Two-parametric PT-symmetric quartic family

Transcription

1 Two-parametri PT-symmetri quarti family Alexandre Eremenko and Andrei Gabrielov Otober 17, 211 Abstrat We desribe a parametrization of the real spetral lous of the twoparametri family of PT-symmetri quarti osillators. For this family, we find a parameter region where all eigenvalues are real, extending the results of Dorey, Dunning, Tateo and Shin. MSC: 81Q5, 34M6, 34A5 Key words: one-dimensional Shrödinger operators, PT-symmetry, quasi-exat solvability, Darboux transform. 1 A family of quarti osillators We onsider the eigenvalue problem in the omplex plane w + (ζ 4 + 2bζ 2 + 2iJζ + λ)w =, w(te πi/2±πi/3 ), t +. (1) Here J and b are parameters. This two-parametri family is interesting for several reasons. When 2J is an integer, 2J < 1, and b, the problem has the same spetrum as a spherially symmetri quarti osillator in R d. In this ase 2J = 2 2l d, where l is the angular momentum quantum number [5, 4]. When J is a positive integer, problem (1) is quasi-exatly solvable (QES) [2]. This means that there are J eigenfuntions of the form w(ζ) = p(ζ) exp( iζ 3 /3 ibζ), Both authors are supported by NSF grant DMS

2 where p is a polynomial of degree J 1 in ζ whose oeffiients are algebrai funtions in b. When J and b are real, the problem is PT-symmetri. The eigenvalues of a PT-symmetri problem an be either real or ome in omplex onjugate pairs. Both possibilities an be present for J > 1. A very interesting feature is level rossing in the real domain: for some real b and J the graphs of the eigenvalues λ k (b) an be real and ross eah other. This phenomenon was disovered by Bender and Boetther [2] numerially, then it was studied in [12], where the presene of infinitely many suh real level rossing points was proved for positive odd J. When J +, the QES part of the spetral lous approximates the whole spetral lous of the PT-symmetri ubi family w + (iz 3 + iaz)w = λw, w(± ) =, (2) whih was subjet of intensive researh, see, for example, [3, 7, 8, 11, 16, 11, 2, 23]. By the hange of the independent variable z = iζ problem (1) is equivalent to L b,j (y) = y + (z 4 2bz 2 + 2Jz)y = λy, y(te ±πi/3 ), t +. (3) Shin s theorem [2] applies to these eigenvalue problems when J, and implies that for J all eigenvalues are real. The proof of Shin s theorem is based on the remarkable ODE-IM orrespondene of Dorey, Dunning and Tateo [8]. Here we extend this result of Shin. Theorem 1. All eigenvalues of (1) or (3) are real for J 1. The ondition J 1 is exat, beause it is known that for every J > 1 there are non-real eigenvalues [12]. Our proof of Theorem 1 is based on purely topologial arguments. Using the formulation (3), we establish a ertain property of eigenfuntions for J =, and then show that this property persists for J < 1 and prevents level rossing. The real spetral lous Z(R) R 3 is defined as the set of all real triples (b,j,λ) for whih there exists y satisfying (3). This is a non-singular analyti surfae in R 3. The main result of this paper is a parametrization of a part of Z(R) orresponding to integer J in terms of Nevanlinna parameters. In [11] we obtained similar parametrization of the real spetral lous of (2) and another two-parametri family of quartis. 2

