Weighted Hardy-Sobolev spaces and complex scaling of differential equations with operator coefficients

TL;DR

This paper introduces weighted Hardy-Sobolev spaces of vector-valued functions to facilitate complex scaling in linear differential equations with operator coefficients, with applications to boundary value problems in cylindrical domains.

Contribution

It develops a new functional analytic framework using weighted Hardy-Sobolev spaces for complex scaling of differential equations with unbounded operator coefficients.

Findings

Weighted Hardy-Sobolev spaces are effective for complex scaling.

Application to boundary value problems in cylindrical domains.

Framework handles operator coefficients with dilation analyticity.

Abstract

In this paper we study weighted Hardy-Sobolev spaces of vector valued functions analytic on double-napped cones of the complex plane. We introduce these spaces as a tool for complex scaling of linear ordinary differential equations with dilation analytic unbounded operator coefficients. As examples we consider boundary value problems in cylindrical domains and domains with quasicylindrical ends.