In this article we consider the system of operator equations T_iX=U_i for i=1,2,...,n and give necessary and suffcient conditions for the existence of common Hermitian solutions to this system of operator equations for arbitrary operators without the closedness condition. Also we study the Moore-penrose inverse of a ncross 1 block operator matrix and. then gi...

For a given standard Hamiltonian H with arbitrary complex scalar and vector potentials in one-dimension, we construct an invertible antilinear operator τ such that H is τ -anti-pseudo-Hermitian, i.e., H = τHτ−1. We use this result to give the explicit form of a linear Hermitian invertible operator with respect to which any standard PT symmetric Hamiltonian with a real degree of freedom is pseud...

Classes of non-Hermitian operators that have only real eigenvalues are presented. Such operators appear in quantum mechanics and are expressed in terms of the generators of the Weyl-Heisenberg algebra. For each non-Hermitian operator A, a Hermitian involutive operator Ĵ such that A is Ĵ-Hermitian, that is, ĴA = AĴ , is found. Moreover, we construct a positive definite Hermitian Q such that A is...

Classes of non-Hermitian operators that have only real eigenvalues are presented. Such operators appear in quantum mechanics and are expressed in terms of the generators of the Weyl-Heisenberg algebra. For each non-Hermitian operator A, a Hermitian involutive operator Ĵ such that A is Ĵ-Hermitian, that is, ĴA = AĴ , is found. Moreover, we construct a positive definite Hermitian Q such that A is...

we observe that the variable φ enters the expression in the form of the operator ∂2/∂φ2. This operator—1D Laplace operator with respect to the variable φ—is a Hermitian operator in the space of single-valued functions of the angle φ, because single-valuedness is equivalent to the 2π-periodicity, and Laplace operator is Hermitian in the space of periodic functions. Hence, we have ONB of the eige...

For a given standard Hamiltonian H = [p − A(x)]2/(2m) + V (x) with arbitrary complex scalar potential V and vector potential A, with x ∈ R, we construct an invertible antilinear operator τ such that H is τ -anti-pseudo-Hermitian, i.e., H = τHτ−1. We use this result to give the explicit form of a linear Hermitian invertible operator with respect to which any standard PT -symmetric Hamiltonian wi...

To find the Hermitian phase operator of a single-mode electromagnetic field in quantum mechanics, the Schrödinger representation is extended to a larger Hilbert space augmented by states with infinite excitation by nonstandard analysis. The Hermitian phase operator is shown to exist on the extended Hilbert space. This operator is naturally considered as the controversial limit of the approximat...