3 The family onsidered in this paper is muh more ompliated beause of the presene of QES part and of the real level rossings. In [13] we parametrized the real quasi-exatly solvable lous Z QES (R) of (3) whih onsists of all triples (b,j,λ) R 3 for whih there exists a funtion y(z) = p(z) exp(z 3 /3 bz) satisfying (3) with a polynomial p. The plan of the paper is the following. In the next setion we introdue Nevanlinna parameters and state our prinipal result, Theorem 2, about the orrespondene between parameters (b, J, λ) Z(R) and Nevanlinna parameters. In setion 3 we prove the easy, algebrai part of this orrespondene. In setion 4 we study the ase J = and in setion 6 we prove Theorem 1. Then in setion 7 we desribe the parametrization of the part of the real spetral lous Z(R) where J is an integer in terms of Nevanlinna parameters and omplete the proof of Theorem 2. In a forthoming paper we will parametrize the whole two-dimensional real spetral lous of the family (3) in terms of Nevanlinna parameters. 2 Nevanlinna parameters The referenes for this setion are [18, 21, 9, 1, 11]. Suppose that b, J and λ are real. Let y be an eigenfuntion of (3). Then y (z) = y(z) satisfies the same differential equation and the same boundary onditions (3), so y = y. We an hoose y so that y(x ) R\{} for some real x. Substituting this x to y = y we onlude that = 1. So our eigenfuntion y is real. It is defined up to multipliation by a real onstant. Let y 1 be a solution of the differential equation in (3), whih is real and linearly independent of y. Consider the meromorphi funtion f = y/y 1. It is a Nevanlinna funtion, whih means that f has no ritial points in C and the only singularities of f 1 are finitely many logarithmi branh points. In the ase that y 1 is normalized by y 1 (x), x +, x R, we all f a normalized Nevanlinna funtion of (3). The normalized Nevanlinna funtion is defined up to a real multiple. Existene of y 1 with suh normalization is guaranteed by a theorem of Sibuya [21]. Different Nevanlinna funtions assoiated with the same point (b,j,λ) Z(R) are related by a real frational-linear transformation of the form f αf/(f β) where α and β are real. Nevanlinna funtions f of (3) have no ritial points, beause f = (y y 1 3

4 yy 1)/y 2 1 and y y 1 yy 1 = onst. They have 6 asymptoti values in the setors S j = {te iθ : t >, θ πj/3 < π/6}, j =,...,5. In what follows j is understood as a residue modulo 6. These setors are in one-to-one orrespondene with logarithmi branh points of f 1. The asymptoti values of the normalized Nevanlinna funtion are in S 1 and S 1, beause of the boundary ondition, and in S beause of the normalization of y 1. We denote by and a the asymptoti values of the normalized Nevanlinna funtion in S 2 and S 3, respetively. As f is real, the asymptoti value in S 2 is, and a is real. These asymptoti values a, are alled the Nevanlinna parameters. They are related to the Stokes multipliers by simple formulas [21, 17]. Normalized funtion f and the Nevanlinna parameters are defined modulo multipliation by a real non-zero number, so we an further normalize them. We will use different normalizations, depending on the situation. Nevanlinna parameters (, a) C (R { }) are subjet to the following onditions:, a, (4) and the set {,,a,,} ontains at least 3 distint points, in other words, the ombination =, a = is prohibited [18, 21]. Nevanlinna parameters (modulo multipliation by real onstants) serve as loal oordinates on the real spetral lous Z(R), [1, 17]. Notie that the set of pairs satisfying (4) modulo proportionality is a non-hausdorff manifold. But this will ause no diffiulties as we always work in loal harts. Relation between (b, J, λ) and (a, ) is very ompliated: Nevanlinna s onstrution of the map (a,) (b,j,λ) involves the uniformization theorem. So it is interesting and hallenging to establish any expliit orrespondenes between the sets in the spae of Nevanlinna parameters and the sets in the (b,j,λ) spae. Our main result in this diretion is Theorem 2. J is an integer if and only if a = or =. In the next setion we prove the only if part. 4

5 3 QES lous and Darboux transform In this setion we prove the easy part of Theorem 2: if J is an integer, then either a = or =. Suppose that for some (b,j,λ) Z(R) we have a =. This means that the eigenfuntion y tends to zero in S 3 (and also in S 1,S 1 ), so y is an elementary funtion of the form pe q with polynomials p and q. Substitution to (3) gives y(z) = p(z) exp(z 3 /3 bz), where p is a polynomial. It is known that suh eigenfuntions exist if and only if J is a positive integer [2, 12]. Suh points (b,j,λ) form the QES spetral lous Z QES (R) whih onsists of smooth algebrai urves Q J (b,λ) =, J = n + 1, n, where Q J are real polynomials of degree J in λ. QES spetral lous was studied in [12, 13]; in the seond paper it was parametrized in terms of Nevanlinna parameters. The onverse is evident: if (b,j,λ) Z QES (R), then a =. In [12] we obtained the following results: (i) If J is a positive integer, and b is real, then all non-qes eigenvalues are real. Let Z J (R) = {(b,λ) R 2 : (b,j,λ) Z(R)} and let Z QES J (R) be similarly defined. Let ZJ be the losure of Z J (R)\Z QES J (R) in R 2. (ii) When J is even, ZJ(R) Z QES J (R) =. When J is odd, then = holds at all points (b,λ) of this intersetion. (iii) If J is a positive integer, then Z J = Z J. To deal with the ondition =, we use the Darboux transform [6, 14, 12], whih we reall. Let ψ,...,ψ n be some eigenfuntions of a differential operator L = d 2 /dz 2 + V (z) with eigenvalues λ,...,λ n. Then the differential operator d2 dz 2 + V 2 (log W(ψ,...,ψ n )), (5) where W is the Wronski determinant, has the same eigenvalues as L, exept λ,...,λ n. 5

