Maximal determinants of Schr\"odinger operators on bounded intervals

TL;DR

This paper investigates extremal potentials for the functional determinant of 1D Schrödinger operators on bounded intervals, extending the concept to $L^q$ potentials and solving the associated optimal control problem.

Contribution

It introduces a new extension of the functional determinant to $L^q$ potentials and characterizes extremal solutions, including explicit cases for $q=1$ and $q=2$.

Findings

Existence and uniqueness of extremal potentials for all $q\,geq 1$.

Complete characterization of extremals for $q=1$ and $q=2$.

Reduction of the problem to an optimal control framework.

Abstract

We consider the problem of finding extremal potentials for the functional determinant of a one-dimensional Schr\"odinger operator defined on a bounded interval with Dirichlet boundary conditions under an $L^q$-norm restriction ($q\geq 1$). This is done by first extending the definition of the functional determinant to the case of $L^q$ potentials and showing the resulting problem to be equivalent to a problem in optimal control, which we believe to be of independent interest. We prove existence, uniqueness and describe some basic properties of solutions to this problem for all $q\geq 1$, providing a complete characterization of extremal potentials in the case where $q$ is one (a pulse) and two (Weierstrass's $\wp$ function).