Tamás Fülöp

A free particle on a circle with point interaction

Abstract The quantum dynamics of a free particle on a circle with point interaction is described by a U (2) family of self-adjoint Hamiltonians. We provide a classification of the family by introducing a number of subfamilies and thereby analyze the spectral structure in detail. We find that the spectrum depends on a subset of U (2) parameters rather than the entire U (2) needed for the Hamiltonians, and that in particular there exists a subfamily in U (2) where the spectrum becomes parameter-independent. We also show that, in some specific cases, the WKB semiclassical approximation becomes exact (modulo phases) for the system.

Symmetry, Duality, and Anholonomy of Point Interactions in One Dimension

Abstract We analyze the spectral structure of a one-dimensional quantum mechanical system with point interaction, which is known to be parametrized by the group U(2). Based on the classification of the interactions in terms of symmetries, we show, on a general basis, how the fermion–boson duality and the spectral anholonomy recently discovered can arise. A vital role is played by a hidden su(2) formed by a certain set of discrete transformations, which becomes a symmetry if the point interaction belongs to a distinguished U(1) subfamily in which all states are doubly degenerate. Within the U(1), there is a particular interaction that admits the interpretation of the system as a supersymmetric Witten model.

Inequivalent quantizations of the three-particle Calogero model constructed by separation of variables

Abstract We quantize the 1-dimensional 3-body problem with harmonic and inverse square pair potential by separating the Schrodinger equation following the classic work of Calogero, but allowing all possible self-adjoint boundary conditions for the angular and radial Hamiltonians. The inverse square coupling constant is taken to be g = 2 ν ( ν − 1 ) with 1 2 ν 3 2 and then the angular Hamiltonian is shown to admit a 2-parameter family of inequivalent quantizations compatible with the dihedral D 6 symmetry of its potential term 9 ν ( ν − 1 ) / sin 2 3 ϕ . These are parametrized by a matrix U ∈ U ( 2 ) satisfying σ 1 U σ 1 = U , and in all cases we describe the qualitative features of the angular eigenvalues and classify the eigenstates under the D 6 symmetry and its S 3 subgroup generated by the particle exchanges. The angular eigenvalue λ enters the radial Hamiltonian through the potential ( λ − 1 4 ) / r 2 allowing a 1-parameter family of self-adjoint boundary conditions at r = 0 if λ 1 . For 0 λ 1 our analysis of the radial Schrodinger equation is consistent with previous results on the possible energy spectra, while for λ 0 it shows that the energy is not bounded from below rejecting those Us admitting such eigenvalues as physically impermissible. The permissible self-adjoint angular Hamiltonians include, for example, the cases U = ± 1 2 , ± σ 1 , which are explicitly solvable and are presented in detail. The choice U = − 1 2 reproduces Calogeros quantization, while for the choice U = σ 1 the system is smoothly connected to the harmonic oscillator in the limit ν → 1 .

Connection conditions and the spectral family under singular potentials

To describe a quantum system whose potential is divergent at one point, one must provide proper connection conditions for the wavefunctions at the singularity. Generalizing the scheme used for point interactions in one dimension, we present a set of connection conditions which are well defined even if the wavefunctions and/or their derivatives are divergent at the singularity. Our generalized scheme covers the entire U(2) family of quantizations (self-adjoint Hamiltonians) admitted for the singular system. We use this scheme to examine the spectra of the Coulomb potential V(x) = −e2/|x| and the harmonic oscillator with square inverse potential V(x) = (mω2/2)x2 + g/x2, and thereby provide a general perspective for these models which have previously been treated with restrictive connection conditions resulting in conflicting spectra. We further show that, for any parity invariant singular potential V(−x) = V(x), the spectrum is determined solely by the eigenvalues of the characteristic matrix U U(2).

Möbius structure of the spectral space of Schrödinger operators with point interaction

The Schrodinger operator with point interaction in one dimension has a U(2) family of self-adjoint extensions. We study the spectrum of the operator and show that (i) the spectrum is uniquely determined by the eigenvalues of the matrix U∈U(2) that characterizes the extension, and that (ii) the space of distinct spectra is given by the orbifold T2/Z2 which is a Mobius strip with boundary. We employ a parametrization of U(2) that admits a direct physical interpretation and furnishes a coherent framework to realize the spectral duality and anholonomy recently found. This allows us to find that (iii) physically distinct point interactions form a three-parameter quotient space of the U(2) family.

Deviation from the Fourier law in room-temperature heat pulse experiments

Abstract We report heat pulse experiments at room temperature that cannot be described by Fouriers law. The experimental data are modeled properly by the Guyer–Krumhansl equation, in its over-diffusion regime. The phenomenon may be due to conduction channels with differing conductivities and parallel to the direction of the heat flux.

Spectral Properties on a Circle with a Singularity

We investigate the spectral and symmetry properties of a quantum particle moving on a circle with a pointlike singularity (or point interaction). We find that, within the U (2) family of the quantum mechanically allowed distinct singularities, a U (1) equivalence (of duality-type) exists, and accordingly the space of distinct spectra is U (1) ×[ SU (2) / U (1)], topologically a filled torus. We explore the relationship of special subfamilies of the U (2) family to corresponding symmetries, and identify the singularities that admit an N = 2 supersymmetry. Subfamilies that are distinguished in the spectral properties or the WKB exactness are also pointed out. The spectral and symmetry properties are also studied in the context of the circle with two singularities, which provides a useful scheme to discuss the symmetry properties on a general basis.

Classical aspects of quantum walls in one dimension

We investigate the system of a particle moving on a half line x{>=}0 under the general walls at x=0 that are permitted quantum mechanically. These quantum walls, characterized by a parameter L, are shown to be realized as a limit of regularized potentials. We then study the classical aspects of the quantum walls by seeking a classical counterpart that admits the same time delay in scattering with the quantum wall, and also by examining the WKB exactness of the transition kernel based on the regularized potentials. It is shown that no classical counterpart exists for walls with L<0, and that the WKB exactness can hold only for L=0 and L={infinity}.

Weakly non-local fluid mechanics: the Schrödinger equation

A weakly non-local extension of ideal fluid dynamics is derived from the Second Law of thermodynamics. It is proved that in the reversible limit, the additional pressure term can be derived from a potential. The requirement of the additivity of the specific entropy function determines the quantum potential uniquely. The relation to other known derivations of the Schrödinger equation (stochastic, Fisher information, exact uncertainty) is clarified.

One-dimensional drift-diffusion between two absorbing boundaries: application to granular segregation

Motivated by a novel method for granular segregation, we analyse the one-dimensional drift-diffusion between two absorbing boundaries. The time evolution of the probability distribution and the rate of absorption are given by explicit formulae; the splitting probability and the mean first-passage time are also calculated. Applying the results we find optimal parameters for segregating binary granular mixtures.