Citation Context

...sume that c = eiβ , 0 ≤ β ≤ π, where the points 0 and π can be identified. The asymptotic value c is called the Nevanlinna 2 parameter. There is a simple relation between c and the Stokes multipliers =-=[11, 8]-=-. The sectors Sj correspond to logarithmic singularities of the inverse function f−1. Thus f−1 has 6 logarithmic singularities that lie over 4 points if c 6= c, or over 3 points if c = c. The map (b, ...

Citation Context

... always understood as a residue modulo 6. Function f has asymptotic values∞, 0, c, 0, c, 0 in the sectors S0, . . . S5, where c ∈ C. It is known that f must have at least 3 distinct asymptotic values =-=[9]-=-, so c 6= 0,∞. Function f is defined up to multiplication by a non-zero real number, so we can always assume that c = eiβ , 0 ≤ β ≤ π, where the points 0 and π can be identified. The asymptotic value ...

Citation Context

...the Nevanlinna parametrization. MSC: 81Q05, 34M60, 34A05. Keywords: one-dimensional Schrödinger operators, quasi-exact solvability, PT-symmetry, singular perturbation. Following Bender and Boettcher =-=[3]-=-, we consider the eigenvalue problem in the complex plane w′′ + (ζ4 + 2bζ2 + 2iJζ + λ)w = 0, w(te−πi/2±πi/3)→ 0, t→ +∞, (1) where J is a positive integer. This problem is quasi-exactly solvable [3]: t...

Citation Context

... a certain cell decomposition of the plane described below. Once f is known, b and λ are found from the formula f ′′′ f ′ − 3 2 ( f ′′ f ′ )2 = −2(z4 − 2bz2 + 2Jz − λ). (3) Now we describe, following =-=[4]-=-, the cell decompositions needed to recover f from c. Suppose first that c /∈ R. 4 c - c 2 1 4 c - c -1-2 g 4 g c - g’ c - g’c g c g 4 ’ L L’ Fig. 1. Cell decompositions Φ and Φ′ of the sphere (solid ...

Citation Context

...) ∈ Γn,m+1 and b is sufficiently large, then µ > λ. This theorem establishes the main features of ZQESn+1 (R) which can be seen in the computer-generated figure in [3]. Similar results were proved in =-=[5]-=- for two other PT-symmetric eigenvalue problems. Our theorem parametrizes all polynomials P of degree 4 with the property that the differential equation y′′+Py = 0 has a solution with n zeros, n−2m of...

Citation Context

...R) is ZJ ∩R2. The quasi-exactly solvable spectral locus ZQESJ is the set of all (b, λ) ∈ ZJ for which there exists an elementary solution y of (2). This is a smooth irreducible algebraic curve in C2, =-=[1, 2]-=-. In this paper we describe ZQESJ (R) = Z QES J ∩R2. We prove a result announced in [6]: Theorem 1. For n ≥ 0, ZQESn+1 (R) consists of [n/2] + 1 disjoint analytic curves Γn,m, 0 ≤ m ≤ [n/2] (analytic ...

Citation Context

...sume that c = eiβ , 0 ≤ β ≤ π, where the points 0 and π can be identified. The asymptotic value c is called the Nevanlinna 2 parameter. There is a simple relation between c and the Stokes multipliers =-=[11, 8]-=-. The sectors Sj correspond to logarithmic singularities of the inverse function f−1. Thus f−1 has 6 logarithmic singularities that lie over 4 points if c 6= c, or over 3 points if c = c. The map (b, ...

Citation Context

...R) is ZJ ∩R2. The quasi-exactly solvable spectral locus ZQESJ is the set of all (b, λ) ∈ ZJ for which there exists an elementary solution y of (2). This is a smooth irreducible algebraic curve in C2, =-=[1, 2]-=-. In this paper we describe ZQESJ (R) = Z QES J ∩R2. We prove a result announced in [6]: Theorem 1. For n ≥ 0, ZQESn+1 (R) consists of [n/2] + 1 disjoint analytic curves Γn,m, 0 ≤ m ≤ [n/2] (analytic ...

Citation Context

...J for which there exists an elementary solution y of (2). This is a smooth irreducible algebraic curve in C2, [1, 2]. In this paper we describe ZQESJ (R) = Z QES J ∩R2. We prove a result announced in =-=[6]-=-: Theorem 1. For n ≥ 0, ZQESn+1 (R) consists of [n/2] + 1 disjoint analytic curves Γn,m, 0 ≤ m ≤ [n/2] (analytic embeddings of R to R2). For (b, λ) ∈ Γn,m, the eigenfunction has n zeros, n− 2m of them...

Citation Context

...her PT-symmetric eigenvalue problems. Our theorem parametrizes all polynomials P of degree 4 with the property that the differential equation y′′+Py = 0 has a solution with n zeros, n−2m of them real =-=[10, 7, 5]-=-. Suppose that (b, λ) ∈ ZQESJ (R). Then the corresponding eigenfunction y of (2) can be always chosen real. Let y1 be a real solution of the differential equation in (2) normalized by y1(x) → 0 as x →...

Citation Context

...heir asymptotic values and cell decompositions. Assume that one vertex v0 of Ψ is placed at z = 0 and normalize so that f ′(0) = 1. The class of normalized functions 7 obtained in this way is compact =-=[12]-=-. Let fν → f0 be a converging sequence.2 The 1-skeletons of the corresponding cell decompositions Ψ(ν) converge to the 1-skeleton of the cell decomposition Ψ(0) as embedded graphs with a marked vertex...

Citation Context

...her PT-symmetric eigenvalue problems. Our theorem parametrizes all polynomials P of degree 4 with the property that the differential equation y′′+Py = 0 has a solution with n zeros, n−2m of them real =-=[10, 7, 5]-=-. Suppose that (b, λ) ∈ ZQESJ (R). Then the corresponding eigenfunction y of (2) can be always chosen real. Let y1 be a real solution of the differential equation in (2) normalized by y1(x) → 0 as x →...