsoid of revolution with eccentricit y ε = 0 . 5). In tegrating (41) w e get, P ( ξ ) =            0 ξ ≥ ξ max , 1 2 P    n ( m ) − mn ′ ( m ) n ( m )+ mn ′ ( m )    ξ = mn ( m ) A ξ < ξ max . (42) The sum is o ve r the real v alues of m > 0 which satisfy ξ = mn ( m ) A . In the vicinit y of ξ max , ty pically t wo solutions coa lesce, leading to a square ro ot singularit y of P ( ξ ) at that p oin t. The v anishing o f P ( ξ ) in t he inte rv a l [ ξ max , 1], a nd the square ro ot singularit y at ξ max are the univers al f eat ur es whic h c ha r a cterize the no dal domain distributions f or separable systems (in 2-d) in general, and simple surfaces of revolution, in particular. Using (41), it is easy t o c heck normalization, R 1 0 P ( ξ )d ξ = 1. The form (42) for the ξ distribution can b e further simplified since for simple surfaces of revolution the twis t condition (12) is satisfied, and there are only tw o v alues of m whic h solv e ξ = m n ( m ) A . Denote t hem b y m − ( ξ ) and m + ( ξ ) ( m − ( ξ ) ≤ m + ( ξ )). They coa lesce a t ξ = ξ max . The v alue of m where the sole maxim um o f mn ( m ) o ccurs is denoted b y m 0 = m ± ( ξ max ). The t wo functions m + ( ξ ) and m − ( ξ ), together, provide a parametric represen tatio n of the curve Γ, since n ± ( ξ ) = A ξ m ± ( ξ ) where 0 < ξ < ξ max . This parametrization will b e used often in t he subsequen t discussion. Counting no da l doma ins on surfac es of r evolution 15 T o leading order, m ± ( ξ ) ∼ m 0 (1 ± p | ζ | ), where ζ = ξ − ξ max ξ max , a nd so from ( 4 2) w e deduce that, in the left neigh b orho o d of ξ max P ( ξ ) ∼ 1 p 1 − ξ /ξ max . (43) 0 0.5 1 0 1 2 3 4 5 6 Ξ P H Ξ L Figure 3. P ( ξ ) for ellipsoids of revolution with eccen tricities ε = 0 . 5 , 1 and 2 (spa rsely dashed line, solid line and das he d line resp.) One can re-ar range (4 2 ) to obta in another expression for the limiting distribution. F o r ξ < ξ max , P ( ξ ) = ξ d d ξ log m − ( ξ ) m + ( ξ ) . (44) This expression is quite rev ealing, b ecause it can be in v erted to prov ide the function m − ( ξ ) m + ( ξ ) based o n infor ma t io n derive d only from the no da l sequence. W e shall show b elow that the next to leading expres sion in the e xpansion of P ( ξ , I K ) provid es another relation b et wee n m − ( ξ ) and m + ( ξ ). Solving t he tw o equation w e obtain a complete parametric represen tation of t he curv e Γ (or n ( m )). W e shall also sho w that Γ defines uniquely the function f ( x ), when f ( x ) is symmetric ab out x = 0 . This will pro v e our claim that the no dal sequence for symmetric surfaces of revolution can b e inv erted and prov ide the “shap e of the drum”! The b ehavior of P ( ξ ) near ξ = 0 can b e easily extracted using (44). Since ξ → 0 implies either m → 0 or n ( m ) → 0 along Γ, w e may use the linear a pproximations Counting no da l doma ins on surfac es of r evolution 16 for n ( m ) as g iven b y (8) and ( 9 ), r espective ly . The equation m n ( m ) A = ξ reduces to a quadratic equation, and its solutions define the tw o branc hes m − ( ξ ) and m + ( ξ ) as m − ( ξ ) ∼ L 2 π 1 − r 1 − 2 ξ 2 A π 2 L 2 ! m + ( ξ ) ∼ m max 2   1 + s 1 − 2 ξ A √ 2 ω m 2 max   . (45) Substituting in (44) w e get P ( ξ ) ∼ 1 2   1 q 1 − 2 ξ η − + 1 q 1 − 2 ξ η +   , (46) where, η − = n (0) 2 2 A = L 2 2 π 2 A ; η + = | n ′ ( m max ) | m 2 max 2 A = m 2 max √ 2 ω A ; η 0 = m 2 max 2 A . (47) The relation (46) show s that lim ξ → 0+ P ( ξ ) = 1, independen tly of the surface under consideration - another univ ersal feature to b e added to the aforemen tioned ones. Moreo ve r, it sho ws that one can extract the dimensionless geometric parameters η − and η + from P ( ξ ) in the neigh b o rho o d of ξ = 0. They are directly related to the pro p erties of the line which generates M thro ug h its length, maxim um distance from the axis of rev olution and its curv ature at the maxim um. Note, how ev er, that the no dal sequence is comp osed o f in tegers, and in con trast to the sp ectral sequence it is in v ariant under isotropic scalings on M . Therefore, only dimensionless quan tities can b e extracted from it. Here, all the lengths are expressed in units of √ A . (The para meter η 0 in (47) is another dimensionless parameter whic h w e define here ev en thoug h it will a pp ear only later). The limit distributions for three ellipsoids of rev olution are sho wn in Figure 3. Computing ξ max as a function of the eccen tr icity r ev eals t ha t ξ max ( ε ) is a monotonically decreasing function whic h v ar ies b et w een ξ max (0) ≈ . 550 and ξ max ( ∞ ) ≈ . 