In this paper we consider the solutions of linear systems of saddle point problems‎. ‎By using the spectrum of a quadratic matrix polynomial‎, ‎we study the eigenvalues of the iterative matrix of the Hermitian and skew Hermitian splitting method‎.

In the presence of CP violation, the effective Hamiltonian matrix describing a neutral meson anti-meson system does not commute with its hermitian conjugate. As a result, this matrix cannot be diagonalized by a unitary transformation and one needs to introduce a reciprocal basis. Although known, this fact is seldom discussed and almost never used. Here, we use this concept to highlight a parame...

We show that a Dicke-type non-Hermitian Hamiltonian admits entirely real spectra by mapping it to the "dressed Dicke model" through a similarity transformation. We find a positive-definite metric in the Hilbert space of the non-Hermitian Hamiltonian so that the time evolution is unitary and allows a consistent quantum description. We then show that this non-Hermitian Hamiltonian describing nond...

In multivariate statistical analysis, several authors have studied the total positivity properties of the generalized (0F1) hypergeometric function of two real symmetric matrix arguments. In this paper, we make use of zonal polynomial expansions to obtain a new proof of a result that these 0F1 functions fail to satisfy certain pairwise total positivity properties; this proof extends both to arb...

New non Hermitian Hamiltonians are generated, as isospectral partners of the generalized Swanson model, viz., H− = A†A + αA2 + βA† 2, where α , β are real constants, with α 6= β, and A† and A are generalized creation and annihilation operators. It is shown that the initial Hamiltonian H−, and its partner H+, are related by pseudo supersymmetry, and they share all the eigen energies except for t...

We show that some non-Hermitian Hamiltonian operators with tridiagonal matrix representation may be quasi Hermitian or similar to operators. In the class of discussed here transformation is given by a Hermitian, positive-definite, diagonal operator. there an important difference between open boundary conditions and periodic ones. illustrate theoretical results means two simple, widely used, mod...

Backward errors and condition numbers are defined and evaluated for eigenvalues and eigenvectors of generalized eigenvalue problems. Both normwise and componentwise measures are used. Unstructured problems are considered first, and then the basic definitions are extended so that linear structure in the coefficient matrices (for example, Hermitian, Toeplitz, Hamiltonian, or band structure) is pr...

In this paper, we present new results relating the numerical range of a matrix A with generalized Levinger transformation L(A, α, β) = αH A + βS A , where H A and S A , are, respectively the Hermitian and skew-hermitian parts of A. Using these results, we, then derive expressions for eigenvalues and eigenvectors of the perturbed matrix A + L(E, α, β), for a fixed matrix E and α, β are real para...

To lowest order of perturbation theory we show that an equivalence can be established between a PT -symmetric generalized quartic anharmonic oscillator model and a Hermitian position-dependent mass Hamiltonian h. An important feature of h is that it reveals a domain of couplings where the quartic potential could be attractive, vanishing or repulsive. We also determine the associated physical qu...

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...