Within the framework of quantum mechanics working with one-dimensional, manifestly non-Hermitian Hamiltonians $H=T+V$ the traditional class of the exactly solvable models with local point interactions $V=V(x)$ is generalized. The consequences of the
use of the nonlocal point interactions such that $(V f)(x) = int K(x,s) f(s) ds$ are discussed using the suitably adapted formalism of boundary triplets.
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-s
ymmetric 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.
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.
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 scatterin
g methods to the investigation of PT-symmetric operators is illustrated by considering the case of 0-perturbed operators.
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 arb
itrary 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.
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>.
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.
Gauged PT quantum mechanics (PTQM) and corresponding Krein space setups are studied. For models with constant non-Abelian gauge potentials and extended parity
For a nonnegative self-adjoint operator $A_0$ acting on a Hilbert space $mathfrak{H}$ singular perturbations of the form $A_0+V, V=sum_{1}^{n}{b}_{ij}<psi_j,cdot>psi_i$ are studied under some additional requirements of symmetry imposed on the initia
l operator $A_0$ and the singular elements $psi_j$. A concept of symmetry is defined by means of a one-parameter family of unitary operators $sU$ 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 $A_0+V$. The results are applied for the investigation of singular perturbations of the Schr{o}dinger operator in $L_2(dR^3)$ and for the study of a (fractional) textsf{p}-adic Schr{o}dinger type operator with point interactions.