464 as sho wn in Figure 4. Thu s, one can deduce the eccen tricit y fr o m the noda l seq uence just b y determining the supp ort of P ( ξ ). Counting no da l doma ins on surfac es of r evolution 17 0 1 2 3 4 5 0.45 0.5 0.55 ¶ Ξ max H ¶ L Figure 4. ξ max as a monotonically decreas ing function of the eccentricit y ε for ellipsoids of rev olutio n. The da shed line mar ks ε = 1 - the spher e - for whic h ξ max = . 5 as in (48). As another application of (44), and as an illustration, we deriv e P ( ξ ) for the s phere. F ro m (A.12) we get that for the sphere n ( m ) = 1 − m whic h immediately giv es P ( ξ ) = 1 √ 1 − 2 ξ ; 0 < ξ < 1 2 . (48) The same result can also b e obta ined directly . The sp ectrum consists o f the v alues E n,m = n ( n + 1) whic h are indep enden t of m ( | m | ≤ n ) and are (2 n + 1) - fold degenerate. The eigenfunctions are the spherical harmonics, and for the sak e of coun ting their no dal domains, w e consider them in their separable basis. The n umber of zeros of P m n (cos θ ) is n when m = 0, a nd n − | m | + 2 otherwise. The n umber of no dal domains are resp ectiv ely n + 1 a nd 2 | m | ( n − | m | + 1). The counting function for the 2 n + 1 degenerate states with a given n satisfies n 2 ≤ N ( n, m ) < ( n + 1) 2 . (49) The ordering within the set is a rbitrary , but in t he limit o f large n it is immaterial, since to leading o r der we can write N ( n, m ) = n ( n + 1)(1 + O ( 1 n )). Hence, the contribution of t he n - fold degenerate set to P ( ξ ) is P ( ξ ; n ) = 1 n n X m =1 δ  ξ − 2 m ( n − m + 1) n ( n + 1)  + O ( 1 n ) Counting no da l doma ins on surfac es of r evolution 18 → Z 1 0 δ  ξ − 2 µ (1 − µ )  d µ = 1 √ 1 − 2 ξ . (50) 3.2. Th e next to le ading terms Sev eral terms con tribute O ( 1 √ K ) corrections. In the f ollo wing w e shall address them in detail. The le ading c orr e ction to ¯ P ( ξ , I K ). Starting again from ( 39), w e see that the m - in tegrat io n results in a sum of t wo terms, coming from the tw o branc hes of solutions of ξ = m n ( m ) A + m 2 √ E A + O ( 1 E ) whic h constitute the t wo-po in t supp or t of the δ function in the integral. They differ b y a moun t δ m ± ( ξ ) of O ( 1 √ E ) from the v alues m ± ( ξ ) whic h w ere intro duced for the computation (44) of the limit distribution, δ m ± ( ξ ) = − 1 2 √ E m mn ′ ( m ) + n ( m ) + O ( 1 E ) , computed at m = m ± ( ξ ) . (51) As long as ( m + δ m ) ± remain inside the in tegra tion range, one can pro ceed with the computation tow ards the semi-classical asymptotic expansion ¯ P ( I K ) = P + 1 √ K P 1 + O ( 1 K ) . (52) Of course, P is the limit distribution (42), and P 1 ( ξ ) = − r A 2 √ 1 + g − 1 g  p − ( ξ ) + p + ( ξ )  , (53) where p ± ( ξ ) are g iv en by p ± ( ξ ) = 1 n ( m ) 1 + m n ′ ( m ) n ( m )      1 − m n ′ ( m ) n ( m ) 1 + m n ′ ( m ) n ( m )         − 1 + 2 m  m n ′ ( m ) n ( m )  ′ 1 − ( m n ′ ( m ) n ( m ) ) 2    , computed at m = m ± ( ξ ) . (54) These e xpressions can b e further simplified. T o do so, we differen tia te m ± ( ξ ) n ( m ± ( ξ )) = A ξ , with resp ect to ξ and get n = A ξ m ; n ′ = n  1 ξ d m d ξ − 1 m  ; n ′′ = n  2 m 2 − 2 ξ m d m d ξ − d 2 m d ξ 2 ξ ( d m d ξ ) 3  , (55) computed at m = m ± ( ξ ). By substituting the abov e in the defining expressions fo r p ± ( ξ ), w e obtain after some straigh tforw a r d calculations p ± ( ξ ) = ± 1 A  d m ± d ξ + 2 ξ d 2 m ± d ξ 2  = ± 2 A p ξ d d ξ  p ξ d m ± d ξ  . (56) Counting no da l doma ins on surfac es of r evolution 19 Hence, P 1 ( ξ ) = − r 2 A √ 1 + g − 1 g p ξ d d ξ  p ξ d d ξ [ m + ( ξ ) − m − ( ξ )]  . (57) This is an explicit expression whic h provide s the leading correction in t erms of the difference m + ( ξ ) − m − ( ξ ) b etw een the tw o branc hes of the parametric represen tation of Γ. T o gether with (44) it fo rms the basis for the in v ersion pro cedure which will b e discusse d in detail in the next section. The conditions for the v alidit y of the ab ov e approximation are no t satisfied if ξ is in the O ( 1 √ K ) neigh b orho o d of either 0 or ξ max . Near ξ max (51) div erges, while near ξ = 0, ( m + δ m ) ± ma y lie outside of the integration range [ 1 2 √ E , m max ]. T o get the correct expressions f or ¯ P ( ξ , I K ) in the vicinity of the extreme v alues of ξ , w e use sev eral v a r ia tions of the same tric k: within the problematic domains of in tegrat io n, w e appro ximate n ( m ) as a linear function of m . The argumen t of the δ functions b ecome quadratic functions of m . The supp ort o f the δ functions can b e ev aluated explicitly , and the m -inte grations can b e p erformed exactly . The remaining E -in tegrat io ns turn out