6 Let J be an integer, (b,λ) Z J(R), and let y be the eigenfuntion orresponding to (b,j,λ). We apply the Darboux transform to our operator L b,j in (3), taking all QES eigenfuntions as ψ,...,ψ n. If J = the Darboux transform does not hange anything. It is easy to see, [12] that the transformed operator (5) is L b, J in this ase. But L b,j (y) = λy, and if we define y (z) = y( z), then L b, J (y ) = λy. However, this y is not an eigenfuntion of L b, J, beause it does not satisfy the normalization ondition in (3). Instead it tends to zero in S 2 and in S 2. This means that y is linearly independent of the eigenfuntion y of L b, J, and g = y /y has asymptoti values in S 2 and in S 2. Sine g has equal asymptoti values in S 2 and S 2, any other Nevanlinna funtion for the same (b,j,λ) has the same property. So = at the point (b, J,λ). The ase of negative J is treated similarly, applying the inverse Darboux transform. Thus = on Z J(R) when J is an integer. This proves the only if part of Theorem 2. As a byprodut we obtain the following: (iv) ondition = at the point (b,λ) Z J (R) holds if and only if J > is odd and (b,λ) Z QES J (R) ZJ(R) Indeed, the eigenfuntion y is elementary if and only if the asymptoti values of f in S 3,S 1,S 1 are zero. This happens if and only if the funtion g defined above has asymptoti values in S,S 2,S 2. To prove the seond part of Theorem 2, we need to find whih part of the spetral lous orresponds to real. 4 Line omplexes Assigning asymptoti values in setors S j is not enough to define a Nevanlinna funtion f, one needs additional information about the topology of the overing f : C\f 1 (asymptoti values) C\{asymptoti values}. Suh information is enoded in the following way [18, 9, 1]. One hooses a ell deomposition Φ of the Riemann sphere C suh that eah 2-ell ontains one asymptoti value, and takes the f-preimage of this ell deomposition. This preimage is a ell deomposition Ψ of the plane whih loally looks 6

7 like Φ. There are many ways to hoose Φ, and here we desribe ell deompositions used by Nevanlinna, see also [15, Ch. 7] for a omprehensive treatment. First we fix two points in C whih are distint from the asymptoti values. We all these points and. They are the verties of Φ. Suppose that we have q asymptoti values. We onnet and with q edges whih do not interset exept at the ends. These edges and verties form the 1-skeleton of Φ; we require that faes of Φ ontain one asymptoti value eah. This hoie of Φ is fixed in this setion. The preimage Ψ = f 1 (Φ) is alled the line omplex. It has the following property: the 1-skeleton of Ψ is a bipartite onneted graph embedded in C whose all verties have degree q, and eah omponent of the omplement has either two or infinitely many edges on the boundary. Moreover, the number of omponents having infinitely many boundary edges is finite. These properties ompletely haraterize all possible line omplexes arising from Nevanlinna funtions. This means that for arbitrary line omplex, and asymptoti values (Naevanlinna parameters), satisfying the restritions stated in setion 2, there exists a normalized Nevanlinna funtion with this line omplex and these asymptoti values, and it is unique up to an affine hange of the independent variable. The Shwarzian derivative of this funtion is a polynomial and we ompose with an affine hange of the independent variable so that this polynomial has leading oeffiient 2 and next oeffiient vanishing, so that f f 3 2 ( ) f 2 = 2(z d + a d 2z d a ). (6) f In this paper, d = 4, (a 2,a 1,a ) = (2b, 2iJ,λ) and Nevanlinna parameters are asymptoti values a and as desribed in setion 2. Funtion f does not hange if a and are multiplied by the same number. For a fixed line omplex, the map that sends the parameter t = /a to the Nevanlinna triple (b,j,λ) is alled the Nevanlinna map. It is smooth, injetive and has non-zero derivative. It gives a parametrization of a part of the spetral lous. We label the faes of Φ and Ψ with the asymptoti values, so that a fae of Ψ has the same label as its image. Two ell deompositions Ψ and Ψ are onsidered equivalent if one an be mapped onto another by a homeomorphism of the plane preserving orientation and labels of faes and verties. If two Nevanlinna funtions f and g have equivalent ell deompositions, then f(z) = g(αz + β), α. 7

