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T. Shigehara

Publications and source records attributed to T. Shigehara.

15 recordsLinked to original sources
arXiv2006

Continuous Spectra of Generalized Kronig-Penney Model

The standard Kronig-Penney model with periodic $δ$ potentials is extended to the cases with generalized contact interactions. The eigen equation which determines the dispersion relation for one-dimensional periodic array of the generalized contact interactions is deduced with the transfer matrix formalism. Numerical results are presented which reveal unexpected band spectra with broader band gap in higher energy region for generic model with generalized contact interaction.

quant-ph↗
arXiv1999

Generalized One-Dimensional Point Interaction in Relativistic and Non-relativistic Quantum Mechanics

We first give the solution for the local approximation of a four parameter family of generalized one-dimensional point interactions within the framework of non-relativistic model with three neighboring $δ$ functions. We also discuss the problem within relativistic (Dirac) framework and give the solution for a three parameter family. It gives a physical interpretation for so-called $ε$ potential. It will be also shown that the scattering properties at high energy substantially differ between non-relativistic and relativistic cases.

quant-ph↗
arXiv1999

Realization of a Four Parameter Family of Generalized One-Dimensional Contact Interactions by Three Nearby Delta Potentials with Renormalized Strengths

We propose a new method to construct a four parameter family of quantum-mechanical point interactions in one dimension, which is known as all possible self-adjoint extensions of the symmetric operator $T=-Δ\lceil C^{\infty}_{0}({\bf R} \backslash\{0\})$. It is achieved in the small distance limit of equally spaced three neighboring Dirac's $δ$ potentials. The strength for each $δ$ is appropriately renormalized according to the distance and it diverges, in general, in the small distance limit. The validity of our method is ensured by numerical calculations. In general cases except for usual $δ$, the wave function discontinuity appears around the interaction and one can observe such a tendency even at a finite distance level.

quant-ph↗
arXiv1998

Equivalence of Local and Separable Realizations of the Discontinuity-Inducing Contact Interaction and Its Perturbative Renormalizability

We prove that the separable and local approximations of the discontinuity-inducing zero-range interaction in one-dimensional quantum mechanics are equivalent. We further show that the interaction allows the perturbative treatment through the coupling renormalization. Keywords: one-dimensional system, generalized contact interaction, renormalization, perturbative expansion. PACS Nos: 3.65.-w, 11.10.Gh, 31.15.Md

quant-ph↗
arXiv1998

Some Aspects of Generalized Contact Interaction in One-Dimensional Quantum Mechanics

We construct a one-dimensional contact interaction ($ε$ potential) which induces the discontinuity of the wave function while keeping its derivative continuous. By combining the $ε$ potential and the Dirac's $δ$ function, we construct most general one-dimensional contact interactions allowable under the time reversal symmetry. We present some elementary results for the scattering problem which suggest a dual relation between $δ$ and $ε$ potentials.

quant-ph↗
arXiv1998

Fermion-Boson Duality of One-dimensional Quantum Particles with Generalized Contact Interaction

For a system of spinless one-dimensional fermions, the non-vanishing short-range limit of two-body interaction is shown to induce the wave-function discontinuity. We prove the equivalence of this fermionic system and the bosonic particle system with two-body $δ$-function interaction with the reversed role of strong and weak couplings. KEYWORDS: one-dimensional system, $ε$-interaction, solvable many-body problem, exact bosonization

quant-ph↗
arXiv1998

Realizing Discontinuous Wave Functions with Renormalized Short-Range Potentials

It is shown that a potential consisting of three Dirac's delta functions on the line with disappearing distances can give rise to the discontinuity in wave functions with the proper renormalization of the delta function strength. This can be used as a building block, along with the usual Dirac's delta, to construct the most general three-parameter family of point interactions, which allow both discontinuity and asymmetry of the wave function, as the zero-size limit of self-adjoint local operators in one-dimensional quantum mechanics. Experimental realization of the Neumann boundary is discussed. KEYWORDS: point interaction, self-adjoint extension, $δ'$ potential, wave function discontinuity, Neumann boundary PACS Nos: 3.65.-w, 11.10.Gh, 68.65+g

quant-ph↗
arXiv1997

Spectral Properties of the Two-Dimensional Laplacian with a Finite Number of Point Interactions

