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...ation of eigenvalues as parameters vary in the complex plane was made for the first time. With such boundary conditions, the problem has an infinite discrete spectrum and eigenvalues tend to infinity =-=[20]-=-. To each eigenvalue corresponds a one-dimensional eigenspace. Suppose that the polynomial potential depends analytically on a parameter a ∈ Cn. Then the spectral locus Z is defined as the set of all ...

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...s an irreducible component of ZJ . It is not known whether ZJ\Z QES J is irreducible. A similar phenomenon occurs in degree 6: there are one-parametric families of even quasi-exactly solvable sextics =-=[22, 23]-=-, and for each such family the quasi-exactly solvable part of the spectral locus is a smooth irreducible algebraic curve [9]. When J → ∞, an appropriate rescaling of ZQESJ tends to the spectral locus ...

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...tely many level crossing points with bk < 0 and real λk. We have bk ∼ −((3/4)pik) 2/3, k →∞. The only known general result of reality of eigenvalues of PT -symmetric operators is a theorem of K. Shin =-=[19]-=-, which for our quartic of type II implies that all eigenvalues are real if J ≤ 0. We have the following extensions of this result. Theorem 8 [13] For every positive integer J , all non-QES eigenvalue...

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...inct. In the opposite direction: if we have a meromorphic function inC without critical points and with finitely many asymptotic values, then it satisfies a Schwarz equation whose RHS is a polynomial =-=[18]-=-. The degree of this polynomial is the number of asymptotic tracts minus 2. (Asymptotic tracts are in one-to-one correspondence with the logarithmic singularities of the inverse function). Asymptotic ...

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...s an irreducible component of ZJ . It is not known whether ZJ\Z QES J is irreducible. A similar phenomenon occurs in degree 6: there are one-parametric families of even quasi-exactly solvable sextics =-=[22, 23]-=-, and for each such family the quasi-exactly solvable part of the spectral locus is a smooth irreducible algebraic curve [9]. When J → ∞, an appropriate rescaling of ZQESJ tends to the spectral locus ...

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...trary degree if we use all coefficients as parameters [14, 1, 2]. However if we consider a subfamily of all polynomials of given degree, then the spectral locus can be reducible in an interesting way =-=[3]-=-. One example is the family of quasi-exactly solvable quartics LJ − y′′ + (z4 − 2bz2 + 2Jz)y = λy, y(re±pii/3)→ 0, r → +∞. (2) When J is a positive integer, this problem has J elementary eigenfunction...

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...bic. Consider the PT-symmetric quartic family (3) of type I: It is equivalent to the PT -symmetric family −w′′ + (z4 + az2 + icz)w = λw, w(±∞) = 0, studied by Bender, et al [3] and Delabaere and Pham =-=[7]-=-. Theorem 5 [10] The real spectral locus of (3) consists of disjoint smooth analytic properly embedded surfaces Sn ⊂ R 3, n ≥ 0, homeomorphic to a punctured disk. For (a, c, λ) ∈ Sn, the eigenfunction...

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...d function λ(a) defined by F (a, λ) = 0 has the following property: its only singularities are algebraic ramification points, and there are finitely many of them over every compact set in the a-space =-=[8]-=-. Next we discuss connectedness of the spectral locus. Theorem 1 [9] For the cubic oscillator − y′′ + (z3 − az)y = −λy, y(±i∞) = 0, (1) the spectral locus is a smooth irreducible curve in C2. 2 Theore...

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...rs were studied extensively from the very beginning of quantum mechanics, mostly by perturbative methods. Cubic oscillator arises in quantum field theory [24] and in the theory of Painlevé equations =-=[15, 16]-=-. ∗Both authors are supported by NSF grant DMS-1067886. 1 We normalize the equation by an affine transformation of the independent variable to obtain: P (z) = zd +O(zd−2), z →∞. Every solution y of Ly...

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... these sectors. Those λ for which such y 6= 0 exists are called eigenvalues, and the corresponding y’s eigenfunctions. The interest of physicists to such eigenvalue problems originates from the paper =-=[4]-=- where a systematic study of analytic continuation of eigenvalues as parameters vary in the complex plane was made for the first time. With such boundary conditions, the problem has an infinite discre...

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...)→ 0, r → +∞. (4) The real spectral locus Z(R) is the subset of the spectral locus Z which consists of points with real coordinates. We begin with the cubic PT-symmetric spectral locus (1). Theorem 4 =-=[10]-=- For every integer n ≥ 0, there exists a simple curve Γn ⊂ R 2, that is the image of a proper analytic embedding of a line, and which has these properties: (i) For every (a, λ) ∈ Γn problem (1) has an...

