This lecture takes a closer look at Hermitian matrices and at their eigenvalues. After a few generalities about Hermitian matrices, we prove a minimax and maximin characterization of their eigenvalues, known as Courant–Fischer theorem. We then derive some consequences of this characterization, such as Weyl theorem for the sum of two Hermitian matrices, an interlacing theorem for the sum of two ...

Let G be a noncompact simple Lie group with finite center, let K be a maximal compact subgroup, and suppose that rank G = rank K. If G/K is not Hermitian symmetric, then a theorem of Borel and de Siebenthal gives the existence of a system of positive roots relative to a compact Cartan subalgebra so that there is just one noncompact simple root and it occurs exactly twice in the largest root. Le...

We provide time-evolution operators, gauge transformations and a perturbative treatment for non-Hermitian Hamiltonian systems, which are explicitly timedependent. We determine various new equivalence pairs for Hermitian and non-Hermitian Hamiltonians, which are therefore pseudo-Hermitian and in addition in some cases also invariant under PT-symmetry. In particular, for the harmonic oscillator p...

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...

We introduce the space $X$ of quaternion hermitian forms size $n$ on a ${\mathfrak p}$-adic field with odd residual characteristic, and define typical spherical functions $\omega(x;s)$ give their induction formula sizes by using local densities forms. Then we functional equation respect to $S_n$, determine explicit formulas $\omega(x;s)$. On other hand, transform Schwartz ${\mathcal S}(K\backsl...

Let K be a number field and OK the ring of integers of K. Let (E, h) be a hermitian finitely generated flat OK-module. For an OK-submodule F of E, let us denote by hF →֒E the submetric of F induced by h. It is well known that the set of all saturated OK-submodules F with d̂eg(F, hF →֒E) ≥ c is finite for any real numbers c (for details, see [4, the proof of Proposition 3.5]). In this note, we woul...

A Hermitian and an anti-Hermitian first-order intertwining operators are introduced and a class of η-weak-pseudo-Hermitian positiondependent mass (PDM) Hamiltonians are constructed. A corresponding reference-target η-weak-pseudo-Hermitian PDM – Hamiltonians’ map is suggested. Some η-weak-pseudo-Hermitian PT -symmetric Scarf II and periodic-type models are used as illustrative examples. Energy-l...

A positive definite even Hermitian lattice is called 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 positive square-free integers m and we list optimal criterions on even universality of Hermitian lattices over Q( √ −m) which admits even universal binary Hermit...

In this paper we study a nonlinear elliptic system of equations imposed on a map from a complete Hermitian (non-Kähler) manifold to a Riemannian manifold. This system is more appropriate to Hermitian geometry than the harmonic map system since it is compatible with the holomorphic structure of the domain manifold in the sense that holomorphic maps are Hermitian harmonic maps. It was first studi...

let a be a banach algebra and e be a banach a-bimodule then s = a  e, the l1-direct sum of a and e becomes a module extension banach algebra when equipped with the algebras product (a; x):(a′; x′) = (aa′; a:x′ + x:a′). in this paper, we investigate △-amenability for these banach algebras and we show that for discrete inverse semigroup s with the set of idempotents es, the module extension bana...