8 The following properties are evident. The yli order of fae labels around a vertex of Ψ is the same as for the image vertex in Φ. Two faes of Ψ with the same labels have disjoint losures. Eah fae is bounded either by two edges of by infinitely many edges. By erasing multiple edges of the 1-skeleton of Ψ and disarding the labels of bounded faes, we obtain a new ell deomposition with labeled faes and verties, whose 1-skeleton is a tree. The line omplex an be uniquely reovered from its assoiated tree. If f is real, so asymptoti values are symmetri with respet to omplex onjugation, sometimes it is possible to hoose a symmetri ell deomposition Φ. Then Ψ is also symmetri. And onversely: if the labeled ell deompositions Φ and Ψ are symmetri with respet to omplex onjugation then f an be hosen real by pre-omposing with an affine map of C. Line omplexes are onvenient for study of the limits of families of Nevanlinna funtions when two asymptoti values ollide. We have the following ompatness theorem [22]. Fix a ell deomposition Φ. Let f n be a sequene of Nevanlinna funtions with line omplexes Ψ n = fn 1 (Φ). Suppose that v n = is a vertex of Ψ n, and that f n are normalized by onditions f n() = 1. Then one an hoose a subsequene from f n that tends to a limit, and this limit is a Nevanlinna funtion. We need only the speial ase when the ell deompositions Ψ n orresponding to f n are all equivalent. If distint asymptoti values of f n tend to distint limits, then f has the same ell deomposition Ψ. If two asymptoti values of f n whih are labeling adjaent faes of Φ, ollide in the limit, then one has to erase from the 1-skeleton of Ψ all edges on the ommon boundaries of faes with these ollided asymptoti values. The omponent of the remaining graph ontaining the vertex v is the ell deomposition of the limit funtion. Now we return to Nevanlinna funtions orresponding to problem (3). One tehnial problem we are faing is that it is not always possible to hoose a symmetri Φ. However this is possible when is real, and in the next setions we onsider this ase. We begin with the simplest ase when J =. In this ase the Nevanlinna funtion has an additional symmetry. The ell deompositions onsidered in [13] are different from line omplexes, beause in the situation onsidered in that paper, it is impossible to define a line omplex with the required symmetry properties. 8

9 5 Subfamily J = In this setion we begin to prove Theorem 1. Let y be an eigenfuntion of L b,, where b C. Funtion y 1 (z) = y( z) satisfies the differential equation in (3) with J =, but does not satisfy the boundary onditions. So y 1 is linearly independent of y, and we onsider the Nevanlinna funtion f = y/y 1. (7) It is not normalized in the sense of setion 2. This funtion f has the following symmetry property: f( z) = 1/f(z). (8) The asymptoti values are in S 1 and S 1, in S 2 and S 2, A in S and 1/A in S 3. This A is the Nevanlinna parameter. It follows from (6) that the ondition (8) is equivalent to J =. If b is real, we know from the results of [4] and [2] that all eigenvalues of L,b are real. Hene f is a real funtion and A R. As f is defined up to multipliation by a real non-zero number, we an normalize so that A >. Proposition 1. For b R, we have A (, 1). Eah eigenfuntion has at most one zero on the real line. Proof. Funtion f has the property that f : C\f 1 ({,A, 1/A, }) C\{,A, 1/A, } is a overing. To onstrut the line omplex, we hoose the ell deomposition Φ of the target sphere shown in Fig. 1. A 1/A Fig. 1. Cell deomposition Φ of the Riemann sphere. 9

10 It has two verties, four edges, and four faes labeled by the asymptoti values. We denote by A the asymptoti value whih is in (, 1), so that A is either A or A 1, and our first goal is to find out whih of these possibilities holds. The line omplex Ψ = f 1 (Φ) is a labeled ell deomposition of C. It has 6 unbounded faes. Moreover, Ψ is symmetri with respet to the real line and with respet to the imaginary line. The symmetry with respet to the real line does not hange the labels, while the symmetry with respet to the imaginary line interhanges with, with and A with 1/A. Unbounded faes of Ψ are asymptoti to the setors S j. For any pair of verties of Ψ onneted by several edges, we replae these several edges with one edge. The result is a simpler ell deomposition T whose 1-skeleton is a tree. The faes of T are labeled with asymptoti values, and T has all symmetry properties desribed above. The label of a fae asymptoti to S j is the asymptoti value in S j. It is easy to lassify all possible labeled trees satisfying the above onditions. They all have two verties of order 4 and the number k of edges between these two verties is odd. These trees depend on one non-negative integer parameter m, suh that k = 2m+1, and we denote them by T m. The tree T 1 and the orresponding line omplex are shown in Fig. 2. 1

