Introduction. Let D be a first-order differential operator of Dirac-type from C(X,E1) to C (X,E2) (E1 and E2 Hermitian N -dimensional vector bundles over a compact n-dimensional C ∞ manifold X with boundary ∂X = X ), and let D≥ be the L2-realization defined by the boundary condition Π≥(u|X′) = 0; here Π≥ is the orthogonal projection onto the nonnegative eigenspace for a certain selfadjoint oper...

The set of continuous norm-preserving stochastic Schrödinger equations associated with the Lindblad master equation is introduced. This set is used to describe the localization properties of the state vector toward eigenstates of the environment operator. Particular focus is placed on determining the stochastic equation which exhibits the highest rate of localization for wide open systems. An e...

We investigate the distribution of the spacings of adjacent eigenvalues of the lattice Dirac operator. At zero chemical potential μ, the nearest-neighbor spacing distribution P (s) follows the Wigner surmise of random matrix theory both in the confinement and in the deconfinement phase. This is indicative of quantum chaos. At nonzero chemical potential, the eigenvalues of the Dirac operator bec...

Here we give brief account of hermitian symplectic spaces, showing that they are intimately connected to symmetric as well as self-adjoint extensions of a symmetric operator. Furthermore we find an explicit parameterisation of the Lagrange Grassmannian in terms of the unitary matrices U(n). This allows us to explicitly describe all self-adjoint boundary conditions for the Schrödinger operator o...

The overlap Dirac operator at nonzero quark chemical potential involves the computation of the sign function of a non-Hermitian matrix. In this talk we present iterative Krylov subspace approximations, with deflation of critical eigenvalues, which we developed to compute the operator on large lattices. We compare the accuracy and efficiency of two alternative approximations based on the Arnoldi...

Let M be a complete Riemannian manifold and D : C∞ 0 (E) → C∞ 0 (F ) a first order differential operator acting between sections of the hermitian vector bundles E, F . Moreover, let V : C∞(E) → L∞ loc (E) be a self–adjoint zero order differential operator. We give a sufficient condition for the Schrödinger operator H = DD + V to be essentially self–adjoint. This generalizes recent work of I. Ol...

In this work we first propose a method for the derivation of general continuous antilinear time-dependent (TD) symmetry operator I(t) I(t) TD non-Hermitian Hamiltonian H(t) disp...

The quantum phase operator has been studied by various authors in the past[1-4]. We here remark on the absence of a hermitian phase operator and the lack of a mathematical basis for ∆N∆φ ≥ 1/2, on the basis of a notion of index or an index theorem. To be specific, we study the simplest one-dimensional harmonic oscillator defined by h = aa+ 1/2 where a and a stand for the annihilation and creati...

We define pseudo-reality and pseudo-adjointness of a Hamiltonian, H, as ρHρ−1 = H∗ and μHμ−1 = H ′, respectively. We prove that the former yields the necessary condition for spectrum to be real whereas the latter helps in fixing a definition for inner-product of the eigenstates. Here we separate out adjointness of an operator from its Hermitian-adjointness. It turns out that a Hamiltonian posse...

Lower bounds on the magnitude of the spectrum of the Hermitian WilsonDirac operator H(m) have previously been derived for 0 < m < 2 when the lattice gauge field satisfies a certain smoothness condition. In this paper lower bounds are derived for 2p−2 < m < 2p for general p = 1, 2, . . . , d where d is the spacetime dimension. The bounds can alternatively be viewed as localisation bounds on the ...