Abstract
It has been recently realized that, in the case of polynomial potentials, the exact WKB method can be reformulated in terms of a system of TBA equations. In this paper we study this method in various examples. We develop a graphical procedure due to Toledo, which provides a fast and simple way to study the wall-crossing behavior of the TBA equations. When complemented with exact quantization conditions, the TBA equations can be used to solve spectral problems exactly in Quantum Mechanics. We compute the quantum corrections to the all-order WKB periods in many examples, as well as the exact spectrum for many potentials. In particular, we show how this method can be used to determine resonances in unbounded potentials.
Highlights
The one-dimensional time independent Schrödinger equation, describing a non-relativistic particle with energy E in a potential V (q),− 2ψ (q) + (V (q) − E)ψ(q) = 0 (1.1)has always been an inexhaustible source of useful toy models in physics
Later, when we lose the ordering property of the turning points (2.29), we will find out that new relevant cycles are appearing into thermodynamic Bethe ansatz (TBA) systems and having two indices will be helpful in order to link together arbitrary turning points instead of limiting ourselves to consecutive ones
All the terms are correctly predicted by our two diagrammatic rules, except two terms with coefficient ±2. If we repeat this exercise with other TBA systems, the same phenomenon occurs each time we have to wall-cross two successive edges attached to the same vertex, creating intersecting graphs
Summary
The one-dimensional time independent Schrödinger equation, describing a non-relativistic particle with energy E in a potential V (q),. In order to understand what motivated [15], a good starting point is the seminal works of Gaiotto, Moore and Neitzke [17, 18] in the context of four-dimensional N = 2 gauge theories They are deriving integral equations solving a Riemann-Hilbert problem in term of the X map defined in [17], the discontinuities of which are given by Kontsevich-Soibelman symplectomorphisms when crossing a BPS ray. More closely related to the TBA system of interest in this paper, is the study of minimal surfaces in AdS3 delimited by a polygonal closed contour on the boundary This problem simplifies and reduces to the Z2 projection of a SU(2) Hitchin problem arising in [17, 18] in the context of four-dimensional N = 2 gauge theories, as described in the previous paragraph. In appendix D, we describe how to extract the bounded or resonant spectrum from a Hamiltonian with polynomial potential by expressing it in the harmonic oscillator basis
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