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 $$2\times 2$$ 2 × 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 $$\mathrm {III}_3$$ 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 Kyiv formula for the isomonodromic tau functions, we obtain the spectrum of such operators in terms of self-dual Nekrasov functions ($$\epsilon _1+\epsilon _2=0$$ ϵ 1 + ϵ 2 = 0 ). Through blowup relations, we also find Nekrasov–Shatashvili type of quantizations ($$\epsilon _2=0$$ ϵ 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 $$\epsilon _2\rightarrow 0$$ ϵ 2 → 0 limit of the blowup relations from the regularized action functional and CFT arguments.

Exponential Networks, WKB and Topological String

“…In this case the WKB solution to the quantum curve is encoded in the Nekrasov-Shatashvili phase ϵ 2 = 0, ϵ 1 = ℏ [1] while the non-perturbative corrections are encoded in the ϵ 2 = −ϵ 1 = 1 ℏ phase [26,54,84,85]. One may argue in favour of a connection between the NS and the GV phase using blowup equation as in [13,51,71,79,89]. However this is much more subtle than for the case of the resolved conifold.…”

“…In higher genus geometries it could be that this relation is more complicated. Nevertheless we know that these two Ω background phases are related by blowup equations [51], see also [13,89] and references there. Hence it should be possible to find a relation between the two Borel planes.…”

“…13 Thus that under analytic continuation x → x + 2πi, (3.15) stays fixed and (3.19) is multiplied by e − 2π(x+πi) …”

Perturbative connection formulas for Heun equations

“…This has brought a new surge of interest in the study of equations of Heun type and their applications in the past decade, see e.g. [AGH,BGG,CC21,CN,LeNo,LiNa,NC,PP14,PP17].…”