Related papers: WKB Method and Quantum Periods beyond Genus One
We show that the all-orders WKB periods of one-dimensional quantum mechanical oscillators are governed by the refined holomorphic anomaly equations of topological string theory. We analyze in detail the double-well potential and the cubic…
In many physical problems it is not possible to find an exact solution. However, when some parameter in the problem is small, one can obtain an approximate solution by expanding in this parameter. This is the basis of perturbative methods,…
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…
Quantum mechanical WKB-method is elaborated for the known quantum Kepler problem in curved 3-space models Euclide, Riemann and Lobachevsky in the framework of the complex variable function theory. Generalized Schr\"{o}dinger, Klein-Fock…
We study the WKB periods for the $(r+1)$-th order ordinary differential equation (ODE) which is obtained by the conformal limit of the linear problem associated with the $A_r^{(1)}$ affine Toda field equation. We compute the quantum…
We derive the semiclassical WKB quantization condition for obtaining the energy band edges of periodic potentials. The derivation is based on an approach which is much simpler than the usual method of interpolating with linear potentials in…
We apply exact WKB analysis to the spectral problem arising in black hole perturbation theory. The boundary conditions for quasinormal modes lead to exact quantization conditions for the complex frequencies. To solve these conditions, one…
Higher-order WKB methods are used to investigate the border between the solvable and insolvable portions of the spectrum of quasi-exactly solvable quantum-mechanical potentials. The analysis reveals scaling and factorization properties that…
Periods of rational integrals appear in quantum mechanics through asymptotic expansions of traces computed with the semiclassical symbol calculus. We develop a novel formal series expansion for the trace of the Dirac delta of a differential…
We consider the moduli space of holomorphic principal bundles for reductive Lie groups over Riemann surfaces (possibly with boundaries) and equipped with meromorphic connections. We associate to this space a point-wise notion of quantum…
We explore the possibility of modifying the Lewis-Riesenfeld method of invariants developed originally to find exact solutions for time-dependent quantum mechanical systems for the situation in which an exact invariant can be constructed,…
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…
The phenomenon of quantum tunneling is reviewed and an overview of applying approximate methods for studying this effect is given. An approach to a time-dependent formalism is proposed in one dimension and generalized to higher dimensions.…
We give an algorithm to compute the periods of smooth projective hypersurfaces of any dimension. This is an improvement over existing algorithms which could only compute the periods of plane curves. Our algorithm reduces the evaluation of…
Quantum-mechanical WKB-method is elaborated for the known quantum oscillator problem in curved 3-spaces models Euclid, Riemann, and Lobachevsky E_{3}, H_{3}, S_{3} in the framework of the complex variable function theory. Generalized…
In the present work the conditions appearing in the WKB approximation formalism of quantum mechanics are analyzed. It is shown that, in general, a careful definition of an approximation method requires the introduction of two length…
The Picard-Fuchs equation is a powerful mathematical tool which has numerous applications in physics, for it allows to evaluate integrals without resorting to direct integration techniques. We use this equation to calculate both the…
It is well known in classical mechanics that, the frequencies of a periodic system can be obtained rather easily through the action variable, without completely solving the equation of motion. The equivalent quantum action variable…
Statistical mechanics of 1D multivalent Coulomb gas may be mapped onto non-Hermitian quantum mechanics. We use this example to develop instanton calculus on Riemann surfaces. Borrowing from the formalism developed in the context of…
We define a class of quantum systems called regular quantum graphs. Although their dynamics is chaotic in the classical limit with positive topological entropy, the spectrum of regular quantum graphs is explicitly computable analytically…