Exact WKB Method Research Articles

Semiclassical Representation of the Scattering Matrix by a Feynman Integral

We study the scattering problem for one-dimensional Schrodinger equations in the semiclassical limit when the energy level is close to the quadratic maxima of the potential. Starting from the formula of the scattering matrix obtained by the exact WKB method, we represent its principal term in a reduced form of the Feynman integral, an absolutely convergent sum of suitably defined probability amplitudes over countably many trajectories on ℝ generated by classical trajectories and tunneling effects.

Unfolding the Quartic Oscillator

The “exact WKB method” is applied to the general quartic oscillator, yielding rigorous results on the ramification properties of the energy levels when the coefficients of the fourth degree polynomial are varied in the complex domain. Simple though exact “model forms” are given for the avoided crossing phenomenon, easily interpreted in terms of complex branch points in the “asymmetry parameter.” In the almost symmetrical situation, this gives a generalization of the Zinn–Justin quantization condition. The analogous “model quantization condition” near unstable equilibrium is thoroughly analysed in the symmetrical case, yielding complete confirmation of the branch point structure discovered by Bender and Wu. The numerical results of this analysis are in excellent agreement with those computed by Shanley, overtaking the most optimistic expectations of the realm of validity of semiclassical models.

Semiclassical study of quantum scattering on the line

We study the well-known problem of 1-d quantum scattering by a potential barrier in the semiclassical limit. Using the so-called exact WKB method and semiclassical microlocal analysis techniques, we get a very precise and complete description of the scattering matrix, in particular when the energy is very close to a unique, quadratic maximum of the potential. In our one-dimensional setting, we also recover the Bohr-Sommerfeld quantization condition for the resonances generated by such a maximum.

Higher length-twist coordinates, generalized Heun’s opers, and twisted superpotentials

In this paper we study a proposal of Nekrasov, Rosly and Shatashvili that describes the effective twisted superpotential obtained from a class S theory geometrically as a generating function in terms of certain complexified length-twist coordinates, and extend it to higher rank. First, we introduce a higher rank analogue of Fenchel-Nielsen type spectral networks in terms of a generalized Strebel condition. We find new systems of spectral coordinates through the abelianization method and argue that they are higher rank analogues of the Nekrasov-Rosly-Shatashvili Darboux coordinates. Second, we give an explicit parametrization of the locus of opers and determine the generating functions of this Lagrangian subvariety in terms of the higher rank Darboux coordinates in some specific examples. We find that the generating functions indeed agree with the known effective twisted superpotentials. Last, we relate the approach of Nekrasov, Rosly and Shatashvili to the approach using quantum periods via the exact WKB method.