The Cauchy problem for Schr\"odinger-type partial differential operators with generalized functions in the principal part and as data

TL;DR

This paper develops a framework for solving Schr"odinger-type equations with distributional coefficients using Colombeau generalized functions, addressing models in seismology and linking solutions to classical and distributional cases.

Contribution

It introduces a method to establish existence and uniqueness of solutions for Schr"odinger operators with generalized coefficients, including novel initial value constructions.

Findings

Proved existence and uniqueness of Colombeau solutions.

Connected generalized solutions with classical and distributional solutions.

Constructed generalized initial values related to probability measures.

Abstract

We set-up and solve the Cauchy problem for Schr\"odinger-type differential operators with generalized functions as coefficients, in particular, allowing for distributional coefficients in the principal part. Equations involving such kind of operators appeared in models of deep earth seismology. We prove existence and uniqueness of Colombeau generalized solutions and analyze the relations with classical and distributional solutions. Furthermore, we provide a construction of generalized initial values that may serve as square roots of arbitrary probability measures.