to b e straightforw ard, so that explicit expressions for ¯ P ( ξ , I K ) in the vicinit y of ξ = 0 and ξ = ξ max are obtained. Behavior of ¯ P ( ξ , I K ) in the neig hb o rho o d of ξ = ξ max . W e recall that ξ max is defined as the maxim um v alue of m n ( m ) A , whic h o ccurs a t m 0 , where n ( m 0 ) + m 0 n ′ ( m 0 ) = 0. Th us, in the neigh b orho o d of m 0 w e can appro ximate n ( m ) ∼ n 0 − n 0 m 0 ( m − m 0 ) where n 0 = n ( m 0 ) = A ξ max m 0 (the non-existenc e of other critical po ints is guaran teed b y the t wist co dition). With this appro ximation, the argumen t of the δ function in (39) is quadratic in m . The integrations ov er m a nd E hav e to b e carried out with atten tion to the requiremen t that the supp o rt of the δ remains within the integration rang e. After some lengthy but straigh tfo rw ard manipulat io ns one gets, ¯ P ( ξ , I K ) ∼ 2 g K η max Z q 1 K η max q 1 (1+ g ) K η max Θ  y − ζ 2  y − 3 d y p y 2 + 2 y − ζ (58) and η max = (4 n 0 ) 2 2 A ; ζ = ξ − ξ max ξ max . (59) Counting no da l doma ins on surfac es of r evolution 20 η max is a nother dimensionless parameter whic h c haracterizes ¯ P ( ξ , I K ) near the b orders of it s supp o rt. Pe rforming the integral w e get, ¯ P ( ξ , I K ) ∼ 1 g K η max  F ( y ↑ , ζ ) − F ( y ↓ , ζ )  (60) F ( y , ζ ) =  1 ζ y 2 + 3 ζ 2 y  p y 2 + 2 y − ζ +  1 ζ + 3 ζ 2       − 1 √ | ζ | arctanh | ζ | + y √ | ζ | √ y 2 +2 y − ζ ζ ≤ 0 1 √ ζ arctan − ζ + y √ ζ √ y 2 +2 y − ζ ζ > 0 y ↑ = r 1 K η max ; y ↓ = s 1 (1 + g ) K η max for ζ ≤ 0 , y ↑ = r 1 K η max ; y ↓ = max ( ζ 2 , s 1 (1 + g ) K η max ) for ζ > 0 . This expression includes the corr ections to the limit distribution at its most noticeable feature, namely its singularit y at ξ max . F or finite K , the square-ro ot singularity is replaced b y a con tinuous function whic h reac hes b ey ond ξ max , shifts the maxim um from ξ max to ξ max ( K , g ) = ξ max + 2 q 1 (1+ g ) K η max and extends the supp ort of P ( ξ , I K ) up to ξ max + 2 q 1 K η max . As K → ∞ , the expression con v erges to the limit. The a pplication of the ab o ve theory fo r an ellipsoid of rev o lut io n for tw o v alues of K are shown in Figures 5. and 6. 0.527 0.5275 0.528 0.5285 0.529 0.5295 0.53 Ξ 0 10 20 30 40 50 60 P H Ξ ,I K L Figure 5. Lo cal b ehavior o f P ( ξ , I K ) at ξ = ξ max for an ellipsoid o f revolution with eccentricit y ε = 0 . 5 ( K = 24000 and g = 1). Counting no da l doma ins on surfac es of r evolution 21 0.527 0.5275 0.528 0.5285 0.529 0.5295 0.53 Ξ 0 20 40 60 80 P H Ξ ,I K L Figure 6. Lo cal b ehavior o f P ( ξ , I K ) at ξ = ξ max for an ellipsoid o f revolution with eccentricit y ε = 0 . 5 ( K = 48000 and g = 0 . 5). Behavior of ¯ P ( ξ , I K ) in the neighb orho o d of ξ = 0 . In t he neighborho o d of ξ = 0 w e ha ve con tributions from t he supp ort of the delta function in (39 ) f rom the neighborho o d of m = 0 - provide d tha t ( m + δ m ) − > 1 2 √ E - and m = m max - pro vided that ξ > 1 A  mn ( m ) + m 2 √ E  m max . Putting all contributions tog ether, we hav e ¯ P ( ξ , I K ) ∼ 1 2            0 for 0 < ξ ≤ q η − K (1+ g ) 1 g 1 1 − ξ /η − h (1 + g ) − η − K ξ 2 i for q η − K (1+ g ) < ξ ≤ p η − K 1 1 − ξ /η − for p η − K < ξ + 1 2            0 for 0 < ξ ≤ q η 0 K (1+ g ) 1 g 1 1 − ξ /η + h (1 + g ) − η 0 K ξ 2 i for q η 0 K (1+ g ) < ξ ≤ p η 0 K 1 1 − ξ /η + for p η 0 K < ξ . (61) The most imp or t an t feature in (61) is that it shifts the supp ort of ¯ P ( ξ , I K ) aw a y from ξ = 0 to min n q η − K (1+ g ) , q η 0 K (1+ g ) o . Note that most of the expressions whic h mak e up P ( ξ , I K ) in the neighborho o d of ξ = 0 are confined to a ξ in terv al of size O ( 1 √ K ). Bey ond Counting no da l doma ins on surfac es of r evolution 22 this in terv al ( 1 √ K ≪ ξ < ξ max ) P ( ξ , I K ) takes the form P ( ξ , I K ) ∼ 1 2 1 1 − ξ η − + 1 1 − ξ η + ! , (62) whic h coincides with the small ξ expression of t he limit distribution (46) to leading order in ξ . T o obtain the do minant b eha vior of P ( ξ , I K ) near ξ = 0, one should add P m =0 ( ξ , I K ) whic h w e compute now. The c ontribution of the m = 0 term in ( 30) . F o llo wing the same steps as ab ov e, the leading appro ximation to P m =0 ( ξ , I K ) reads, P m =0 ( ξ , I K ) ∼ 1 g K Z ∞ − 1 2 χ I K  2 A H ( n + 1 2 , 0))  δ  ξ − n + 1 2 A H ( n + 1 2 , 0)  d n. (63) This b eing already a correction term, w e are a llow ed to neglect the se mi-classical correction 1 2 to n in the argument of H ( n + 1 2 , 0). With H ( n, 0) =  π n L  2 from (20), w e get P m =0 ( ξ , I K ) ∼ 1 g K Z L π q K (1+ g ) 2 A L π √ K 2 A δ  ξ − 1 n L 2 2 A π 2  d n . (64) Therefore, P m =0 ( ξ , I K ) ∼      1 ξ 2 η − g K if q η − K (1+ g ) ≤ ξ < p η − K 0 otherwise , (65) where η − w as defined in ( 4 7). Both the shift of the supp o r t a wa y from the origin ξ = 0 and its size, decrease as 1 √ K . In its supp ort, P m =0 ( ξ , I K ) is b ounded b et w een the v alues 1 g and 1 + 1 g . Ho w ev er, although the contribution of this term t o the probabilit y densit y is O (1), its effect on the probability is O ( 1 K ), o r in other w ords, P m =0 ( ξ , I K ) = O ( 1 K ) in the w eak sense. In this vicinit y , P m =0 ( ξ , I K ) dep ends only on a single geometric parameter, η − . Note that a priori , one w ould hav e expected the low er limit of the support of ¯ P ( ξ , I K ) to b e O ( 1 K ), and not O ( 1 √ K ). This prediction stems from the definition of the normalized no dal sequence (26); for N > 1 w e hav e the lo we r b ound ξ N ≥ 2 N , so P ( ξ , I K ) ≡ 0 for ξ < 2 (1+ g ) K , and no t for ξ < O ( 1 √ K ) as observ ed. The oscillatory terms Q ( I K ) con t ribute terms o f order 1 √ K in the ξ = 0 vicinit y . They orig inate f r o m a Gibbs phenomenon and they will b e discusse d in App endix B. Counting no da l doma ins on surfac es of r evolution 23 Figures 6. a nd 7. compare t he results of n umerical sim ulations with the expres sions deriv ed a b o v e in the vicinit y of ξ = 0, and the theory includes also the oscillatory corrections (to b e precise, with the integrated density Π( ξ , I K ) := R ξ 0 P ( ξ ′ , I K )d ξ ′ ). 0 0.01 0.02 0.03 0.04 0.05 0.06 Ξ 0 0.01 0.02 0.03 0.04 0.05 0.06 P H Ξ ,I K L Figure 7. Lo c a l b ehavior of Π( ξ , I K ) a t ξ = 0 for a n ellipsoid o f revolution with eccentricit y ε = 0 . 5 ( K = 24000, g = 1 ). 0 0.01 0.02 0.03 0.04 Ξ 0 0.01 0.02 0.03 0.04 P H Ξ ,I K L Figure 8. Lo c a l b ehavior of Π( ξ , I K ) a t ξ = 0 for a n ellipsoid o f revolution with eccentricit y ε = 0 . 5 ( K = 48000, g = 0 . 5). Counting no da l doma ins on surfac es of r evolution 24 3.3. Th e oscil latory c on tributions Q ( ξ , I K ) We ak estimate of Q ( I K ). In this section we estimate the oscillatory part Q ( ξ , I K ), defined in (32). W e sho w that it is of smaller o r der in 1 √ K than the leading order correction of the corresp onding smo ot h part. This j ustifies the preceding analysis where w e considered only the smo oth part, which giv es the only contribution of O ( 1 √ K ). The n umerical results also g ive evidence o f this self-a veraging pro cess in the semi-classical limit. T o b e more precise, w e shall show that Z 1 0 Q ( ξ , I K ) ϕ ( ξ )d ξ = o ( 1 K ) , (66) for any test function ϕ . W e shall follo w a series of na tural r egula r izations whic h are justified in studying this w eak limit. W e shall also discuss the lo cal b ehavior of Q ( ξ , I K ) in the neigh b orho o d of ξ = 0, in order to compare with the n umerical results. W e ha v e Q ( ξ , I K ) = P ( N ,M ) 6 =0 Q N ,M ( ξ , I K ), where the F ourier comp o nen ts (33) ar e appro ximated as Q N ,M ( ξ , I K ) ∼ 1 g K Z (1+ g ) K / 2 A K/ 2 A d E Z m max 0 δ ( ξ − mn ( m ) A )    n ( m ) − mn ′ ( m )    e 2 π i √ E ( M m + N n ( m )) d m . (67 ) W e no w turn to a smo othing of this distribution b y adding a small imaginary part to the arg ument of the delta function, sa y ε > 0. This amoun ts to replacing the δ function b y the Loren tzian δ ε ( ξ ) = 2 ℜ Z ∞ 0 e 2 π iξ x e − 2 π εx d x = 1 π ε ε 2 + ξ 2 . (68) And so, Q ε N ,M ( ξ , I K ) ∼ ε π g K Z (1+ g ) K / 2 A K/ 2 A e 2 π i √ E ( M m + N n ( m )) ε 2 + ( ξ − ξ n ( m ) ,m ) 2    n ( m ) − mn ′ ( m )    d m . (69) W e m ust notice here a qualitative difference b et w een the semiclassical theory of the sp ectral densit y a nd the no dal domain distribution. The index set I K = [ K , K + ∆ K ], corresp onds to a spectral in terv al [ E K , E K +∆ K ]. In the later case, as K → ∞ , the eigen v alues are distributed in an ev er growing interv al, while the normalized no dal Counting no da l doma ins on surfac es of r evolution 25 sequence is distributed in (0 , 1], b ecoming arbitrarily dense. So, in con trast to the regularization of the sp ectral densit y , in g eneral w e do not exp ect the limits ε → 0+ and K → ∞ to comm ute for the no dal domain distribution. W e shall first consider the semi-classical limit. Once again, b y the line ar approximation, w e hav e n ( m ) = n ± + n ′ ± ( m − m ± )+ O ( 1 √ E ), and mn ( m ) / A = 1 A ( n ± + m ± n ′ ± )( m − m ± ) + O ( 1 √ E ) =: ξ ′ ± ( m − m ± ) + O ( 1 √ E ). By taking the whole real line as the m -integration range and shifting m 7→ x = m − m σ ( σ = ± ), w e ha ve Q ε N ,M ( ξ , I K ) ∼ X σ = ± ε π g K ( n σ − m σ n ′ σ ) Z (1+ g ) K / 2 A K/ 2 A d E e 2 π iS √ E Z ∞ −∞ e 2 π i √ E Rx ε 2 + ( ξ − ξ ′ σ x ) 2 d x = = X σ 1 g K n σ − m σ n ′ σ n σ + m σ n ′ σ Z (1+ g ) K / 2 A K/ 2 A e 2 π iS √ E − 2 π ε √ E | R ξ ′ σ | d E = = X σ 2 g K n σ − m σ n ′ σ n σ + m σ n ′ σ Z √ (1+ g ) K / 2 A √ K/ 2 A e 2 π ( iS − ε | R ξ ′ σ | ) k k d k , (70) with    R S    =    n ′ σ 1 n σ m σ       N M    . No w, instead of a sharp unifor m windo w (taking in to a ccoun t only those states whose index lie in the in terv al I K ), w e consider a G a ussian regularization, h K ( k ) = 1 √ π v exp  − ( k − µ ) 2 /v 2  (71) with µ = √ 1+ g +1 2 q K 2 A and v = ( √ 1 + g − 1 ) q K 2 A , and extend the in t egr a tion ov er the whole real line ( o f course, this regularizat io n do es not affect the asymptotic b eha vior w e study), Q ε N ,M ( ξ , I K ) ∼ X σ = ± 2 √ π v g K n σ − m σ n ′ σ n σ + m σ n ′ σ Z ∞ −∞ e 2 π ( iS − ε | R ξ ′ σ | ) k − ( k − µ ) 2 /v 2 k d k = = X σ 2 √ π v g K n σ − m σ n ′ σ n σ + m σ n ′ σ e − µ 2 /v 2 Z ∞ −∞ e − k 2 /v 2 +(2 πiS − 2 π ε | R ξ ′ σ | +2 µ/v 2 ) k k d k = = X σ 2 v 2 g K n σ − m σ n ′ σ n σ + m σ n ′ σ  π iS − π ε    R ξ ′ σ    + µ v 2  × Counting no da l doma ins on surfac es of r evolution 26 × exp  − π 2 v 2 S 2 + π 2 ε 2 v 2 R 2 ξ ′ 2 σ − 2 π 2 iεv 2    R ξ ′ σ    S + 2 µ π iS − 2 π µε    R ξ ′ σ     . (72) W e pro ceed tow ards a crude, y et sufficien t for our purp o se, estimate for Q ( ξ , I K ), | Q ε ( ξ , I K ) | ≤ X ( N ,M ) 6 =0 X σ = ± 2 v 2 g K    n σ − m σ n ′ σ n σ + m σ n ′ σ     π | S | + π ε    R ξ ′ σ    + µ v 2  × × exp  − π 2 v 2 S 2 + π 2 ε 2 v 2 R 2 ξ ′ 2 σ − 2 π µε    R ξ ′ σ     . (73) By taking in to account that P γ ∈ Z 2 e − t ( α · γ ) 2 + √ tβ · γ ≍ 1 t and P γ ∈ Z 2 | γ i | e − t ( α · γ ) 2 + √ tβ · γ ≍ 1 t 3 / 2 , t → ∞ , w e ha ve | Q ε ( ξ , I K ) | ≤ O ( 1 K 3 / 2 ) , (74) uniformly in ε (b efor e the limit ε → 0+ is tak en, all t hr ee terms in the amplitude of the summed quan tity con tribute t o this same order; µ v 2 ≍ 1 √ K ). L o c al estimate of Q ( ξ , I K ) in the neighb orho o d of ξ = 0. F ro m the num erical in ve stigations, a clear oscillatory b ehavior o f the distribution is observ ed in the neigh b o rho o d of ξ = 0, whic h implies the imp ortance of the oscillatory part in that region. This is due to a Gibbs phenomenon, a nd is not in contradiction to the fact that the oscillatory part Q ( I K ) is of less order in the w eak sense than the smo oth part ¯ P ( I K ) - t his is a lo cal con tributio n. W e find the manifestation in the dominan t terms P N Q N , 0 and P M Q 0 ,M , which agrees with the num erical results. The explicit for m of these con tr ibutions are presen ted in App endix B. 4. The geometric information st ored in the no dal sequence The deriv ation o f the probability densit y P ( ξ , I K ) in the preceding section is based on functions whic h are obtained directly from the dynamical relations em b edded in the Hamiltonian and its dep endence on the action v ariables. These, in turn a re computed from the profile curv e y = f ( x ), whic h defines M . Here, w e would lik e to in v estigate the p ossibilit y of inv erting this relationship, and ask what can b e said ab out the surface once t he no dal sequenc e is kno wn. Counting no da l doma ins on surfac es of r evolution 27 A few parameters can b e easily extracted from the probability densit y P ( ξ , I K ). T aking the limit K → ∞ w e get the limit distribution P ( ξ ). Its supp ort pro vides the para meter ξ max . In the vicinit y of ξ = 0, P ( ξ ) dep ends symmetrically on the tw o parameters η − and η + . This pro vides a useful relationship, whic h can b e com bined with the inf o rmation f r om P ( ξ , I K ) near ξ = 0, for finite K , which dep ends o n all four (indep enden t) geometric para meters, η − , η + , η 0 and η max . These parameters a re related to the g eometry of the surface (47). As w as commen ted in the previous sec tion, the relations ( 4 4) and (57), can b e solv ed to obtain the tw o branches m − ( ξ ) and m + ( ξ ). Conseque n tly , the function n ( m ) is determined, since n ( m ± ( ξ )) = A ξ m ± ( ξ ) . This will b e the first step to w ards the in version of t he no dal coun ting data whic h will b e explained in the next subsection. 