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 give a necessary and sufficient condition for the reality of the spectrum of a non-Hermitian Hamiltonian admitting a complete set of biorthonormal eigenvectors. Recently, we have explored in [1] the basic mathematical structure underlying the spectral properties of PT -symmetric Hamiltonians [2]. In particular, we have shown that these properties are associated with a class of more general (...

A generalized anti-hermitian staggered Dirac operator is formulated. Its relation with noncommutative geometry is briefly reviewed. Once this antihermitian operator is modified to be “γ5-hermitian”, it will provide a new solution to Ginsparg-Wilson relation, basing on an abstract algebraic analysis of Neuberger’s overlap construction and a redefinition of chirality.

We develop a duality theory for unbounded Hermitian operators with dense domain in Hilbert space. As is known, the obstruction for a Hermitian operator to be selfadjoint or to have selfadjoint extensions is measured by a pair of deficiency indices, and associated deficiency spaces; but in practical problems, the direct computation of these indices can be difficult. Instead, in this paper we ide...

In quantum mechanics, any quantum state ‘vector’ | i may be expanded as a linear superposition of the eigenvectors of any Hermitian operator | i = P n an|ni. This is the Expansion Principle of quantum mechanics. For most problems, the Hermitian operator of choice is the Hamiltonian operator Ĥ = (p̂/2m0) + V (r), but it need not be. We choose the Hamiltonian operator since there exist a few probl...

A non-Hermitian operator with a real spectrum and a complete set of eigenvectors may serve as the Hamiltonian operator for a unitary quantum system provided that one makes an appropriate choice for the defining the inner product of physical Hilbert state. We study the consequences of such a choice for the representation of states in terms of projection operators and the geometry of the state sp...

I extend the formulation of pseudo-Hermitian quantum mechanics to η(+)-pseudo-Hermitian Hamiltonian operators H with an unbounded metric operator η(+). In particular, I give the details of the construction of the physical Hilbert space, observables and equivalent Hermitian Hamiltonian for the case that H has a real and discrete spectrum and its eigenvectors belong to the domain of η(+) and cons...

In this thesis algorithmic improvements and variants for two-flavor lattice QCD simulations with dynamical fermions are studied using the O(a) improved Dirac-Wilson operator in the Schrödinger functional setup and employing a hybrid Monte Carlo-type (HMC) update. Both, the Hermitian and the Non-Hermitian operator are considered. For the Hermitian Dirac-Wilson operator we investigate the advanta...

Relativistic massive bosons with spin one are considered in several quantization schemes. In all of them the system is shown described by a non-Hermitian Hamiltonian H 6= H and helicity operator Λ. Constructively we show that in all of the contemplated schemes both these operators H and Λ prove simultaneously PT −symmetric, i.e., pseudo-Hermitian with respect to a certain not too complicated in...

We investigate the spectral flows of the hermitian Wilson-Dirac operator in the fundamental and adjoint representations on two ensembles of pure SU(2) gauge field configurations at the same physical volume. We find several background gauge field configurations where the index of the hermitian Wilson-Dirac operator in the adjoint representation is not four times the index in the fundamental repr...