Exact WKB Method Research Articles

TBA equations and exact WKB analysis in deformed supersymmetric quantum mechanics

We study the spectral problem in deformed supersymmetric quantum mechanics with polynomial superpotential by using the exact WKB method and the TBA equations. We apply the ODE/IM correspondence to the Schrödinger equation with an effective potential deformed by integrating out the fermions, which admits a continuous deformation parameter. We find that the TBA equations are described by the ℤ4-extended ones. For cubic superpotential corresponding to the symmetric double-well potential, the TBA system splits into the two D3-type TBA equations. We investigate in detail this example based on the TBA equations and their analytic continuation as well as the massless limit. We find that the energy spectrum obtained from the exact quantization condition is in good agreement with the diagonalization approach of the Hamiltonian.

An anharmonic alliance: exact WKB meets EPT

Certain quantum mechanical systems with a discrete spectrum, whose observables are given by a transseries in ħ, were shown to admit ħ0-deformations with Borel resummable expansions which reproduce the original model at ħ0 = ħ. Such expansions were dubbed Exact Perturbation Theory (EPT). We investigate how the above results can be obtained within the framework of the exact WKB method by studying the spectrum of polynomial quantum mechanical systems. Within exact WKB, energy eigenvalues are determined by exact quantization conditions defined in terms of Voros symbols aγi\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {a}_{\\gamma_i} $$\\end{document}, γi being their associated cycles, and generally give rise to transseries in ħ. After reviewing how the Borel summability of energy eigenvalues in the quartic anharmonic potential emerges in exact WKB, we extend it to higher order anharmonic potentials with quantum corrections. We then show that any polynomial potential can be ħ0-deformed to a model where the exact quantization condition reads simply aγ = −1 and leads to the EPT Borel resummable series for all energy eigenvalues.

Semiclassical WKB problem for the non-self-adjoint Dirac operator with an analytic rapidly oscillating potential

In this paper we examine the semiclassical behavior of the scattering data of a non-self-adjoint Dirac operator with a rapidly oscillating potential that is complex analytic in some neighborhood of the real line. Some of our results are rigorous and quite general. On the other hand, complete and concrete understanding requires the investigation of the WKB geometry of specific examples. For such detailed computations we use a particular example that has been investigated numerically more than 20 years ago by Bronski and Miller and rely heavily on their numerical computations. Mostly employing the exact WKB method, we provide the complete rigorous uniform semiclassical analysis of the Bohr-Sommerfeld condition for the location of the eigenvalues across unions of analytic arcs as well as the associated norming constants. For the reflection coefficient as well as the eigenvalues near 0 in the spectral plane, we employ instead an older theory that has been developed in great detail by Olver. Our analysis is motivated by the need to understand the semiclassical behavior of the focusing cubic NLS equation with initial data Aexp⁡{iS/ϵ}, in view of the well-known fact discovered by Zakharov and Shabat that the spectral analysis of the Dirac operator enables the solution of the NLS equation via inverse scattering theory.

Painlevé I and exact WKB: Stokes phenomenon for two-parameter transseries

For more than a century, the Painlevé I equation has played an important role in both physics and mathematics. Its two-parameter family of solutions was studied in many different ways, yet still leads to new surprises and discoveries. Two popular tools in these studies are the theory of isomonodromic deformation that uses the exact WKB method, and the asymptotic description of transcendents in terms of two-parameter transseries. Combining methods from both schools of thought, we follow work by Takei and collaborators to formulate complete, two-parameter connection formulae for solutions when they cross arbitrary Stokes lines in the complex plane. These formulae allow us to study Stokes phenomenon for the full two-parameter family of transseries solutions. In particular, we recover the exact expressions for the Stokes data that were recently found by Baldino, Schwick, Schiappa and Vega and compare our connection formulae to theirs. We also explain several ambiguities in relating transseries parameter choices to actual Painlevé transcendents, study the monodromy of formal solutions, and provide high-precision numerical tests of our results.