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...its only singularities are algebraic ramification points, and there are finitely many of them over every compact set in the a-space [8]. Next we discuss connectedness of the spectral locus. Theorem 1 =-=[9]-=- For the cubic oscillator − y′′ + (z3 − az)y = −λy, y(±i∞) = 0, (1) the spectral locus is a smooth irreducible curve in C2. 2 Theorem 2 [8] For the even quartic oscillator −y′′ + (z4 + az2)y = λy, y(±...

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...rs were studied extensively from the very beginning of quantum mechanics, mostly by perturbative methods. Cubic oscillator arises in quantum field theory [24] and in the theory of Painlevé equations =-=[15, 16]-=-. ∗Both authors are supported by NSF grant DMS-1067886. 1 We normalize the equation by an affine transformation of the independent variable to obtain: P (z) = zd +O(zd−2), z →∞. Every solution y of Ly...

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... irreducible curves in C2, one corresponding to even eigenfunctions, another to odd ones. These theorems can be generalized to polynomials of arbitrary degree if we use all coefficients as parameters =-=[14, 1, 2]-=-. However if we consider a subfamily of all polynomials of given degree, then the spectral locus can be reducible in an interesting way [3]. One example is the family of quasi-exactly solvable quartic...

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... R≥0, (a, λ) 7→ a is a 2-to-1 covering. (iv) For a ≥ 0, (a, λ) ∈ Γn and (a, µ) ∈ Γn+1 imply µ > λ. The following computer-generated plot of the real spectral locus of (1) is taken from Trinh’s thesis =-=[21]-=-. Theorem 4 rigorously establishes some features of this picture. At the points whose abscissas are marked in Fig. 1, pairs of real eigenvalues collide and escape to the complex plane. Theorem 4 prove...

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... another interesting feature of the real spectral locus is present: for some parameter values the QES spectral locus crosses the rest of the spectral locus. This is called “level crossing”. Theorem 7 =-=[11]-=- The points (b, λ) ∈ ZQESJ where the level crossing occurs are the intersection points of ZQESJ with Z−J . For each J ≥ 1 there are 6 infinitely many such points, in general, complex. When J is odd, t...

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...he PT-symmetric quartic family of the type II is more complicated, due to the presence of the QES spectrum. Let ZQESJ (R) be the real QES spectral locus of the operator LJ , defined in (2). Theorem 6 =-=[12]-=- For J = n+ 1 > 0, ZQESn+1 (R) consists of [n/2] + 1 disjoint analytic curves Γn,m, 0 ≤ m ≤ [n/2]. For (b, λ) ∈ Γn,m, the eigenfunction has n zeros, n− 2m of them real. If n is odd, then b → +∞ on bot...

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...s d = 3 and d = 4. Cubic and quartic oscillators were studied extensively from the very beginning of quantum mechanics, mostly by perturbative methods. Cubic oscillator arises in quantum field theory =-=[24]-=- and in the theory of Painlevé equations [15, 16]. ∗Both authors are supported by NSF grant DMS-1067886. 1 We normalize the equation by an affine transformation of the independent variable to obtain:...

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... irreducible curves in C2, one corresponding to even eigenfunctions, another to odd ones. These theorems can be generalized to polynomials of arbitrary degree if we use all coefficients as parameters =-=[14, 1, 2]-=-. However if we consider a subfamily of all polynomials of given degree, then the spectral locus can be reducible in an interesting way [3]. One example is the family of quasi-exactly solvable quartic...

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...es of PT -symmetric operators is a theorem of K. Shin [19], which for our quartic of type II implies that all eigenvalues are real if J ≤ 0. We have the following extensions of this result. Theorem 8 =-=[13]-=- For every positive integer J , all non-QES eigenvalues of LJ are real. Theorem 9 [13] All eigenvalues of LJ are real for every real J ≤ 1 (not necessarily integer). E0,0 E 0,1 E 0,2 E 0,3 Fig. 3. Z0(...

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... irreducible curves in C2, one corresponding to even eigenfunctions, another to odd ones. These theorems can be generalized to polynomials of arbitrary degree if we use all coefficients as parameters =-=[14, 1, 2]-=-. However if we consider a subfamily of all polynomials of given degree, then the spectral locus can be reducible in an interesting way [3]. One example is the family of quasi-exactly solvable quartic...