Quantum spectral problems and isomonodromic deformations

Abstract

We develop a self-consistent approach to study the spectral properties of a class of quantum mechanical operators by using the knowledge about monodromies of $2\times 2$ linear systems (Riemann-Hilbert correspondence). Our technique applies to a variety of problems, though in this paper we only analyse in detail two examples. First we review the case of the (modified) Mathieu operator, which corresponds to a certain linear system on the sphere and makes contact with the Painlevé ${\mathrm{III}}_3$ equation. Then we extend the analysis to the 2-particle elliptic Calogero-Moser operator, which corresponds to a linear system on the torus. By using the Kiev formula for the isomonodromic tau functions, we obtain the spectrum of such operators in terms of self-dual Nekrasov functions (${\epsilon }_1+{\epsilon }_2=0$). Through blowup relations, we also find Nekrasov-Shatashvili type of quantizations (${\epsilon }_2=0$). In the case of the torus with one regular singularity we obtain certain results which are interesting by themselves. Namely, we derive blowup equations (filling some gaps in the literature) and we relate them to the bilinear form of the isomonodromic deformation equations. In addition, we extract the ${\epsilon }_2\to 0$ limit of the blowup relations from the regularized action functional and CFT arguments.

62 pages, minor corrections, references added

References in corpus (11)

• Defects and Quantum Seiberg-Witten Geometry
• Small Instantons, Little Strings and Free Fermions
• Differential equation for four-point correlation function in Liouville field theory and elliptic four-point conformal blocks
• Nekrasov prepotential with fundamental matter from the quantum spin chain
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• Classical conformal blocks from TBA for the elliptic Calogero-Moser system
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• Poles of Painlevé IV Rationals and their Distribution
• Backlund transformation of Painleve III($D_8$) tau function
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• Scalar/Dirac perturbations of the Reissner-Nordström black hole and Painlevé transcendents
• Massive scalar perturbations in Kerr Black Holes: near extremal analysis
• Expansions for semiclassical conformal blocks
• Quantum Representation of Affine Weyl Groups and Associated Quantum Curves
• Painlevé/CFT correspondence on a torus
• Painlevé Kernels and Surface Defects at Strong Coupling