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...

Let Λ be a ring endowed with an involution a 7→ ã. We say that two units a and b of Λ fixed under the involution are congruent if there exists an element u ∈ Λ× such that a = ubũ. We denote by H(Λ) the set of congruence classes. In this paper we consider the case where Λ is an order with involution in a semisimple algebraA over a local field and study the question whether the natural map H(Λ)→ ...

This lecture takes a closer look at Hermitian matrices and at their eigenvalues. After a few generalities about Hermitian matrices, we prove a minimax and maximin characterization of their eigenvalues, known as Courant–Fischer theorem. We then derive some consequences of this characterization, such as Weyl theorem for the sum of two Hermitian matrices, an interlacing theorem for the sum of two ...

Let X be a compact manifold of dimension 2n equipped with an almost complex structure J : TX → TX, E a Hermitian vector bundle on X, and gX a Riemannian metric on X. Assume that the almost complex structure J is compatible with gX . Consider a Hermitian line bundle L over X endowed with a Hermitian connection ∇L such that its curvature R = (∇L)2 is nondegenerate. Thus, ω = i 2πR L is a symplect...

We provide time-evolution operators, gauge transformations and a perturbative treatment for non-Hermitian Hamiltonian systems, which are explicitly timedependent. We determine various new equivalence pairs for Hermitian and non-Hermitian Hamiltonians, which are therefore pseudo-Hermitian and in addition in some cases also invariant under PT-symmetry. In particular, for the harmonic oscillator p...

Abstract A non-Hermitian Hamiltonian that has an unbroken PT symmetry can be converted by means of a similarity transformation to a physically equivalent Hermitian Hamiltonian. This raises the following question: In which form of the quantum theory, the non-Hermitian or the Hermitian one, is it easier to perform calculations? This paper compares both forms of a non-Hermitian ix quantum-mechanic...