4.1. I nversion of the No dal Se quenc e In this section w e discuss no dal domain in v ersion in detail. The first result is: with the distribution ¯ P ( I K ) as given data, the “sc ale -invariant” action n ( m ) / √ A is determine d uniquely. F or simple surfaces of rev o lutio n (separable systems in general with the Ha milto nian satisfying the ho mogeneit y condition), n ( m ) incorp orates a ll the information ab out the dynamics, a nd as w e shall see in some cases, the actual geometry . Our starting p oint is the semi-classical asymptotic expansion (52), ¯ P ( ξ , I K ) ∼ P ( ξ ) + 1 √ K P 1 ( ξ ) , (75) whic h, f o r conv enience, we solv e for P 1 ( ξ ), P 1 ( ξ ) = √ K  ¯ P ( ξ , I K ) − P ( ξ )  + O ( 1 √ K ) . (76) Ob viously , P and P 1 are determined from the giv en data, lim K →∞ ¯ P ( I K ) = P , and lim K →∞ √ K  ¯ P ( I K ) − P  = P 1 . (77) F ro m relation (44) , P ( ξ ) = ξ d d ξ log m − ( ξ ) m + ( ξ ) ⇒ m + ( ξ ) m − ( ξ ) = exp Z ξ max ξ P ( ξ ′ ) ξ ′ d ξ ′ , (78) Counting no da l doma ins on surfac es of r evolution 28 the ratio m + ( ξ ) /m − ( ξ ) is determined, giv en P . Accompan ying the ab ov e, w e ha ve the relation (57), P 1 ( ξ ) = − r 2 A √ 1 + g − 1 g p ξ d d ξ  p ξ d d ξ [ m + ( ξ ) − m − ( ξ )]  . (79) whic h can b e in verted to giv e u ( ξ ) := 1 √ A  m + ( ξ ) − m − ( ξ )  = − 1 √ 2 g √ 1 + g − 1 Z ξ 1 √ ξ ′  Z ξ ′ P 1 ( ξ ′′ ) √ ξ ′′ d ξ ′′  d ξ ′ . (80) T o determine u ( ξ ) uniquely , tw o in tegration constan ts should b e provide d, and they are giv en in terms of the initial conditions for u ( ξ ) a nd d u d ξ at the p oin t ξ = 0 or ξ = ξ max . They ar e expressed in terms of the geometric parameters { η − , η + , η 0 , η max } , whic h are extracted from the form of P ( ξ ) near ξ = 0 and ξ = ξ max . Th us the ratio and the difference b et w een m + ( ξ ) ar e m − ( ξ ) given. T ogether with n ± ( ξ ) = ξ A m ± ( ξ ) they g iv e the parametric represen t a tion of Γ. Not e that w e require no information from the statistical prop erties of the no dal counts of the isotropic quantum states - i.e. P m =0 ( I K ). The significance o f this result b ecomes apparen t in the next section, where w e confine ourselv es to mirror-symmetric surfaces. 4.2. Mirr or-symmetric f ( x ) is uniquely de termine d by the n ( m ) F o llowing the preceding section, w e prov e that if the generating curv e f is mirror - symmetric, (i.e. f ( − x ) = f ( x )), the action v ariable n ( m ) determines the f uniquely . Since f is an ev en function, w e ma y write n ( m ) = 2 π Z x + 0 p f ( x ) 2 − m 2 p 1 + f ′ ( x ) 2 f ( x ) d x , (81) where f ( x + ) = m , or x + = f − 1 ( m ). Changing the v ariable x 7→ u = f ( x ) /m max , n ( m ) = 2 π Z 1 m/m max p m 2 max u 2 − m 2 u p 1 + f ′ ◦ f − 1 ( m max u ) 2 | f ′ ◦ f − 1 ( m max u ) | d u . (82) Supp ose that f , g are differen t curv es of this class which give the same a ction v aria ble, i.e. Z 1 m/m max ( f ) p m max ( f ) 2 u 2 − m 2 u p 1 + f ′ ◦ f − 1 ( m max ( f ) u ) 2 | f ′ ◦ f − 1 ( m max ( f ) u ) | d u = Counting no da l doma ins on surfac es of r evolution 29 = Z 1 m/m max ( g ) p m max ( g ) 2 u 2 − m 2 u p 1 + g ′ ◦ g − 1 ( m max ( g ) u ) 2 | g ′ ◦ g − 1 ( m max ( g ) u ) | d u , (83) while f 6 = g . Firstly , note that m max ( f ) = m max ( g ). This is b ecause m max is the sole real ro ot of n ( m ), th us, since b oth sides of the ab o ve equalit y are prop ortional t o n ( m ) they mus t ha ve a common ro o t, denoted simply b e m max , since n ( m ) is analytic on (0 , m max ]. Th us, since the integration limits are iden tical, the in tegr a nds m ust b e equal. It suffi ces to sho w that √ 1+ f ′ ◦ f − 1 ( y ) 2 | f ′ ◦ f − 1 ( y ) | = √ 1+ g ′ ◦ g − 1 ( y ) 2 | g ′ ◦ g − 1 ( y ) | , for y ∈ [0 , m max ], implies f = g . F rom the ab ov e we hav e f ′ ◦ f − 1 ( y ) = g ′ ◦ g − 1 ( y ). The problem has b een reduced to sho wing that t he nonlinear op erator Af = f ′ ◦ f − 1 acting on our function class, p ossesses an inv erse, i.e. Af = Ag ⇔ f = g . Consider the inhomogeneous “functional” equation f ′ ◦ f − 1 = h , for some h in some other a ppro priate function class. W e shall sho w that for given h , this determines, along with some initial condition, a unique f , fo r ma lly f = A − 1 h . W e hav e f ′ ( f − 1 ( y )) = h ( y ), or f ′ ( x ) = h ( f ( x )), since y = f ( x ). Th us, w e hav e reduced this to a first o r der ordinary differen tial equation, y ′ = h ( y ). Accompanied by the initial condition f (0) = m max , this b ecomes an initial v alue pro blem on [0 , 1] with a unique solution ( h is smo oth on [0 , 1)). Thus, A − 1 exists. In our problem the initial condition is pro vided by kno wledge of the ro ot of n ( m ), so w e hav e reac hed the conclusion that the tw o in t egr a ls cannot equal if f 6 = g . 