arXiv subjects
Publications and source records attributed to Tomas Ya. Azizov.
Given two possibly unbounded selfadjoint operators A and G such that the resolvent sets of AG and GA are non-empty, it is shown that the operator AG has a spectral function on IR with singularities if there exists a non-zero polynomial p such that the symmetric operator Gp(AG) is non-negative. This result generalizes a well-known theorem for definitizable operators in Krein spaces.
In the recent years a generalization of Hermiticity was investigated using a complex deformation H=p^2 +x^2(ix)^εof the harmonic oscillator Hamiltonian, where εis 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: εeven. 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)^ε. Surprisingly it turns out that there are a large class of self-adjoint operators associated to H=p^2 +x^2(ix)^εwhich are not PT symmetric.