Linear nonlocal problem for the abstract time-dependent non-homogeneous Schr\"odinger equation

TL;DR

This paper investigates a nonlocal-in-time Schr"odinger problem, deriving an exact solution representation, establishing well-posedness conditions, and extending results to unbounded nonlocal parameters, with practical examples.

Contribution

It provides a novel exact solution representation and extends well-posedness results for nonlocal Schr"odinger equations with unbounded parameters.

Findings

Derived an exact solution operator representation.

Established necessary and sufficient conditions for well-posedness.

Extended existence results to cases with unbounded nonlocal parameters.

Abstract

A nonlocal-in-time problem for the abstract Schr\"odinger equation is considered. By exploiting the linear nature of nonlocal condition we derive an exact representation of the solution operator under assumptions that the spectrum of Hamiltonian is contained in the horizontal strip of complex plane. The derived representation permits us to establish the necessary and sufficient conditions for the problem's well-posedness and the existence of its mild, strong solutions. Furthermore, we present new sufficient conditions for the existence of solution which extend the available results to the case when some nonlocal parameters are unbounded. Two examples are provided.