J. R. Klauder | Searcharxiv

arXiv2005

Family of solvable generalized random-matrix ensembles with unitary symmetry

We construct a very general family of characteristic functions describing Random Matrix Ensembles (RME) having a global unitary invariance, and containing an arbitrary, one-variable probability measure which we characterize by a `spread function'. Various choices of the spread function lead to a variety of possible generalized RMEs, which show deviations from the well-known Gaussian RME originally proposed by Wigner. We obtain the correlation functions of such generalized ensembles exactly, and show examples of how particular choices of the spread function can describe ensembles with arbitrary eigenvalue densities as well as critical ensembles with multifractality.

cond-mat.other↗

arXiv2004

Conductance distribution in 3D Anderson insulators: deviation from log-normal form

We show how a recent proposal to obtain the distribution of conductances in three dimensions (3D) from a generalized Fokker-Planck equation for the joint probability distribution of the transmission eigenvalues can be implemented for all strengths of disorder by numerically evaluating certain correlations of transfer matrices. We then use this method to obtain analytically, for the first time, the 3D conductance distribution in the insulating regime and provide a simple understanding of why it differs qualitatively from the log-normal distribution of a quasi one-dimensional wire.

cond-mat.mes-hall↗

arXiv2002

Relativistic coherent states and charge structure of the coordinate and momentum operators

We consider relativistic coherent states for a spin-0 charged particle that satisfy the next additional requirements: (i) the expected values of the standard coordinate and momentum operators are uniquely related to the real and imaginary parts of the coherent state parameter; (ii) these states contain only one charge component. Three cases are considered: free particle, relativistic rotator, and particle in a constant homogeneous magnetic field. For the rotational motion of the two latter cases, such a description leads to the appearance of the so-called nonlinear coherent states.

quant-ph↗

arXiv2000

Temporally stable coherent states for infinite well and Pöschl-Teller potentials

This paper is a direct illustration of a construction of coherent states which has been recently proposed by two of us (JPG and JK). We have chosen the example of a particle trapped in an infinite square-well and also in Pöschl-Teller potentials of the trigonometric type. In the construction of the corresponding coherent states, we take advantage of the simplicity of the solutions, which ultimately stems from the fact they share a common SU(1,1) symmetry à la Barut--Girardello. Many properties of these states are then studied, both from mathematical and from physical points of view.

math-ph↗

arXiv2000

On the Implementation of Constraints through Projection Operators

Quantum constraints of the type Q ψ= 0 can be straightforwardly implemented in cases where Q is a self-adjoint operator for which zero is an eigenvalue. In that case, the physical Hilbert space is obtained by projecting onto the kernel of Q, i.e. H_phys = ker(Q) = ker(Q*). It is, however, nontrivial to identify and project onto H_phys when zero is not in the point spectrum but instead is in the continuous spectrum of Q, because in this case the kernel of Q is empty. Here, we observe that the topology of the underlying Hilbert space can be harmlessly modified in the direction perpendicular to the constraint surface in such a way that Q becomes non-self-adjoint. This procedure then allows us to conveniently obtain H_phys as the proper Hilbert subspace H_phys = ker(Q*), on which one can project as usual. In the simplest case, the necessary change of topology amounts to passing from an L^2 Hilbert space to a Sobolev space.

quant-ph↗

arXiv1999

Generalized Fokker-Planck Equation For Multichannel Disordered Quantum Conductors

The Dorokhov-Mello-Pereyra-Kumar (DMPK) equation, which describes the distribution of transmission eigenvalues of multichannel disordered conductors, has been enormously successful in describing a variety of detailed transport properties of mesoscopic wires. However, it is limited to the regime of quasi one dimension only. We derive a one parameter generalization of the DMPK equation, which should broaden the scope of the equation beyond the limit of quasi one dimension.

cond-mat.dis-nn↗

arXiv1996

Dynamics as Shadow of Phase Space Geometry

Starting with the generally well accepted opinion that quantizing an arbitrary Hamiltonian system involves picking out some additional structure on the classical phase space (the {\sl shadow} of quantum mechanics in the classical theory), we describe classical as well as quantum dynamics as a purely geometrical effect by introducing a {\sl phase space metric structure}. This produces an ${\cal O}(\hbar)$ modification of the classical equations of motion reducing at the same time the quantization of an arbitrary Hamiltonian system to standard procedures. Our analysis is carried out in analogy with the adiabatic motion of a charged particle in a curved background (the additional metric structure) under the influence of a universal magnetic field (the classical symplectic structure). This allows one to picture dynamics in an unusual way, and reveals a dynamical mechanism that produces the selection of the right set of physical quantum states.

quant-ph↗

arXiv1995

The Schroedinger Problem, Levy Processes Noise in Relativistic Quantum Mechanics

The main purpose of the paper is an essentially probabilistic analysis of relativistic quantum mechanics. It is based on the assumption that whenever probability distributions arise, there exists a stochastic process that is either responsible for temporal evolution of a given measure or preserves the measure in the stationary case. Our departure point is the so-called Schrödinger problem of probabilistic evolution, which provides for a unique Markov stochastic interpolation between any given pair of boundary probability densities for a process covering a fixed, finite duration of time, provided we have decided a priori what kind of primordial dynamical semigroup transition mechanism is involved. In the nonrelativistic theory, including quantum mechanics, Feyman-Kac-like kernels are the building blocks for suitable transition probability densities of the process. In the standard "free" case (Feynman-Kac potential equal to zero) the familiar Wiener noise is recovered. In the framework of the Schrödinger problem, the "free noise" can also be extended to any infinitely divisible probability law, as covered by the Lévy-Khintchine formula. Since the relativistic Hamiltonians $|\nabla |$

chao-dyn↗

arXiv1995

The Schroedinger Problem, Levy Processes Noise in Relativistic Quantum Mechanics

The main purpose of the paper is an essentially probabilistic analysis of relativistic quantum mechanics. It is based on the assumption that whenever probability distributions arise, there exists a stochastic process that is either responsible for temporal evolution of a given measure or preserves the measure in the stationary case. Our departure point is the so-called Schrödinger problem of probabilistic evolution, which provides for a unique Markov stochastic interpolation between any given pair of boundary probability densities for a process covering a fixed, finite duration of time, provided we have decided a priori what kind of primordial dynamical semigroup transition mechanism is involved. In the nonrelativistic theory, including quantum mechanics, Feyman-Kac-like kernels are the building blocks for suitable transition probability densities of the process. In the standard "free" case (Feynman-Kac potential equal to zero) the familiar Wiener noise is recovered. In the framework of the Schrödinger problem, the "free noise" can also be extended to any infinitely divisible probability law, as covered by the Lévy-Khintchine formula. Since the relativistic Hamiltonians $|\nabla |$ and $\sqrt {-\triangle +m^2}-m$ are known to generate such laws, we focus on them for the analysis of probabilistic phenomena, which are shown to be associated with the relativistic wave (D'Alembert) and matter-wave (Klein-Gordon) equations, respectively. We show that such stochastic processes exist and are spatial jump processes. In general, in the presence of external potentials, they do not share the Markov property, except for stationary situations. A concrete example of the pseudodifferential Cauchy-Schrödinger evolution is analyzed in detail. The relativistic covariance of related wave

quant-ph↗