Exact WKB and Abelianization for the $$T_3$$ Equation

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.

“…This particular exercise was already covered (partially 16 ) in [15]. In the recent [31], they are covering the cubic case from the point of view of abelianization.…”

Section: The Cubic Potentialmentioning

confidence: 99%

Section: Jhep07(2021)171mentioning

confidence: 99%

TBA equations and quantization conditions

Emery

2021

“…In the exact-WKB analysis [129,130,[132][133][134][135][136][137][138][139][140][141][142][143][144][145][146], which has been studied mainly by mathematicians, one investigates the properties of solutions to certain differential equations using the Borel summation. 3 Our main interest is in its application to the Schrödinger equation,…”

Section: Jhep12(2020)114mentioning

confidence: 99%

“…In this section, we review the exact WKB method [50,129,130,[132][133][134][135][136][137][138][139][140][141][142][143][144] and the related techniques, including Borel resummation, Stokes curves and monodromy matrices. 9 For simplicity, we focus on the one-dimensional Schrödinger equation, and assume that the potential V (x) doesn't include , i.e.…”

Section: Exact Wkbmentioning

confidence: 99%

On exact-WKB analysis, resurgent structure, and quantization conditions

Sueishi

Kamata

Misumi

et al. 2020

There are two well-known approaches to studying nonperturbative aspects of quantum mechanical systems: saddle point analysis of the partition functions in Euclidean path integral formulation and the exact-WKB analysis based on the wave functions in the Schrödinger equation. In this work, based on the quantization conditions obtained from the exact-WKB method, we determine the relations between the two formalism and in particular show how the two Stokes phenomena are connected to each other: the Stokes phenomenon leading to the ambiguous contribution of different sectors of the path integral formulation corresponds to the change of the “topology” of the Stoke curves in the exact-WKB analysis. We also clarify the equivalence of different quantization conditions including Bohr-Sommerfeld, path integral and Gutzwiller’s ones. In particular, by reorganizing the exact quantization condition, we improve Gutzwiller’s analysis in a crucial way by bion contributions (incorporating complex periodic paths) and turn it into an exact result. Furthermore, we argue the novel meaning of quasi-moduli integral and provide a relation between the Maslov index and the intersection number of Lefschetz thimbles.

“…19 For the rigorous and detailed explanation of exact WKB analysis see [37,38]. WKB parameterization of monodromies is also very well-known in the context of the four-dimensional supersymmetric gauge theories, see [27,39,40] and references therein. The WKB graphs.…”

Section: Monodromies From Exact Wkbmentioning

confidence: 99%

Irregular conformal blocks, Painlevé III and the blow-up equations

Gavrylenko

Marshakov

Stoyan

2020

We study the relation of irregular conformal blocks with the Painlevé III3 equation. The functional representation for the quasiclassical irregular block is shown to be consistent with the BPZ equations of conformal field theory and the Hamilton-Jacobi approach to Painlevé III3. It leads immediately to a limiting case of the blow-up equations for dual Nekrasov partition function of 4d pure supersymmetric gauge theory, which can be even treated as a defining system of equations for both c = 1 and c → ∞ conformal blocks. We extend this analysis to the domain of strong-coupling regime where original definition of conformal blocks and Nekrasov functions is not known and apply the results to spectral problem of the Mathieu equations. Finally, we propose a construction of irregular conformal blocks in the strong coupling region by quantization of Painlevé III3 equation, and obtain in this way a general expression, reproducing c = 1 and quasiclassical c → ∞ results as its particular cases. We have also found explicit integral representations for c = 1 and c = −2 irregular blocks at infinity for some special points.