We study large, sparse generalized eigenvalue problems for matrix pencils, where one of the matrices is Hamiltonian and the other skew Hamiltonian. Problems of this form arise in the numerical simulation of elastic deformation of anisotropic materials, in structural mechanics and in the linear-quadratic control problem for partial diierential equations. We develop a structure-preserving skew-Ha...

A class η-weak-pseudo-Hermiticity generators for spherically symmetric non-Hermitian Hamiltonians are presented. An operators-based procedure is introduced so that the results for the 1D Schrödinger Hamiltonian may very well be reproduced. A generalization beyond the nodeless states is proposed. Our illustrative examples include η-weak-pseudo-Hermiticity generators for the non-Hermitian weakly ...

We compute the leading contribution to the effective Hamiltonian of SU(N) matrix theory in the limit of large separation. We work with a gauge fixed Hamiltonian and use generalized Born-Oppenheimer approximation, extending the recent work of Halpern and Schwartz for SU(2) [8]. The answer turns out to be a free Hamiltonian for the coordinates along the flat directions of the potential. Applicati...

The Hamiltonian of a quantum mechanical system has an affiliated spectrum. If this spectrum is the sequence of prime numbers, a connection between quantum mechanics and the nontrivial zeros of the Riemann zeta function can be made. In this case, the Riemann zeta function is analogous to chaotic quantum systems, as the harmonic oscillator is for integrable quantum systems. Such quantum Riemann z...

Non-Hermitian skin effect, namely that the eigenvalues and eigenstates of a non-Hermitian tight-binding Hamiltonian have significant differences under open or periodic boundary conditions, is remarkable phenomenon systems. Inspired by presence we study evolution wave-packets in systems, which can be determined using single-particle Green's function. Surprisingly, find thermodynamical limit, fun...

The inertia of a Hermitian matrix is defined to be a triplet composed by the numbers of the positive, negative and zero eigenvalues of the matrix counted with multiplicities, respectively. In this paper, we give various closed-form formulas for the maximal and minimal values for the rank and inertia of the Hermitian expression A + X, where A is a given Hermitian matrix and X is a variable Hermi...

A class of spherically symmetric non-Hermitian Hamiltonians and their η-pseudo-Hermiticity generators are presented. An operators-based procedure is introduced so that the results for the 1D Schrödinger Hamiltonian may very well be reproduced. A generalization beyond the nodeless states is proposed. Our illustrative examples include η-pseudo-Hermiticity generators for the non-Hermitian weakly p...

Exact solvability of some non-Hermitian η-weak-pseudo-Hermitian Hamiltonians is explored as a byproduct of η-weak-pseudo-Hermiticity generators. A class of Veff (x) = V (x) + iW (x) potentials is considered, where the imaginary part W (x) is used as an η-weak-pseudo-Hermiticity generator to obtain exactly solvable η-weak-pseudo-Hermitian Hamiltonian models. PACS numbers: 03.65.Ge, 03.65.Fd,03.6...

The paper deals with the associated algebraic matrix Riccati equation (AAMRE), closely related to the standard algebraic matrix Riccati equation arising in the theory of linear-quadratic optimisation and filtering. The sensitivity of the AAMRE relative to perturbations in its coefficients is studied. Both linear local (norm-wise and componentwise) and non-linear non-local perturbation bounds ar...

The potential V (x) = −x4, which is unbounded below on the real line, can give rise to a well-posed bound state problem when x is taken on a contour in the lower-half complex plane. It is then PT symmetric rather than Hermitian. Nonetheless it has been shown numerically to have a real spectrum, and a proof of reality, involving the correspondence between ordinary differential equations and inte...