Solvable spectral problems from 2d CFT and N=2 gauge theories

TL;DR

This paper explores how dualities between 2d conformal field theories, N=2 gauge theories, and integrable systems can be used to analyze spectral problems of Schrödinger operators with various complex potentials.

Contribution

It summarizes recent results on applying 2d/4d dualities to solve spectral problems for Schrödinger operators with special potentials, highlighting new connections and open questions.

Findings

Established links between 2d CFT, gauge theories, and spectral problems.

Applied dualities to Mathieu-type, PT-symmetric, and elliptic potentials.

Outlined open problems for future research.

Abstract

The so-called 2d/4d correspondences connect two-dimensional conformal field theory (2d CFT), N=2 supersymmetric gauge theories and quantum integrable systems. The latter in the simplest case of the SU(2) gauge group are nothing but the quantum-mechanical systems. In the present article we summarize our recent results and list open problems concerning an application of the aforementioned dualities in the studies of spectral problems for some Schrodinger operators with Mathieu-type periodic, periodic PT-symmetric and (Heun's) elliptic potentials.