11 1/A A 1/A A Fig. 2. The tree T 1 (left) and the orresponding line omplex (right). Comparing the yli order of labels of faes adjaent to a vertex of order 4 in Fig. 1 and Fig. 2 we onlude that A = A so A (, 1). This proves the first part of the proposition. The seond part is immediately lear from the lassifiation of the trees T m : the ell deomposition Ψ has at most one fae labeled with whih intersets the real line. (A similar argument was used in [9].) Notie that the number of real zeros of y is if m is even and 1 if m is odd. (9) Lemma 1. For b R, eigenvalues of L,b are distint. Proof. The subset of the spetral lous Z parametrized by the Nevanlinna funtions satisfying (8) oinides with Z. Sine Nevanlinna parameters (modulo multipliation by non-zero numbers) serve as a loal oordinate system on Z, Z is a smooth one-dimensional subset of Z. Hene Z (R) onsists of smooth real urves and, possibly, isolated points. Sine all eigenvalues of L,b are real for b R, isolated points are not allowed. Hene Z (R) is smooth, and the real eigenvalues of L b, annot ollide as b R hanges. This proves the lemma. 11

12 Thus we an label the eigenvalues in inreasing order, λ (b) < λ 1 (b) <... To eah eigenvalue λ k orresponds one ell deomposition Ψ, and one tree T = T m. So we have some orrespondene m(k), whih is defined so that T m(k) orresponds to λ k. Proposition 2. m(k) = k. So the eigenfuntion y k orresponding to λ k has no real zeros when k is even and one real zero when k is odd. Proof. We use the asymptoti result in [12, 11] to degenerate (3) as b + to the harmoni osillator Y (u) + 4u 2 Y (u) = µy (u), Y (it), t ±. The boundary ondition here omes from the boundary ondition in (3). Eigenfuntions Y k orresponding to eigenvalues µ k, labeled in the inreasing order, have all zeros on the imaginary axis. There is one zero on the real axis (namely at the origin) when k is odd and none if k is even. To analyse the behavior of zeros of eigenfuntions of (3) as b +, we onsider the funtion g = f/(f A). This orresponds to a different hoie of y 1 in the basis (y,y 1 ) of solutions of the differential equation in (3). Asymptoti values of g are in S, in S ±1, 1 in S ±2, and a := 1/(1 A 2 ) in S 3. Aording to the result of [11], to every eigenfuntion Y k with eigenvalue µ k, and to every positive b large enough, orresponds a unique eigenfuntion y of (3) with eigenvalue λ(b) µ k. Let m = m(k) and let T m be the tree orresponding to y. Then T m must ollapse to the tree T k orresponding to Y k (see Fig. 3). 12

13 1 Fig. 3. Tree T 1. By ollapse of a tree we mean the following: two fae labels beome equal, and all edges on the ommon boundary of these faes are erased. Suh ollapse an happen only if 1/(1 A 2 ) 1 or 1/(1 A 2 ). In the seond ase we must multiply g by 1 A 2, so the asymptoti value in S 3 beomes 1 and in S ±2 it beomes 1 A 2. Then as the tree T m (k) ollapses to the tree T k, we must have m(k) = k. This proves the proposition. Proposition 2 gives a parametrization of the spetral lous for J = by Nevanlinna parameter A. For eah m, the orrespondene b A is a real analyti homeomorphism of the real line onto (, 1). 6 Region J < 1 Now we an prove Theorem 1. When b,j and λ are real, the eigenvalue problem is PT-symmetri, and we an define a real eigenfuntion y. We hoose the seond linearly independent solution y 1 of the differential equation in (3) from the onditions that y 1 in S, and y 1 is real. Then f = y/y 1 is real. The asymptoti values of f in S,...,S 5 are,,,a,, in this order, where a R. Thus for every (b,j,λ) on the real spetral lous, Nevanlinna parameters a R and C are defined. 13