5. Summary and conclusions The unique inv ersion of the nodal se quence whic h was demonstrated ab ov e for s ymmetric and “simple” surfaces of rev olutio n pa v es the w a y t o a sequence of problems whic h should now b e addressed. Other families of separable manifolds are known, amongst whic h the Liouville surfaces [30] and the axially symmetric Zoll surfaces [31] are of prime imp ortance. W e b eliev e that the general approach taken in the presen t pap er could b e applied to handle these case, how eve r, mo difications should b e a pplied to tak e care of Counting no da l doma ins on surfac es of r evolution 30 sp ecial problems whic h are intrinsic to these problems, and this remains for a further study . Another class of in tegrable systems cons ists of independen t particle mo dels whic h are commonly used in A tomic and Nuclear ph ysics. The study o f suc h systems extends the researc h o f no dal domains to systems with arbitrary dimensions. The next systems in complexit y are systems whic h are classically in tegrable but are not separable quan t um mech anically . The simplest examples consis t of e.g. the Diric hlet Laplacians in the equilateral o r the isosceles rig ht triangles. Ev en thoug h the sp ectrum can b e expressed precisely in t erms of the “quantum n umbers”, counting of no dal doma ins is difficult, and the study o f the no dal sequences in suc h cases migh t call for other approac hes then the one pursued here. Do no dal sequences in other systems store geometric informa t ion? Is there a w ay to extract this information to determine the g eometry? Thes e a re y et op en problems, and the only hint for an affirmative answ er comes from preliminary n umerical sim ulatio ns whic h indicate tha t “no dal” trace formulae exist for “quan tum graphs” [3 2 ] and for “c haot ic billia rds” [33]. No rigor ous treatment exists so far . 6. Ac kno wledgmen t s The authors w ould lik e to thank Prof. Jon Keating and D r. Sv en Gn utzmann for length y and enlightenin g discussions. This w ork w as supp orted b y the Minerv a Cen ter for non-linear Ph ysics and the Einstein (Minerv a) Cen ter at the W eizmann Institute, b y the ISF and b y grants from the GIF (g ran t I-808 -228.14/ 2003), a nd EPSR C (grant GR/T06872/ 01). App endix A . The regularization of the action integral The action v aria ble ( 5 ) is defined in terms of an integral, whose form is not con ve nien t for further computations, suc h as e.g., the ev aluation of its higher deriv ativ es. This can b e do ne b y regular izing the in tegral in a wa y whic h will b e explained here. Rather than in tro ducing fractional deriv atives as w as done in e.g., [24], we compute t he integrals Counting no da l doma ins on surfac es of r evolution 31 explicitly . It is conv enien t to in tro duce the notatio n q ( x ) = f ( x ) 2 and µ = m 2 . W e start with n ( µ ) = 1 π Z x + x − p q ( x ) − µ p 4 q ( x ) + q ′ ( x ) 2 2 q ( x ) d x , where q ( x ± ) = µ . (A.1) The function q ( x ) is analytic in I and has a single maxim um at x max . W e separate the in tegrat io n interv al in (A.1) to tw o consecutiv e in t erv als [ x − , x max ] a nd [ x max , x + ] a nd write a ccordingly n ( µ ) = n − ( µ ) + n + ( µ ) . (A.2) In eac h of the in terv als [ − 1 , x max ) and ( x max , 1], q ( x ) is a monotonic function. Therefore it can b e inv erted in eac h of the interv als in terms of the corresp onding f unctions x ± ( q ), whic h a re analytic in the in t erv al [0 , q max ). Th us, n ± ( µ ) = 1 π Z q max µ √ q − µw ± ( q )d q (A.3) with w ± ( q ) = 1 2 q s 4 q  d x ± ( q ) d q  2 + 1 . (A.4) The expressions in the square brac k ets ab ov e are analytic in the domain of q where x ± ( q ) are analytic, that is, in [0 , q max ). They can be T aylor expanded with a conv erg ence radius q max so that w ± ( q ) = 1 2 q + ∞ X r =0 τ ± r q r . (A.5) Substituting this expressions in (A.3), p erforming the in tegrals and defining τ r = τ + r + τ − r , w e finally get n ( µ ) = 2 π " ( q max − µ ) 1 2 − µ 1 2 arccos  µ q max  1 2 # + 1 π ∞ X r =0 τ r I r ( µ ) , (A.6) where, I r ( µ ) = Z q max µ q r √ q − µ d q = ( q max − µ ) 3 2 r X k =0  r k  ( q max − µ ) r − k µ k ( r − k ) + 3 2 . (A.7) Using (7) we find n (0) = L π = 1 π √ q max " 2 + ∞ X r =0 τ r q r +1 max r + 3 2 # . (A.8) Counting no da l doma ins on surfac es of r evolution 32 Th us in t he vicinity of µ = 0 w e get n ( µ ) ∼ L π − √ µ . (A.9) The b eha vior of n ( µ ) near the other extreme end of the interv al - µ = q max - cannot b e deduced in the same wa y , b ecause x ( q ) is not defined at this p oint. Ho we v er, staring directly from (A.1 ) w e can obta in the b ehavior of n ( µ ) in this domain. F or this purp ose w e write q ( x ) ∼ q max − 1 2 ω ( x − x max ) 2 , ω = | q ′′ ( x max ) | , x → x max . (A.10) T o leading order in ( q max − µ ), (A.1) reduces to n ( µ ) ∼ 2 π √ 2 ω q max Z q max µ r q − µ q max − q d q = q max − µ √ 2 ω q max ∼ r 2 ω ( f max − | m | ) . (A.11) Finally , w e return to the example of the sphere. With f ( x ) 2 = 1 − x 2 the integrals can b e p erformed exactly , n ( m ) = 1 − | m | . (A.12) This is consisten t with the lo cal expressions presen ted in (A.9,A.11) . App endix B . The oscillatory part near the origin W e recov er the oscillatory b eha vior of the distribution near ξ = 0 from the leading order sums Q ( ξ , I K ) ∼ X N ∈ Z ∗ Q N , 0 ( ξ , I K ) + X M ∈ Z ∗ Q 0 ,M ( ξ , I K ) . (B.1) W e b egin with (3 3). As ξ → 0+, the contribution in the ab o ve integral will come from the neigh b orhouds o f m = 0 and m = m max (the delta function of the in tegrand is supp orted o n { m + δ m } ± ). These will b e treated separately , denoted b y Q ± N ,M ( ξ , I K ) resp ectiv ely , so that Q N ,M ( ξ , I K ) = Q − N ,M ( ξ , I K ) + Q + N ,M ( ξ , I K ) . (B.2) In what follows , w e define x := ξ /η − when referring to the Q − terms, a nd x := ξ /η + for the Q + terms. Counting no da l doma ins on surfac es of r evolution 33 Giv en that m − ∼ L 2 π x and δ m − ∼ − 1 2 η x , by c hanging v aria bles to the appropriate dimensionless w a ven umber k := L √ E , we ha v e Q − 0 ,M ( ξ , I K ) ∼ 1 π 2 η − g K e − π iM x/ 2 1 − x Z π √ η − (1+ g ) K π √ η − K Θ  k − π x  e iM xk  k − π 2  d k . (B.3) Similarly , w e carry out the calculation of the integral Q + 0 ,M ( ξ , I K ). Here, the condition that ( m + δ m ) + lies in the integration range reads A ξ >  mn ( m ) + η m  m max = m max η , Q + 0 ,M ( ξ , I K ) ∼ 1 η 0 g K e π iM √ ω / 2(1+ x/ 2) 1 − x Z √ η 0 (1+ g ) K √ η 0 K Θ  k − 1 x  e 2 π iM (1 − x/ 2) k  k − p ω / 2 2  d k . (B.4 ) where the appropriate w av enum b er is k := m max √ E . F o llowing the ab o ve calculations, Q − N , 0 ( ξ , I K ) ∼ 1 π 2 η − g K e π iN x/ 2 1 − x Z π √ η − (1+ g ) K π √ η − K Θ  k − π x  e iN (2 − x ) k  k − π 2  d k , (B.5) and Q + N , 0 ( ξ , I K ) ∼ 1 η 0 g K e − π iN (1+ x/ 2) 1 − x Z √ η 0 (1+ g ) K √ η 0 K Θ  k − 1 x  e π iN √ ω / 2 xk  k − p 2 /ω 2  d k . (B.6) By p erforming the inte grations, w e hav e X M ∈ Z ∗ Q 0 ,M ( ξ , I K ) ∼ 1 π η − g K 1 (1 − x ) x 2            0 for q η − (1+ g ) K ≥ ξ f − 1 ( ξ , I K ) f o r q 1 (1+ g ) η − K < ξ ≤ q 1 η − K f − 2 ( ξ , I K ) f o r q 1 η − K < ξ + 1 π η 0 g K 1 (1 − x )(2 − x ) 2              0 for q ω 2(1+ g ) η 0 K ≥ ξ f + 1 ( ξ , I K ) f o r q ω 2 η 0 (1+ g ) K < ξ ≤ q ω 2 η 0 K f + 2 ( ξ , I K ) f o r q ω 2 η 0 K < ξ , where f − 1 ( ξ , I K ) := α (1 − x 2 ) − α ( x p η − K − x 2 ) + x p η − K β ( x p η − K − x 2 ) − β (1 − x 2 ) , Counting no da l doma ins on surfac es of r evolution 34 f − 2 ( ξ , I K ) := α ( x p (1 + g ) η − K − x 2 ) − α ( x p η − K − x 2 )+ + x p η − K β ( x p η − K − x 2 ) − x p (1 + g ) η − K β ( x p (1 + g ) η − K − x 2 ) , ( B.7 ) and f + 1 ( ξ , I K ) := α  2 x − 1 + (1 + x 2 ) r ω 2  − α  (2 − x ) p η 0 K + (1 + x 2 ) r ω 2  + +2 p η 0 K β  (2 − x ) p η 0 K + (1 + x 2 ) r ω 2  − 2 x β  2 x − 1 + (1 + x 2 ) r ω 2  , f + 2 ( ξ , I K ) := α  (2 − x ) p (1 + g ) η 0 K + (1 + x 2 ) r ω 2  − α  (2 − x ) p η 0 K + (1 + x 2 ) r ω 2  + +2 p η 0 K β  (2 − x ) p η 0 K + (1 + x 2 ) r ω 2  − − 2 p (1 + g ) η 0 K β  (2 − x ) p (1 + g ) η 0 K + (1 + x 2 ) r ω 2  . (B.8) Similarly , X N ∈ Z ∗ Q N , 0 ( ξ , I K ) ∼ 1 π η − g K 1 (1 − x )(2 − x ) 2            0 for q η − (1+ g ) K ≥ ξ g − 1 ( ξ , I K ) f o r q 1 (1+ g ) η − K < ξ ≤ q 1 η − K g − 2 ( ξ , I K ) f o r q 1 η − K < ξ + ω 2 π η 0 g K 1 (1 − x ) x 2              0 for q ω 2(1+ g ) η 0 K ≥ ξ g + 1 ( ξ , I K ) f o r q ω 2 η 0 (1+ g ) K < ξ ≤ q ω 2 η 0 K g + 2 ( ξ , I K ) f o r q ω 2 η 0 K < ξ , (B.9) where g − 1 ( ξ , I K ) := α ( 2 x − 1 + x 2 ) − α  (2 − x ) p η − K + x 2  + +2 p η − K β  (2 − x ) p η − K + x 2  − 2 x β ( 2 x − 1 + x 2 ) , g − 2 ( ξ , I K ) := α  x 2 + (2 − x ) p (1 + g ) η − K  − α  x 2 + (2 − x ) p η − K  + + 2 p η − K β  (2 − x ) p η − K + x 2  − 2 p (1 + g ) η − K β  (2 − x ) p (1 + g ) η − K + x 2  , (B.10) Counting no da l doma ins on surfac es of r evolution 35 and g + 1 ( ξ , I K ) := α ( r 2 ω − 1 − x 2 ) − α  x r 2 η 0 ω K − (1 + x 2 )  + + x r 2 η 0 ω K β  x r 2 η 0 ω K − (1 + x 2 )  − r 2 ω β  r 2 ω − (1 + x 2 )  , g + 2 ( ξ , I K ) := α  x r (1 + g ) 2 η 0 ω K − (1 + x 2 )  − α  x r 2 η 0 ω K − (1 + x 2 )  + + r 2 η 0 ω K xβ  x r 2 η 0 ω K − (1+ x 2 )  − r (1 + g ) 2 η 0 ω K xβ  x r (1 + g ) 2 η 0 ω K − (1+ x 2 )  . 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