Exact WKB and the quantum Seiberg-Witten curve for 4d N = 2 pure SU(3) Yang-Mills. Abelianization

We investigate the exact WKB method for the quantum Seiberg-Witten curve of 4d N = 2 pure SU(3) Yang-Mills in the language of abelianization. The relevant differential equation is a third-order equation on ℂℙ1 with two irregular singularities. We employ the exact WKB method to study the solutions to such a third-order equation and the associated Stokes phenomena. We also investigate the exact quantization condition for a certain spectral problem. Moreover, exact WKB analysis leads us to consider new Darboux coordinates on a moduli space of flat SL(3,ℂ)-connections. In particular, in the weak coupling region we encounter coordinates of the higher length-twist type generalizing Fenchel-Nielsen coordinates. The Darboux coordinates are conjectured to admit asymptotic expansions given by the formal quantum periods series and we perform numerical analysis supporting this conjecture.

TBA equations and quantization conditions

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.

Landau–Zener formula in a “non-adiabatic” regime for avoided crossings

We study a two-level transition probability for a finite number of avoided crossings with a small interaction. Landau–Zener formula, which gives the transition probability for one avoided crossing as \(e^{-\pi \frac{\varepsilon ^{2}}{h}}\), implies that the parameter h and the interaction \(\varepsilon \) play an opposite role when both tend to 0. The exact WKB method produces a generalization of that formula under the optimal regime \(\scriptstyle {\frac{h}{\varepsilon ^2}}\) tends to 0. In this paper, we investigate the case \(\scriptstyle {\frac{\varepsilon ^2}{h}}\) tends to 0, called “non-adiabatic” regime. This is done by reducing the associated Hamiltonian to a microlocal branching model which gives us the asymptotic expansions of the local transfer matrices.

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

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.

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

We describe the exact WKB method from the point of view of abelianization, both for Schr\"odinger operators and for their higher-order analogues (opers). The main new example which we consider is the "$T_3$ equation," an order $3$ equation on the thrice-punctured sphere, with regular singularities at the punctures. In this case the exact WKB analysis leads to consideration of a new sort of Darboux coordinate system on a moduli space of flat $\mathrm{SL}(3)$-connections. We give the simplest example of such a coordinate system, and verify numerically that in these coordinates the monodromy of the $T_3$ equation has the expected asymptotic properties. We also briefly revisit the Schr\"odinger equation with cubic potential and the Mathieu equation from the point of view of abelianization.

Numerical verification of the exact WKB formula for the generalized Landau-Zener-Stueckelberg problem

The WKB analysis for higher order differential equations had not been developed until recent progress of the so-called exact WKB method, so it was difficult to find WKB solutions for multistate nonadiabatic models in general, whereas enormous efforts were made especially for the study of generalized Landau-Zener models. Under such circumstances Aoki-Kawai-Takei provided a closed expression for the $S$ matrix of a time-dependent multistate nonadiabatic system, in which diabatic potential curves are given as polynomial functions of time and they all intersect with each other. Their result is completely explicit, and the model covers much wider classes of systems than multistate nonadiabatic models studied so far. Here we numerically show how their exact WKB formula works as long as crossing points of diabatic potential curves are well separated. The model studied includes the cases where diabatic potential functions are nonlinear, not like generalized Landau-Zener models. We also discuss the situations where virtual turning points and new Stokes curves, new components in the Stokes geometry appearing only for higher order differential equations, come into play. In particular, we examine how the presence of new Stokes curves affects final transition probabilities using a couple of concrete examples.

TBA equations for the Schrödinger equation with a regular singularity

We derive the thermodynamic Bethe ansatz (TBA) equations for the Schrödinger equation with an arbitrary polynomial potential and a regular singular (simple and double pole) term. The TBA equations provide a non-trivial generalization of the ODE/IM correspondence and also give a solution for the Riemann–Hilbert problem in the exact WKB method. We study the TBA equations in detail for the linear and the harmonic oscillator potentials together with inverse and centrifugal terms. As an application, we also compute numerically the Voros spectrum for these potentials using the Bohr–Sommerfeld quantization condition.

Semiclassical WKB problem for the non-self-adjoint Dirac operator with analytic potential

In this paper, we examine the semiclassical behavior of the scattering data of a non-self-adjoint Dirac operator with analytic potential decaying at infinity. In particular, employing the exact WKB method, we provide the complete rigorous uniform semiclassical analysis of the reflection coefficient and the Bohr–Sommerfeld condition for the location of the eigenvalues. Our analysis has some interesting consequences concerning the focusing cubic nonlinear Schrödinger (NLS) equation in view of the well-known fact discovered by Zakharov and Shabat that the spectral analysis of the Dirac operator is the basis of the solution of the NLS equation via inverse scattering theory.

