For a given standard Hamiltonian H = [p − A(x)]2/(2m) + V (x) with arbitrary complex scalar potential V and vector potential A, with x ∈ R, we construct an invertible antilinear operator τ such that H is τ -anti-pseudo-Hermitian, i.e., H = τHτ−1. We use this result to give the explicit form of a linear Hermitian invertible operator with respect to which any standard PT -symmetric Hamiltonian wi...

We are going to study the limiting spectral measure of fixed dimensional Hermitian block-matrices with large dimensional Wigner blocks. We are going also to identify the limiting spectral measure when the Hermitian block-structure is Circulant. Using the limiting spectral measure of a Hermitian Circulant block-matrix we will show that the spectral measure of a Wigner matrix with k−weakly depend...

We endow a non-semisimple category of modules unrolled quantum $${\mathfrak {sl}(2)}$$ with Hermitian structure. also prove that the CGP TQFT constructed in arXiv:1202.3553 using this is Hermitian. This gives rise to projective representations mapping class group indefinite unitary matrices.

The Lax formulation of the hyper-Hermiticity condition in four dimensions is used to derive a potential that generalises Plebanski’s second heavenly equation for hyper-Kahler 4-manifolds. A class of examples of hyper-Hermitian metrics which depend on two arbitrary functions of two complex variables is given. The twistor theory of four-dimensional hyper-Hermitian manifolds is formulated as a com...

We show that if the Stokes matrix of a connection with a pole of order two and no ramification gives rise, when added to its adjoint, to a positive semi-definite Hermitian form, then the associated integrable twistor structure (or TERP structure, or non-commutative Hodge structure) is pure and polarized.

The statistical properties of trajectories of eigenvalues of Gaussian complex matrices whose Hermitian condition is progressively broken are investigated. It is shown how the ordering on the real axis of the real eigenvalues is reflected in the structure of the trajectories and also in the final distribution of the eigenvalues in the complex plane.

In this paper, we study the iterative algorithms for saddle point problems(SPP). Bai, Golub and Pan recently studied a class of preconditioned Hermitian and skew-Hermitian splitting methods(PHSS). By further accelerating it with another parameters, using the Hermitian/skew-Hermitian splitting iteration technique we present the generalized preconditioned Hermitian and skew-Hermitian splitting me...

We show that, under a condition called minimality, if the Stokes matrix of a connection with a pole of order two and no ramification gives rise, when added to its adjoint, to a positive semi-definite Hermitian form, then the associated integrable twistor structure (or TERP structure, or non-commutative Hodge structure) is pure and polarized.

in this paper‎, ‎we study the extremal‎ ‎ranks and inertias of the hermitian matrix expression $$‎ ‎f(x,y)=c_{4}-b_{4}y-(b_{4}y)^{*}-a_{4}xa_{4}^{*},$$ where $c_{4}$ is‎ ‎hermitian‎, ‎$*$ denotes the conjugate transpose‎, ‎$x$ and $y$ satisfy‎ ‎the following consistent system of matrix equations $a_{3}y=c_{3}‎, ‎a_{1}x=c_{1},xb_{1}=d_{1},a_{2}xa_{2}^{*}=c_{2},x=x^{*}.$ as‎ ‎consequences‎, ‎we g...

Comparing the lopsided Hermitian/skew-Hermitian splitting (LHSS) method and Hermitian/skewHermitian splitting (HSS) method, a new criterion for choosing the above two methods is presented, which is better than that of Li, Huang and Liu [Modified Hermitian and skew-Hermitian splitting methods for nonHermitian positive-definite linear systems, Numer. Lin. Alg. Appl., 14 (2007): 217-235]. Key-Word...