Maximal determinants of Schrödinger operators on bounded intervals

Abstract : We consider the problem of finding extremal potentials for the functional determinant of a one-dimensional Schro ̈dinger operator defined on a bounded interval with Dirichlet boundary conditions under an Lq-norm res- triction (q ≥ 1). This is done by first extending the definition of the functional determinant to the case of Lq 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 ≥ 1, providing a complete characterization of extremal potentials in the case where q is one (a pulse) and two (Weierstrass’s ℘ function).
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https://hal.inria.fr/hal-01406270
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Submitted on : Wednesday, July 31, 2019 - 3:57:10 PM
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Clara Aldana, Jean-Baptiste Caillau, Pedro Freitas. Maximal determinants of Schrödinger operators on bounded intervals. 2017. ⟨hal-01406270⟩

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