We consider the solution of complex symmetric shifted linear systems. Such systems arise in large-scale electronic structure simulations and there is a strong need for the fast solution of the systems. With the aim of solving the systems efficiently, we consider a special case of the QMR method for non-Hermitian shifted linear systems and propose its weighted quasi-minimal residual approach. A ...

In arXiv:0709.0483 Günther and Samsonov outline a “generalization” of quantum mechanics that involves simultaneous consideration of Hermitian and non-Hermitian operators and promises to be “capable to produce effects beyond those of standard Hermitian quantum mechanics.” We give a simple physical interpretation of Hermiticity and discuss in detail the shortcomings of the above-mentioned composi...

Using the ranks and Moore-Penrose inverses of involved matrices, in this paper we establish some necessary sufficient solvability conditions for a system Sylvester-type quaternion matrix equations, give an expression general solution to when it is solvable. As application system, consider special symmetry solution, named η-Hermitian equations. Moreover, present algorithm numerical example verif...

We call a positive definite even Hermitian lattice even universal if it represents all even positive integers. We introduce a method to get all even universal binary Hermitian lattices over imaginary quadratic fields Q( √ −m) for all m and we list optimal criterions on even universality of Hermitian lattices over Q( √ −m) which admits even universal binary Hermitian lattices. And we apply our r...

The general solution of SUSY intertwining relations for three-dimensional Schrödinger operators is built using the class of second order supercharges with nondegenerate constant metric. This solution includes several models with arbitrary parameters. We are interested only in quantum systems which are not amenable to separation of variables, i.e. can not be reduced to lower dimensional problems...

For the non-Hermitian and positive semidefinite systems of linear equations, we derive sufficient and necessary conditions for guaranteeing the unconditional convergence of the preconditioned Hermitian and skew-Hermitian splitting iteration methods. These result is specifically applied to linear systems of block tridiagonal form to obtain convergence conditions for the corresponding block varia...

One of the simplest non-Hermitian Hamiltonians, first proposed by Schwartz in 1960, that may possess a spectral singularity is analysed from the point of view of the non-Hermitian generalization of quantum mechanics. It is shown that the η operator, being a second-order differential operator, has supersymmetric structure. Asymptotic behaviour of the eigenfunctions of a Hermitian Hamiltonian equ...

We will introduce a method to get all universal Hermitian lattices over imaginary quadratic fields Q( √ −m) for all m. For each imaginary quadratic field Q( √ −m), we obtain a criterion on universality of Hermitian lattices: if a Hermitian lattice L represents 1, 2, 3, 5, 6, 7, 10, 13, 14 and 15, then L is universal. We call this the fifteen theorem for universal Hermitian lattices. Note that t...

We study the continuity equation of Gauduchon metrics and establish its interval maximal existence, which extends Kahler introduced by La Nave \& Tian for Hermitian Sherman Weinkove. Our method is based on solution to conjecture Szekelyhidi, Tosatti

Potential algebras are extended from Hermitian to non-Hermitian Hamiltonians and shown to provide an elegant method for studying the transition from real to complex eigenvalues for a class of non-Hermitian Hamiltonians associated with the complex Lie algebra A1.