Unbounded C-symmetries and their nonuniqueness

It is shown that if the C operator for a PT-symmetric Hamiltonian with simple eigenvalues is not unique, then it is unbounded. Apart from the special cases of finite-matrix Hamiltonians and Hamiltonians generated by differential expressions with PT-symmetric point interactions, the usual situation is that the C operator is unbounded. The fact that the C operator is unbounded is significant because, while there is a formal equivalence between a PT-symmetric Hamiltonian and a conventionally Hermitian Hamiltonian in the sense that the two Hamiltonians are isospectral, the Hilbert spaces are inequivalent. This is so because the mapping from one Hilbert space to the other is unbounded. This shows that PT-symmetric quantum theories are mathematically distinct from conventional Hermitian quantum theories.

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.

On elements of the Lax-Phillips scattering scheme for PT-symmetric operators

Generalized PT-symmetric operators acting an a Hilbert space $mathfrak{H}$ are defined and investigated. The case of PT-symmetric extensions of a symmetric operator $S$ is investigated in detail. The possible application of the Lax-Phillips scattering methods to the investigation of PT-symmetric operators is illustrated by considering the case of 0-perturbed operators.

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 ${vec{alpha}}inmathbb{S}^2$. 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 $Sigma_{{J_{vec{alpha}}}}$ ($forall{vec{alpha}}inmathbb{S}^2$) 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 ${vec{alpha}}, {vec{beta}}inmathbb{S}^2$ and describe in detail the structure of operators $AinSigma_{{J_{vec{alpha}}}}$ with empty resolvent set.

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 <2,2>.

On a class of $J$-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 $<2,2>$ 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 $AinSigma_J$ 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.

$J$-self-adjoint operators with $mathcal{C}$-symmetries: extension theory approach

A well known tool in conventional (von Neumann) quantum mechanics is the self-adjoint extension technique for symmetric operators. It is used, e.g., for the construction of Dirac-Hermitian Hamiltonians with point-interaction potentials. Here we reshape this technique to allow for the construction of pseudo-Hermitian ($J$-self-adjoint) Hamiltonians with complex point-interactions. We demonstrate that the resulting Hamiltonians are bijectively related with so called hypermaximal neutral subspaces of the defect Krein space of the symmetric operator. This symmetric operator is allowed to have arbitrary but equal deficiency indices $<n,n>$. General properties of the $cC$ operators for these Hamiltonians are derived. A detailed study of $cC$-operator parametrizations and Krein type resolvent formulas is provided for $J$-self-adjoint extensions of symmetric operators with deficiency indices $<2,2>$. The technique is exemplified on 1D pseudo-Hermitian Schrodinger and Dirac Hamiltonians with complex point-interaction potentials.