We discuss spectral properties of the Laplacian with multiple ($N$) point interactions in two-dimensional bounded regions. A mathematically sound formulation for the problem is given within the framework of the self-adjoint extension of a symmetric (Hermitian) operator in functional analysis. The eigenvalues of this system are obtained as the poles of a transition matrix which has size $N$. Closely examining a generic behavior of the eigenvalues of the transition matrix as a function of the energy, we deduce the general condition under which point interactions have a substantial effect on statistical properties of the spectrum.

quant-ph↗
arXiv1997

Spectral Properties of Three-Dimensional Quantum Billiards with a Pointlike Scatterer

We examine the spectral properties of three-dimensional quantum billiards with a single pointlike scatterer inside. It is found that the spectrum shows chaotic (random-matrix-like) characteristics when the inverse of the formal strength $\bar{v}^{-1}$ is within a band whose width increases parabolically as a function of the energy. This implies that the spectrum becomes random-matrix-like at very high energy irrespective to the value of the formal strength. The predictions are confirmed by numerical experiments with a rectangular box. The findings for a pointlike scatterer are applied to the case for a small but finite-size impurity. We clarify the proper procedure for its zero-size limit which involves non-trivial divergence. The previously known results in one and two-dimensional quantum billiards with small impurities inside are also reviewed from the present perspective.

chao-dyn↗
arXiv1996

Resonance Tunneling in Double-Well Billiards with a Pointlike Scatterer

The coherent tunneling phenomenon is investigated in rectangular billiards divided into two domains by a classically unclimbable potential barrier. We show that by placing a pointlike scatterer inside the billiard, we can control the occurrence and the rate of the resonance tunneling. The key role of the avoided crossing is stressed. Keywords: chaotic tunneling, quantum billiard, delta potential, diabolical degeneracy PACS: 3.65.-w, 4.30.Nk, 5.45.+b, 73.40.Gk

chao-dyn↗
arXiv1996

Flux String in Quantum Billiards with Two Particles

We examine the quantum motion of two particles interacting through a contact force which are confined in a rectangular domain in two and three dimensions. When there is a difference in the mass scale of two particles, adiabatic separation of the fast and slow variables can be performed. Appearance of the Berry phase and magnetic flux is pointed out. The system is reduced to a one-particle Aharonov-Bohm billiard in two-dimensional case. In three dimension, the problem effectively becomes the motion of a particle in the presence of closed flux string in a box billiard.

hep-th↗
arXiv1996

Wave Chaos in Quantum Billiards with Small but Finite-Size Scatterer

We study the low energy quantum spectra of two-dimensional rectangular billiards with a small but finite-size scatterer inside. We start by examining the spectral properties of billiards with a single pointlike scatterer. The problem is formulated in terms of self-adjoint extension theory of functional analysis. The condition for the appearance of so-called wave chaos is clarified. We then relate the pointlike scatterer to a finite-size scatterer through the appropriate truncation of basis. We show that the signature of wave chaos in low energy states is most prominent when the scatterer is weakly attractive. As an illustration, numerical results of a rectangular billiard with a small rectangular scatterer inside are exhibited.

chao-dyn↗
arXiv1995

Level spacing distribution of pseudointegrable billiard

In this paper, we examine the level spacing distribution $P(S)$ of the rectangular billiard with a single point-like scatterer, which is known as pseudointegrable. It is shown that the observed $P(S)$ is a new type, which is quite different from the previous conclusion. Even in the strong coupling limit, the Poisson-like behavior rather than Wigner-like is seen for $S>1$, although the level repulsion still remains in the small $S$ region. The difference from the previous works is analyzed in detail.

chao-dyn↗