One-dimensional reflection in the quantum mechanical bootstrap

TL;DR

This paper applies the quantum mechanical bootstrap method to one-dimensional scattering problems, using boundary conditions, semidefinite programming, and WKB approximation to analyze reflection coefficients and scattering behavior.

Contribution

It introduces a novel application of the bootstrap approach to scattering problems with Robin boundary conditions and combines numerical methods with analytical approximations.

Findings

Successfully extracted reflection coefficients for various potentials.

Demonstrated the effectiveness of semidefinite programming in quantum scattering.

Analyzed the scattering of Liouville theory's exponential potential.

Abstract

We describe the application of the quantum mechanical bootstrap to the solution of one-dimensional scattering problems. By fixing a boundary and modulating the Robin parameter of the boundary conditions we are able to extract the reflection coefficient for various potentials and compare to physical expectations. This includes an application of semidefinite programming to solving a half-line Schrodinger problem with arbitrary Robin boundary conditions. Finally, the WKB approximation is used to numerically determine the scattering behavior of the exponential potential of Liouville theory.