On a class of $J$-self-adjoint operators with empty resolvent set

TL;DR

This paper characterizes $J$-self-adjoint extensions of symmetric operators with deficiency indices <2,2> in Krein spaces, linking their properties to Clifford algebras and providing parameterizations relevant to indefinite Sturm-Liouville and Dirac operators.

Contribution

It establishes a connection between $J$-self-adjoint extensions with empty resolvent set and Clifford algebra commutation, offering a new framework for their classification and construction.

Findings

Equivalence between $J$-self-adjoint extensions with empty resolvent set and Clifford algebra commutation.

Construction of operators $C_{ ext{chi}, ext{omega}}$ for stable $C$-symmetry.

Application to indefinite Sturm-Liouville and Dirac operators with point interactions.

Abstract

In the present paper we investigate the set $\Sigma_J$ of all $J$-self-adjoint extensions of a symmetric operator $S$ with deficiency indices $<2,2>$ which commutes with a non-trivial fundamental symmetry $J$ of a Krein space $(\mathfrak{H}, [\cdot,\cdot])$, SJ=JS. Our aim is to describe different types of $J$-self-adjoint extensions of $S$. One of our main results is the equivalence between the presence of $J$-self-adjoint extensions of $S$ with empty resolvent set and the commutation of $S$ with a Clifford algebra ${\mathcal C}l_2(J,R)$, where $R$ is an additional fundamental symmetry with $JR=-RJ$. This enables one to construct the collection of operators $C_{\chi,\omega}$ realizing the property of stable $C$-symmetry for extensions $A\in\Sigma_J$ directly in terms of ${\mathcal C}l_2(J,R)$ and to parameterize the corresponding subset of extensions with stable $C$-symmetry. Such a situation occurs naturally in many applications, here we discuss the case of an indefinite Sturm-Liouville operator on the real line and a one dimensional Dirac operator with point interaction.