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

In the present paper we investigate the set $\Sigma_J$ of all $J$-self-adjoint extensions of a symmetric operator $S$ with deficiency indices $$ 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

On J-Self-Adjoint Operators with Stable C-Symmetry

The paper is devoted to a development of the theory of self-adjoint operators in Krein spaces (J-self-adjoint operators) involving some additional properties arising from the existence of C-symmetries. The main attention is paid to the recent notion of stable C-symmetry for J-self-adjoint extensions of a symmetric operator S. The general results are specialized further by studying in detail the case where S has defect numbers

On self-adjoint operators in Krein spaces constructed by Clifford algebra Cl_2

Let $J$ and $R$ be anti-commuting fundamental symmetries in a Hilbert space $\mathfrak{H}$. The operators $J$ and $R$ can be interpreted as basis (generating) elements of the complex Clifford algebra ${\mathcal C}l_2(J,R):={span}\{I, J, R, iJR\}$. An arbitrary non-trivial fundamental symmetry from ${\mathcal C}l_2(J,R)$ is determined by the formula $J_{\vec{\alpha}}=\alpha_{1}J+\alpha_{2}R+\alpha_{3}iJR$, where ${\vec{\alpha}}\in\mathbb{S}^2$. Let $S$ be a symmetric operator that commutes with ${\mathcal C}l_2(J,R)$. The purpose of this paper is to study the sets $\Sigma_{{J_{\vec{\alpha}}}}$ ($\forall{\vec{\alpha}}\in\mathbb{S}^2$) of self-adjoint extensions of $S$ in Krein spaces generated by fundamental symmetries ${{J_{\vec{\alpha}}}}$ (${{J_{\vec{\alpha}}}}$-self-adjoint extensions). We show that the sets $\Sigma_{{J_{\vec{\alpha}}}}$ and $\Sigma_{{J_{\vec{\beta}}}}$ are unitarily equivalent for different ${\vec{\alpha}}, {\vec{\beta}}\in\mathbb{S}^2$ and describe in detail the structure of operators $A\in\Sigma_{{J_{\vec{\alpha}}}}$ with empty resolvent set

Lax-Phillips scattering theory for PT-symmetric \rho-perturbed operators

The S-matrices corresponding to PT-symmetric \rho-perturbed operators are defined and calculated by means of an approach based on an operator-theoretical interpretation of the Lax-Phillips scattering theory

PT symmetry, Cartan decompositions, Lie triple systems and Krein space related Clifford algebras

Gauged PT quantum mechanics (PTQM) and corresponding Krein space setups are studied. For models with constant non-Abelian gauge potentials and extended parity inversions compact and noncompact Lie group components are analyzed via Cartan decompositions. A Lie triple structure is found and an interpretation as PT-symmetrically generalized Jaynes-Cummings model is possible with close relation to recently studied cavity QED setups with transmon states in multilevel artificial atoms. For models with Abelian gauge potentials a hidden Clifford algebra structure is found and used to obtain the fundamental symmetry of Krein space related J-selfadjoint extensions for PTQM setups with ultra-localized potentials.Comment: 11 page

On symmetries in the theory of finite rank singular perturbations

Abstract. For a nonnegative self-adjoint operator A0 acting on a Hilbert space H singular perturbations of the form A0 + V, V = Pn 1 bij < ψj, ·> ψi are studied under some additional requirements of symmetry imposed on the initial operator A0 and the singular elements ψj. A concept of symmetry is defined by means of a one-parameter family of unitary operators U that is motivated by results due to R. S. Phillips. The abstract framework to study singular perturbations with symmetries developed in the paper allows one to incorporate physically meaningful connections between singular potentials V and the corresponding self-adjoint realizations of A0 + V. The results are applied for the investigation of singular perturbations of the Schrödinger operator in L2(R3) and for the study of a (fractional) p-adic Schrödinger type operator with point interactions. 1