Dyson analysed the low-energy excitations of a ferromagnet using a Hamiltonian that was non-Hermitian with respect to the standard inner product. This allowed for a facile rendering of these excitations (known as spin waves) as weakly interacting bosonic quasi-particles. More than 50 years later, we have the full denouement of the non-Hermitian quantum mechanics formalism at our disposal when c...

A Hermitian matrix X is called a g-inverse of a Hermitian matrix A, denoted by A, if it satisfies AXA = A. In this paper, a group of explicit formulas are established for calculating the global maximum and minimum ranks and inertias of the difference A − PNP , where both A and N are Hermitian g-inverses of two Hermitian matrices A and N , respectively. As a consequence, necessary and sufficient...

We study a non-Hermitian generalization of quantum systems in which an imaginary vector potential is added to the momentum operator. In the tight-binding approximation, we make the hopping energy asymmetric in the Hermitian Hamiltonian. In a previous article, we conjectured that the non-Hermitian critical point where the energy gap vanishes is equal to the inverse correlation length of the Herm...

The expectation values of a Hermitian operator  in (2s + 1)2 specific coherent states of a spin are known to determine the operator unambiguously. As shown here, (almost) any other set of (2s + 1)2 coherent state projectors also provide a basis for self-adjoint operators. This is proved by considering the determinant of the Gram matrix associated with the coherent state projectors as a Hamilto...

Renormalization group limit cycles and more chaotic behavior may be commonplace for quantum Hamiltonians requiring renormalization, in contrast to experience based on classical models with critical behavior, where fixed points are far more common. We discuss the simplest quantum model Hamiltonian identified so far that exhibits a renormalization group with both limit cycle and chaotic behavior....

A Hermitian matrix X is called a g-inverse of a Hermitian matrix A, denoted by A, if it satisfies AXA = A. In this paper, a group of explicit formulas are established for calculating the global maximum and minimum ranks and inertias of the difference A − PNP , where both A and N are Hermitian g-inverses of two Hermitian matrices A and N , respectively. As a consequence, necessary and sufficient...

In conventional quantum mechanics, no-deleting and no-cloning theorems indicate that two different nonorthogonal states cannot be perfectly deterministically deleted cloned, respectively. Here, we investigate the deleting cloning in a pseudo-unitary system. We first present pseudo-Hermitian Hamiltonian with real eigenvalues two-qubit By using operators generated from this Hamiltonian, show it i...

This paper is concerned with the computation of 3D vertex singularities of anisotropic elastic fields. The singularities are described by eigenpairs of a corresponding operator pencil on a subdomain of the sphere. The solution approach is to introduce a modified quadratic variational boundary eigenvalue problem which consists of two self-adjoint, positive definite sesquilinear forms and a skew-...

Recently defect production was investigated during non-unitary dynamics due to non-Hermitian Hamiltonian. By ramping up the coupling linearly in time through an exceptional point, defects are produced much same way as approaching a Hermitian critical point. A generalized Kibble--Zurek scaling accounted for ensuing of density terms speed drive and corresponding exponents. Here we extend this set...