On Domains of PT Symmetric Operators Related to -y''(x) + (-1)^n x^{2n}y(x)

In the recent years a generalization of Hermiticity was investigated using a complex deformation H=p^2 +x^2(ix)^\epsilon of the harmonic oscillator Hamiltonian, where \epsilon is a real parameter. These complex Hamiltonians, possessing PT symmetry (the product of parity and time reversal), can have real spectrum. We will consider the most simple case: \epsilon even. In this paper we describe all self-adjoint (Hermitian) and at the same time PT symmetric operators associated to H=p^2 +x^2(ix)^\epsilon. Surprisingly it turns out that there are a large class of self-adjoint operators associated to H=p^2 +x^2(ix)^\epsilon which are not PT symmetric

PT Symmetric, Hermitian and P-Self-Adjoint Operators Related to Potentials in PT Quantum Mechanics

In the recent years a generalization $H=p^2 +x^2(ix)^\epsilon$ of the harmonic oscillator using a complex deformation was investigated, where \epsilon\ is a real parameter. Here, we will consider the most simple case: \epsilon even and x real. We will give a complete characterization of three different classes of operators associated with the differential expression H: The class of all self-adjoint (Hermitian) operators, the class of all PT symmetric operators and the class of all P-self-adjoint operators. Surprisingly, some of the PT symmetric operators associated to this expression have no resolvent set

On limit point and limit circle classification for PT symmetric operators

A prominent class of PT-symmetric Hamiltonians is H:= 1/2 p^2 + x^2 (ix)^N, for x \in \Gamma$for some nonnegative number N. The associated eigenvalue problem is defined on a contour$\Gamma$ in a specific area in the complex plane (Stokes wedges), see [3,5]. In this short note we consider the case N=2 only. Here we elaborate the relationship between Stokes lines and Stokes wedges and well-known limit point/limit circle criteria from [11, 6, 10]

Small perturbation of selfadjoint and unitary operators in Krein spaces

We investigate the behaviour of the spectrum of selfadjoint operators in Krein spaces under perturbations with uniformly dissipative operators. Moreover we consider the closely related problem of the perturbation of unitary operators with uniformly bi-expansive. The obtained perturbation results give a new characterization of spectral points of positive type and of type pi + of selfadjoint (resp. unitary) operators in Krein spaces

Compact and finite rank perturbations of linear relations in Hilbert spaces

Abstract. For closed linear operators or relations A and B acting between Hilbert spaces H and K the concepts of compact and finite rank perturbations are introduced with the help of the orthogonal projections PA and PB in H©K onto A and B. Various equivalent characterizations for such perturbations are proved and it is shown that these notions are a natural generalization of the usual concepts of compact and finite rank perturbations

Spectral points of definite type and type π for linear operators and relations in Krein spaces

Spectral points of type \pi_+ and type \pi_− for closed linear operators and relations in Krein spaces are introduced with the help of approximative eigensequences. It turns out that these spectral points are stable under compact perturbations and perturbations small in the gap metric

On domains of powers of linear operators and finite rank perturbations

Let S and T be linear operators in a linear space such that S T. In this note an estimate for the codimension of domSn in domTn in terms of the codimension of domS in domT is obtained. An immediate consequence is that for any polynomial p the operator p(S) is a finite-dimensional restriction of the operator p(T) whenever S is a finite-dimensional restriction of T. The general results are applied to a perturbation problem of selfadjoint definitizable operators in Krein spaces