14 As long as a, the number of real zeros annot hange, beause y and f never have multiple zeros. Equality a = an happen on the spetral lous if and only if J is a positive integer and the eigenfuntion y is elementary [2, 12]. Lemma 2. Let γ(t),t [, 1] be a urve in the real (b,j) plane. Suppose that γ() = (, ), and that J 1 on γ. Then for (b,j) = γ(1), all eigenvalues are real. This implies Theorem 1. Proof of Lemma 2. It is suffiient to prove the lemma for urves in the open half-plane J < 1. The general ase follows by ontinuity. For (b,j) = (, ) all eigenvalues are real and distint, so we an order them as λ < λ 1 <... Then eah λ k an be analytially ontinued along γ for t [,t k ) for some t k >. We denote these analyti ontinuations by λ k (t). Aording to theorems of Shin [19] all but finitely many of the λ k are real and by [1, Thm. 1], all but finitely many λ k an be analytially ontinued for t [, 1]. Let G be a bounded simply onneted neighborhood of γ in the half-plane {(b,j) R 2 : J < 1}. Then all but finitely many branhes of λ(b, J) are holomorphi, distint and real in G. The remaining branhes satisfy a minimal algebrai equation of the form λ m + a m 1 (b,j)λ m a m (b,j) =, (1) with a j analyti in G. The zeros of the disriminant of this equation in G form a losed subset K G. We laim that all solutions of equation (1) are distint for all (b,j) γ. Indeed, if any two eigenvalues ollide as t inreases, then some adjaent eigenvalues λ j (t) and λ j+1 (t) must ollide at some point t. The orresponding eigenfuntions will tend to the same limit as t ր t. But this is impossible beause one of them has no real zeros and another has one. Thus K does not interset γ, and there is an analyti ontinuation of eigenfuntions to γ(1). 7 Classifiation of line omplexes In this setion we onsider the general ase of real asymptoti values a and. In this ase the normalized Nevanlinna funtion f has asymptoti values 14

15 (,,,a,, ) in S,...,S 5. We hoose the ell deomposition Φ similar to that in Fig. 1, see Figs. 5,7,9. There are three generi ases: Case L. < < a, Case R. < a <, Case E. < < a. Letters L, R and E stand for left, right and even, the meaning of this notation will be lear later (Figs ). There are also non-generi ases a = and =. Assuming that a, we lassify all possible line omplexes. In all ases L, R and E, we first lassify all possible bipartite trees symmetri with respet to the real line, with 6 faes labeled by,,,a,, in this yli order, the fae labeled biseted by the positive ray, and satisfying the ondition that faes with the same label have disjoint losures. There are three types of suh trees shown in Fig. 4. They depend on two integer parameters, k > and l, where k is the number of edges between two ramified verties as shown in Fig. 4. Parameter l takes all integer values, but l is the number of edges between ramified verties as indiated in Fig. 4. We say that a tree has type if l =, type 1 if l > and type 2 if l <. For the trees that oured in setion 5 we have T n = X 2n+1,. l> a k a k a l l l= l< -l -l k Fig. 4. Three types of trees. 15

16 Now we onsider all ases separately, and argue by the following sheme. First, for a given ase of ordering a,, on the real line, and for eah tree, we deide whether this tree an ome from a line omplex, and if it an, we reover the line omplex. To do this, we begin with a ramified vertex v of the tree. Comparing the yli order of the labels around this vertex v with the yli order around the verties of the ell deomposition Φ of the sphere we determine whether this vertex of the tree is a or, and add the missing edges, step by step. When we ome to another ramified vertex, either the yli order is orret or not. If it is not orret, the tree does not orrespond to a line omplex in the onsidered ase. Otherwise, we reover the line omplex uniquely. We write the Nevanlinna map defined in setion 4 as (b,j,λ) = F(Ψ,t), where Ψ is the line omplex, and t = /a. Then we onsider possible limits as a or on the spetral lous. This gives degenerations to the intersetions of the QES spetral lous with the non-qes spetral lous, and desription of these intersetions in [13] permits to reover the value of J from the tree in ases 1 and 2. Then we onsider the degeneration as or a whenever possible, as it was done in [11, 12, 13]. We will onlude that real values of orrespond to integer J and we obtain the parametrization of the whole spetral lous for integer J. Case L. < < a, see Figs a Fig. 5. Cell deomposition Φ in ase L. 16

