Results on Certain Non-Hermitian Hamiltonians

Abstract

We present a few results on the spectral properties of a class of physically reasonable non-Hermitian Hamiltonians. These theorems relate the spectral properties of a non-self-adjoint operator (of the aforementioned class) in terms of that of a self-adjoint operator. These theorems can be specialized to yield conditions under which the perturbed eigenvalues (of the above class of operators) vary continuously from the eigenvalues of the unperturbed operators. If the Schrödinger equation has to be solved numerically, a knowledge of the spectral properties of the non-Hermitian Hamiltonian would insure when the eigensolutions exist.

Equidistant-state preparation

“…The basis consisting of the states {|φ p k }, k = 0,...,p − 1 that are biorthogonal ( φ p k |ψ p k ∼ δ k ,k ) [9,14] to the set (6), can be obtained in the same way but changing the parameter of the initial state λ…”

Unbiased nonorthogonal bases for tomographic reconstruction

“…where ω = e 2πi/p , while the biorthogonal basis is formed by eigenstates of its Hermitian conjugate [8], Z † |φ n = ω −n |φ n . Now, let us introduce the operator…”

“…Using Eqs. (8) and (9), we can expand the identity operator using the nonorthogonal projectors corresponding to the basis {|ψ s n } as follows:…”

“…, p − 1. Relations (11) and (12) allow the expansion of the density matrix as a sum of projectors on p nonorthogonal equidistant bases (5) and the orthogonal basis (8) and (9) aŝ…”

“…A particular case of such bases belongs to the class of so-called symmetric states [7], which are generated by applying integer powers of a single unitary operator onto any one of the states. An advantage of the equally separated bases is that the corresponding biorthogonal basis [8], {|m } such that m|n = δ mn , can be easily constructed. This opens the possibility of defining an explicit optimal tomographic procedure along the lines of the general approach discussed in [9].…”