On spectral polynomials of the Heun equation. I

Shapiro, Boris; Tater, Miloš

doi:10.1016/j.jat.2009.09.005

Cited by 28 publications

(50 citation statements)

References 7 publications

(22 reference statements)

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“…This is the spirit of some works already present in the literature, e.g. [23,29,37,38]. It is expected that the pairs (R, µ) satisfying (3.17) depend on continuous parameters, but as it follows from the present work, only for a finite number of pairs (R, µ) the measure µ is an equilibrium measure (of some admissible contour) in the external field ϕ = Re V .…”

Section: 3supporting

confidence: 63%

S-curves in polynomial external fields

Kuijlaars

Silva

2015

Journal of Approximation Theory

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Abstract. Curves in the complex plane that satisfy the S-property were first introduced by Stahl and they were further studied by Gonchar and Rakhmanov in the 1980s. Rakhmanov recently showed the existence of curves with the S-property in a harmonic external field by means of a max-min variational problem in logarithmic potential theory. This is done in a fairly general setting, which however does not include the important special case of an external field Re V where V is a polynomial of degree ≥ 2. In this paper we give a detailed proof of the existence of a curve with the S-property in the external field Re V within the collection of all curves that connect two or more pre-assigned directions at infinity in which Re V → +∞. Our method of proof is very much based on the works of Rakhmanov on the max-min variational problem and of Martínez-Finkelshtein and Rakhmanov on critical measures.

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Section: 3supporting

confidence: 63%

S-curves in polynomial external fields

Kuijlaars

Silva

2015

Journal of Approximation Theory

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“…Much of the research in the past several years has focused on the asymptotic properties of the zeros of Stieltjes and Van Vleck polynomials as the degree of the corresponding Stieltjes polynomials tends toward infinity [4,3,16,14,15,20,21]. Interlacing theorems such as the one above are interesting, not only for what they tell us about the zeros for a finite degree, but also because they help us to understand such asymptotic limits.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On the distribution and interlacing of the zeros of Stieltjes polynomials

Bourget

McMillen

2010

Proc. Amer. Math. Soc.

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Abstract. Polynomial solutions to the generalized Lamé equation, the Stieltjes polynomials, and the associated Van Vleck polynomials have been studied since the 1830's, beginning with Lamé in his studies of the Laplace equation on an ellipsoid, and in an ever widening variety of applications since. In this paper we show how the zeros of Stieltjes polynomials are distributed and present two new interlacing theorems. We arrange the Stieltjes polynomials according to their Van Vleck zeros and show, firstly, that the zeros of successive Stieltjes polynomials of the same degree interlace, and secondly, that the zeros of certain Stieltjes polynomials of successive degrees interlace.

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“…In 1878 Heine stated [55] that there are at most [56,57] that there exists an N 0 such that for all N > N 0 we have exactly m solutions (counted with multiplicity). Physically we interpret the result (5.23) as the statement that each vacuum corresponds to one of the possible ways of distributing the N eigenvalues of the matrix X between the n critical points of the one-field superpotential W (z).…”

Section: Heine-stieltjes and Van Vleck Polynomialsmentioning

confidence: 99%

Twistorial topological strings and a $\mathrm{tt}^*$ geometry for $\mathcal{N} = 2$ theories in 4d

Cecotti

Vafa

Neitzke

2016

Advances in Theoretical and Mathematical Physics

113

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We define twistorial topological strings by considering tt * geometry of the 4d N = 2 supersymmetric theories on the Nekrasov-Shatashvili 1 2 Ω background, which leads to quantization of the associated hyperKähler geometries. We show that in one limit it reduces to the refined topological string amplitude. In another limit it is a solution to a quantum Riemann-Hilbert problem involving quantum Kontsevich-Soibelman operators. In a further limit it encodes the hyperKähler integrable systems studied by GMN. In the context of AGT conjecture, this perspective leads to a twistorial extension of Toda. The 2d index of the 1 2 Ω theory leads to the recently introduced index for N = 2 theories in 4d. The twistorial topological string can alternatively be viewed, using the work of Nekrasov-Witten, as studying the vacuum geometry of 4d N = 2 supersymmetric theories on T 2 × I where I is an interval with specific boundary conditions at the two ends. arXiv:1412.4793v1 [hep-th] 15 Dec 2014 42 5.5 Abelian tt * geometries in (R 2 × S 1 ) r 43 5.5.1 Solving Abelian tt * equations in (R 2 × S 1 ) r 44 5.5.2 Constructing new Abelian tt * geometries from old ones 48 5.6 Example: the generalized Penner model 49 5.7 Example: the Double Penner model 50 6 Twistorial Topological Strings for the N = 2 SQED (conifold B-model) 52 6.1 The dictionary between tt * and GMN hyperKähler geometries 52 6.2 Twistorial Double Gamma Function 53 6.3 Relation with SQED amplitudes in 12 Ω-background 55 6.4 The higher-dimensional tt * geometry 56 6.5 SQED twistorial amplitudes and the 1 2 Ω-background W eff (a e , 1 ) 57 6.6 The θ-limit tt * geometry 58 6.7 θ-limit vs. the quantum KS wall crossing formula 597 Twistorial invariant aspects of the tt * geometry 60 7.1 C-limit: the 4d AMNP index vs. the CFIV index 61 7.1.1 4d interpretation of Q C 62 7.1.2 Q C for N = 2 SQED 64 7.2 Twistorial Liouville amplitudes 65 7.2.1 The tt * metric: the twistorial Υ-function 678 The C-limit 68 8.1 The C-limit amplitude in SQED 69 8.2 Review on the X γ 70 8.3 Prequantization and Ψ 72 8.4 Comparing Ψ with ψ C 74 8.5 Ψ as a generating function 75 8.6 Deriving the generating function 76 9 Concluding Remarks 78 A On the R → 0 limit of the connection 79 B Solving the q-TBA equation for Argyres-Douglas models 82 C β-deformed Quiver Matrix LG Models (exact twistorial ADE Toda amplitudes) 85 C.1 The models 86 C.2 The Gaudin model, the Mukhin-Varchenko conjectures, and tt * 87 -1 -C.3 Explicit tt * amplitudes for the A r quiver matrix LG models 91

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On spectral polynomials of the Heun equation. I

Cited by 28 publications

References 7 publications

S-curves in polynomial external fields

S-curves in polynomial external fields

On the distribution and interlacing of the zeros of Stieltjes polynomials

Twistorial topological strings and a $\mathrm{tt}^*$ geometry for $\mathcal{N} = 2$ theories in 4d

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