Why is Landau-Ginzburg link cohomology equivalent to Khovanov homology?

In this note we make an attempt to compare a cohomological theory of Hilbert spaces of ground states in the mathcal{N}=left(2,2right) 2d Landau-Ginzburg theory in models describing link embeddings in ℝ3 to Khovanov and Khovanov-Rozansky homologies. To confirm the equivalence we exploit the invariance of Hilbert spaces of ground states for interfaces with respect to homotopy. In this attempt to study solitons and instantons in the LandauGiznburg theory we apply asymptotic analysis also known in the literature as exact WKB method, spectral networks method, or resurgence. In particular, we associate instantons in LG model to specific WKB line configurations we call null-webs.

Non-perturbative Quantum Mechanics from Non-perturbative Strings

This work develops a new method to calculate non-perturbative corrections in one-dimensional Quantum Mechanics, based on trans-series solutions to the refined holomorphic anomaly equations of topological string theory. The method can be applied to traditional spectral problems governed by the Schrodinger equation, where it both reproduces and extends the results of well-established approaches, such as the exact WKB method. It can be also applied to spectral problems based on the quantization of mirror curves, where it leads to new results on the trans-series structure of the spectrum. Various examples are discussed, including the modified Mathieu equation, the double-well potential and the quantum mirror curves of local $${\mathbb {P}}^2$$ and local $${\mathbb {F}}_0$$ . In all these examples, it is verified in detail that the trans-series obtained with this new method correctly predict the large-order behavior of the corresponding perturbative sectors.

Quantization conditions of eigenvalues for semiclassical Zakharov-Shabat systems on the circle

Bohr-Sommerfeld type quantization conditions of semiclassical eigenvalues for the non-selfadjoint Zakharov-Shabat operator on the circle are derived using an exact WKB method. The conditions are given in terms of the action associated with the unit circle or the action associated with turning points following the absence or presence of real turning points.

Real eigenvalues of a non-self-adjoint perturbation of the self-adjoint Zakharov-Shabat operator

We study the eigenvalues of the self-adjoint Zakharov-Shabat operator corresponding to the defocusing nonlinear Schrödinger equation in the inverse scattering method. Real eigenvalues exist when the square of the potential has a simple well. We derive two types of quantization conditions for the eigenvalues by using the exact WKB method and show, moreover, that the eigenvalues stay real for a sufficiently small non-self-adjoint perturbation when the potential has some PT-like symmetry.

Shatalov‐Sternin's construction of complex WKB solutions and the choice of integration paths

We re‐examine Shatalov‐Sternin's proof of existence of resurgent solutions of a linear ODE. In particular, we take a closer look at the “Riemann surface” (actually, a two‐dimensional complex manifold) whose existence, endless continuability and other properties are claimed by those authors. We present a detailed argument for a part of the “Riemann surface” relevant for the exact WKB method.

Transition probability for multiple avoided crossings with a small gap through an exact WKB method and a microlocal approach

In this Note, we study the adiabatic transition probability for a two-level system in the case of a finite number of avoided crossings. More precisely, we investigate a global change of bases of a first order differential system with respect to a semiclassical “adiabatic” parameter (h↓0) and an interaction parameter (ε↓0). We obtain its asymptotic behaviors by means of an exact WKB method and a microlocal analysis according to the interrelation of the two parameters.

Exact WKB analysis of non-adiabatic transition probabilities for three levels

The Schrödinger equation idψ/dt = ηHψ with a large parameter η and an appropriate 3×3 matrix H is studied by the exact WKB method, i.e. WKB analysis based on the Borel resummation. Transition probabilities of solutions of the equation are explicitly calculated with the help of the connection formula for WKB solutions. Since a 3×3 system is considered, addition of new Stokes curves is necessary to obtain the correct connection formula (Berk-Nevins-Roberts' observation), and we present a practically useful procedure for finding required new Stokes curves.

Eigenvalues of complex Hamiltonians with [formula omitted]-symmetry. I

Using the “exact WKB method” [E. Delabaere et al., J. Math. Phys. 38 (1997) 6126], we prove some reality results on the spectrum of some families of non-Hermitian Hamiltonians having PT -symmetry. This partially solves a conjecture of Zinn-Justin and Bessis. In part II [Phys. Lett. A 250 (1998) 29] we will prove non-reality results for other such families.