On domains of symmetric operators related to −y″(x) + (− 1)nx2ny(x)

Abstract

In recent years a generalization of Hermiticity has been investigated using a complex deformation H = p2 + x2(ix)ϵ of the harmonic oscillator Hamiltonian, where ϵ is a real parameter. These complex Hamiltonians, possessing symmetry (the product of parity and time reversal), can have a real spectrum. We will consider the most simple case: ϵ even. In this paper we describe all self-adjoint (Hermitian) and at the same time symmetric operators associated with H = p2 + x2(ix)ϵ. Surprisingly, it turns out that there is a large class of self-adjoint operators associated with H = p2 + x2(ix)ϵ which are not symmetric.