17 L 1,1 L 1,-1 a a Fig. 6. L m,l omplexes. Trees of type (with l = ) are impossible in this ase. Trees X k,l of type 1 (with l > ) are possible in this ase if and only if k = 2m + 1, m =, 1,... and l 1 is odd. Suh line omplexes will be alled L m,l. When a ց, t = /a, the orresponding Nevanlinna funtion has a limit on the QES lous. This limit funtion has l 1 zeros, none of them real. It follows that J = l for the limit funtion. By the first part of Theorem 2 whih we proved in setion 5, is real for integer J on the non-qes part of the spetral lous, so we onlude that the whole image of the Nevanlinna map t F(L m,l,t) belongs to Z J (R) with J = l. In the limit when ր, that is t = /a ր, we obtain a Nevanlinna funtion for the harmoni osillator with m zeros. Trees of type 2 (with l < ) are possible in this ase if and only if k = 2m + 1 and l 1 is odd. Suh line omplexes will be alled L m,l. The only possible limit on the spetral lous is. This orresponds to an elementary seond solution of the differential equation in (3) (solution whih is linearly independent of the eigenfuntion). These points are marked in Fig. 15. Thus the hart L m,l, l 1 orresponds to the hart L m, l via Darboux transform. We have J = l < in this ase. Case R. < a <, Figs

18 a Fig. 7. Cell deomposition Φ in ase R. R,1 R,-1 a a Fig. 8. R m,l omplexes. Trees of type (with l = ) are impossible in this ase. Trees X k,l of the type 1 (with l > ) are possible with k = 2m + 1 and l positive odd. We all the omplex R m,l. Degeneration a, t = /a + is possible, and the limit belongs to the QES spetral lous. The limit Nevanlinna funtion has l 1 zeros, so J = l for this limit funtion. Again, using the first part of Theorem 2, we onlude that J = l on the whole image of the Nevanlinna map F(R m,l,.) Degeneration + (that is t = /a +) gives a Nevanlinna funtion for the harmoni osillator with J 1 + m zeros. As L m,l and R m,l have ommon limit of the QES spetral lous, their Nevanlinna images form a single urve. The results in 18

19 [13] about QES spetral lous together with ounting of zeros of degeneration to harmoni osillator show that L m,l lies on the left and R m,l lies on the right from the intersetion point with the QES spetral lous. See Figs. 11, 13, where Nevanlinna images of L m,l and R m,l are shown with the solid lines, and the QES lous with the dotted line. Trees X k,l of type 2 (with l < ) give line omplexes when k = 2m + 1 and l negative odd. We all these omplexes R m,l. Degeneration is possible on the spetral lous. These harts orrespond to the harts R m, l by Darboux transform. Thus ases L and R and trees of types 1 and 2 over all ases when J is odd. We onlude from this: a) J is onstant on the Nevanlinna images of L m,l and R m,l, namely J = l. b) Even values of J must be overed by the remaining trees from our lassifiation. Case E. < < a, see Figs a Fig. 9. Cell deomposition Φ in ase E. 19

20 E 1,2 E 1, E 1,-2 a a a Fig. 1. E m,l omplexes. Trees X k, of type have parameter k = 2m+1, m, and we denote the orresponding line omplex by E m,. This line omplex represents Nevanlinna funtions from setion 5 whih orrespond to J =. Trees X k,l of type 2 have k = 2m + 1 and l negative even. We all the orresponding line omplex E m,l. No degeneration on the spetral lous is possible. We know that for even J the non-qes spetral lous onsists of graphs of funtions. Degeneration as and a gives a Nevanlinna funtion for the harmoni osillator with m zeros. Thus these trees parametrize the whole spetrum for negative even J. Trees X k,l of type 1 have k = 2m + 1 and l positive even. We all the orresponding omplex E m,l. It orresponds to E m, l by the Darboux transform. Degeneration as and a gives a Nevanlinna funtion for the harmoni osillator with m and m + l zeros, respetively. These arguments show that real and a orrespond to integer J and that we obtain a parametrization of the whole non-qes spetral lous in this way. This ompletes the proof of Theorem 2. The parametrization of the spetral lous for integer J is represented in Figs The symbols of line omplexes are written below the orresponding urves. The QES spetral lous is shown with dotted lines. It was 2

21 parametrized with different ell deompositions (not with line omplexes!) in [13]. Symbols X k,l in the figures refer to the harts on the QES lous desribed in [13]. X 3, L 1,2 X 2, L 1,1 R 1,3 R1,2 X 1, R1,1 R1, L1, X, Fig. 11. Z 1 (R). E 2,3 E 2,2 E 2,1 E 2, X -1, Fig. 12. Z 2 (R). 21

