Wigner quantization of some one-dimensional Hamiltonians

Recently, several papers have been dedicated to the Wigner quantization of different Hamiltonians. In these examples, many interesting mathematical and physical properties have been shown. Among those we have the ubiquitous relation with Lie superalgebras and their representations. In this paper, we study two one-dimensional Hamiltonians for which the Wigner quantization is related with the orthosymplectic Lie superalgebra osp(1|2). One of them, the Hamiltonian H = xp, is popular due to its connection with the Riemann zeros, discovered by Berry and Keating on the one hand and Connes on the other. The Hamiltonian of the free particle, H_f = p^2/2, is the second Hamiltonian we will examine. Wigner quantization introduces an extra representation parameter for both of these Hamiltonians. Canonical quantization is recovered by restricting to a specific representation of the Lie superalgebra osp(1|2)

Topological quantization of boundary forces and the integrated density of states

For quantum systems described by Schr\"odinger operators on the half-space \RR^{d-1}\times\RR^{leq 0} the boundary force per unit area and unit energy is topologically quantised provided the Fermi energy lies in a gap of the bulk spectrum. Under this condition it is also equal to the integrated density of states at the Fermi energy.Comment: 7 page

Some remarks on quasi-Hermitian operators

A quasi-Hermitian operator is an operator that is similar to its adjoint in some sense, via a metric operator, i.e., a strictly positive self-adjoint operator. Whereas those metric operators are in general assumed to be bounded, we analyze the structure generated by unbounded metric operators in a Hilbert space. Following our previous work, we introduce several generalizations of the notion of similarity between operators. Then we explore systematically the various types of quasi-Hermitian operators, bounded or not. Finally we discuss their application in the so-called pseudo-Hermitian quantum mechanics.Comment: 18page

On the probabilistic description of a multipartite correlation scenario with arbitrary numbers of settings and outcomes per site

We consistently formalize the probabilistic description of multipartite joint measurements performed on systems of any nature. This allows us: (1) to specify in probabilistic terms the difference between nonsignaling, the Einstein- Podolsky-Rosen (EPR) locality and Bell's locality; (2) to introduce the notion of an LHV model for an S_{1}x...xS_{N}-setting N-partite correlation experiment, with outcomes of any spectral type, discrete or continuous, and to prove both general and specific "quantum" statements on an LHV simulation in an arbitrary multipartite case; (3) to classify LHV models for a multipartite quantum state, in particular, to show that any N-partite quantum state, pure or mixed, admits an Sx1x...x1 -setting LHV description; (4) to evaluate a threshold visibility for a noisy bipartite quantum state to admit an S_{1}xS_ {2}-setting LHV description under any generalized quantum measurements of two parties. In a sequel to this paper, we shall introduce a single general representation incorporating in a unique manner all Bell-type inequalities for either joint probabilities or correlation functions that have been introduced or will be introduced in the literature.Comment: 26 pages; added section Conclusions and some references for section

Higher order Schrodinger and Hartree-Fock equations

The domain of validity of the higher-order Schrodinger equations is analyzed for harmonic-oscillator and Coulomb potentials as typical examples. Then the Cauchy theory for higher-order Hartree-Fock equations with bounded and Coulomb potentials is developed. Finally, the existence of associated ground states for the odd-order equations is proved. This renders these quantum equations relevant for physics.Comment: 19 pages, to appear in J. Math. Phy

Green's function for the Hodge Laplacian on some classes of Riemannian and Lorentzian symmetric spaces

We compute the Green's function for the Hodge Laplacian on the symmetric spaces M\times\Sigma, where M is a simply connected n-dimensional Riemannian or Lorentzian manifold of constant curvature and \Sigma is a simply connected Riemannian surface of constant curvature. Our approach is based on a generalization to the case of differential forms of the method of spherical means and on the use of Riesz distributions on manifolds. The radial part of the Green's function is governed by a fourth order analogue of the Heun equation.Comment: 18 page

Neumark Operators and Sharp Reconstructions, the finite dimensional case

A commutative POV measure $F$ with real spectrum is characterized by the existence of a PV measure $E$ (the sharp reconstruction of $F$) with real spectrum such that $F$ can be interpreted as a randomization of $E$. This paper focuses on the relationships between this characterization of commutative POV measures and Neumark's extension theorem. In particular, we show that in the finite dimensional case there exists a relation between the Neumark operator corresponding to the extension of $F$ and the sharp reconstruction of $F$. The relevance of this result to the theory of non-ideal quantum measurement and to the definition of unsharpness is analyzed.Comment: 37 page

Turning big bang into big bounce: II. Quantum dynamics

We analyze the big bounce transition of the quantum FRW model in the setting of the nonstandard loop quantum cosmology (LQC). Elementary observables are used to quantize composite observables. The spectrum of the energy density operator is bounded and continuous. The spectrum of the volume operator is bounded from below and discrete. It has equally distant levels defining a quantum of the volume. The discreteness may imply a foamy structure of spacetime at semiclassical level which may be detected in astro-cosmo observations. The nonstandard LQC method has a free parameter that should be fixed in some way to specify the big bounce transition.Comment: 14 pages, no figures, version accepted for publication in Class. Quant. Gra

Schroedinger operators with singular interactions: a model of tunneling resonances

We discuss a generalized Schr\"odinger operator in $L^2(\mathbb{R}^d), d=2,3$, with an attractive singular interaction supported by a $(d-1)$-dimensional hyperplane and a finite family of points. It can be regarded as a model of a leaky quantum wire and a family of quantum dots if $d=2$, or surface waves in presence of a finite number of impurities if $d=3$. We analyze the discrete spectrum, and furthermore, we show that the resonance problem in this setting can be explicitly solved; by Birman-Schwinger method it is cast into a form similar to the Friedrichs model.Comment: LaTeX2e, 34 page

Long time, large scale limit of the Wigner transform for a system of linear oscillators in one dimension

We consider the long time, large scale behavior of the Wigner transform W_\eps(t,x,k) of the wave function corresponding to a discrete wave equation on a 1-d integer lattice, with a weak multiplicative noise. This model has been introduced in Basile, Bernardin, and Olla to describe a system of interacting linear oscillators with a weak noise that conserves locally the kinetic energy and the momentum. The kinetic limit for the Wigner transform has been shown in Basile, Olla, and Spohn. In the present paper we prove that in the unpinned case there exists $\gamma_0>0$ such that for any $\gamma\in(0,\gamma_0]$ the weak limit of W_\eps(t/\eps^{3/2\gamma},x/\eps^{\gamma},k), as \eps\ll1, satisfies a one dimensional fractional heat equation $\partial_t W(t,x)=-\hat c(-\partial_x^2)^{3/4}W(t,x)$ with $\hat c>0$. In the pinned case an analogous result can be claimed for W_\eps(t/\eps^{2\gamma},x/\eps^{\gamma},k) but the limit satisfies then the usual heat equation