Generalized boundary triples, Weyl functions and inverse problems

TL;DR

This paper develops a comprehensive theory of generalized boundary triples and Weyl functions for symmetric operators, providing new analytic characterizations and resolvent formulas with applications to differential operators.

Contribution

It introduces new classes of generalized boundary triples, characterizes them analytically via Weyl functions, and derives Kren-type resolvent formulas, expanding the framework for inverse problems and PDE applications.

Findings

Characterization of Weyl functions for boundary triples.

Analytic conditions for essential selfadjointness of operators.

Kren-type resolvent formulas for boundary triples.

Abstract

With a closed symmetric operator $A$ in a Hilbert space ${\mathfrak H}$ a triple $\Pi=\{{\mathcal H},\Gamma_0,\Gamma_1\}$ of a Hilbert space ${\mathcal H}$ and two abstract trace operators $\Gamma_0$ and $\Gamma_1$ from $A^*$ to ${\mathcal H}$ is called a generalized boundary triple for $A^*$ if an abstract analogue of the second Green's formula holds. Various classes of generalized boundary triples are introduced and corresponding Weyl functions $M$ are investigated. The most important ones for applications are specific classes of (essentially) unitary boundary triples which guarantee that the Weyl functions of boundary triples are Nevanlinna functions on ${\mathcal H}$, or at least they belong to the class of Nevanlinna families. The boundary condition $\Gamma_0f=0$ determines a reference operator $A_0$. The case where $A_0$ is selfadjoint implies a relatively simple analysis, as the joint domain of the trace mappings $\Gamma_0$ and $\Gamma_1$ admits a von Neumann type decomposition. The case where $A_0$ is only essentially selfadjoint is more involved, but appears to be of great importance, for instance, in applications to PDEs and ODEs. Various classes of generalized boundary triples will be characterized in purely analytic terms via the Weyl function $M$. These characterizations involve solving direct and inverse problems for specific classes of (unbounded) operator functions $M$. One of the main results specifies the analytic properties of $M$ which guarantee that $A_0$ is essentially selfadjoint. In this study we also derive, for instance, Kre\u{\i}n-type resolvent formulas for the most general classes of unitary and isometric boundary triples appearing in the present work. All the main results are shown to have applications in the study of ordinary and partial differential operators.