22 X 3,1 L 3,2 X 2,1 L 3,1 R3,3 R 3,2 X 1,1 R 3,1 R 3, L 3, X X,1-2, Fig. 13. Z 3 (R). E,3 E,2 E,1 E, Fig. 14. Z (R). 22

23 R -1,3 L -1,2 L -1,1 R -1,2 R -1,1 R -1, L -1, Fig. 15. Z 1 (R) superimposed with Z QES 1 (R) (thin dotted line). Referenes [1] I. Bakken, A multiparameter eigenvalue problem in the omplex plane, Amer. J. Math. 99 (1977), no. 5, [2] C. Bender and S. Boetther, Quasi-exatly solvable quarti potential, J. Phys. A 31 (1998), no. 14, L273 L277, arxiv:physis/9817. [3] C. Bender and S. Boetther, Real spetra in non-hermitian Hamiltonians having PT symmetry, Phys. Rev. Lett., 8 (1998) [4] C. Bender, D. Brody, J-H. Chen, H. Jones, K. Milton and C. Ogilvie, Equivalene of a omplex PT-symmetri quarti Hamiltonian and a Hermitian quarti Hamiltonian with an anomaly, arxiv:hep-th/6566v2. [5] V. Buslaev and V. Grehi, Equivalene of unstable anharmoni osillators and double wells, J. Phys. A, 26 (1993) [6] M. Crum, Assoiated Sturm Liouville systems, Quart. J. Math., 6 (1955) [7] E. Delabaere, D. T. Trinh, Spetral analysis of the omplex ubi osillator, J. Phys. A 33 (2), no. 48,

24 [8] P. Dorey, C. Dunning and R. Tateo, The ODE/IM orrespondene. J. Phys. A 4 (27), no. 32, R25 R283. [9] A. Eremenko, A. Gabrielov and B. Shapiro, Zeros of eigenfuntions of some anharmoni osillators, Ann. Inst. Fourier, Grenoble, 58, 2 (28) [1] A. Eremenko and A. Gabrielov, Analyti ontinuation of eigenvalues of a quarti osillator, Comm. Math. Phys., v. 287, No. 2 (29) [11] A. Eremenko and A. Gabrielov, Singular perturbation of polynomial potentials in the omplex domain with appliations to PT-symmetri families, to appear in Mosow Math. J., arxiv: [12] A. Eremenko and A. Gabrielov, Quasi-exatly solvable quarti: elementary integrals and asymptoti, J. Phys. A: Math. Theor. 44 (211) [13] A. Eremenko and A. Gabrielov, Quasi-exatly solvable quarti: real algebrai spetral lous, arxiv: [14] J. Gibbons and A. P. Veselov, On the rational monodromy-free potentials with sexti growth, J. Math. Phys. 5 (29), no. 1, 13513, 25 pp. [15] A. A. Goldberg and I. V. Ostrovskii, Distribution of values of meromorphi funtions, AMS, Providene RI, 28. [16] V. Grehi, M. Maioli and A. Martinez, Padé summability of the ubi osillator, J. Math. Phys. A: Math. Theor. 42 (29) 42528, 17pp. [17] D. Masoero, Y-System and Deformed Thermodynami Bethe Ansatz, Lett. Math. Phys. 94 (21), [18] R. Nevanlinna, Über Riemannshe Flähen mit endih vielen Windungspunkten, Ata Math., 58 (1932) [19] K. Shin, Eigenvalues of PT-symmetri osillators with polynomial potentials, J. Phys. A 38 (25), no. 27, [2] K. Shin, On the reality of the eigenvalues for a lass of PT-symmetri osillators, Comm. Math. Phys. 229 (22), no. 3,

25 [21] Y. Sibuya, Global theory of a seond order linear ordinary differential equation with a polynomial oeffiient, North-Holland, Amsterdam; Amerian Elsevier, NY, [22] L. Volkovyski, Converging sequenes of Riemann surfaes, Mat. Sbornik, 23 (65) N3 (1948) [23] J. Zinn-Justin and U. Jentshura, Imaginary ubi perturbation: numerial and analyti study, J. Phys. A, 43 (21) Department of mathematis, Purdue University, West Lafayette, IN 4797 eremenko@math.purdue.edu agabriel@math.purdue.edu 25