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# Point interactions in two and three dimensions as models of small scatterers

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*in*Physics Letters A 222(1) · October 1996

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Abstract

In addition to the conventional renormalized-coupling-constant picture, point interactions in two and three dimensions are shown to model within a suitable energy range scattering on localized potentials, both attractive and repulsive.

- ... • Diffraction on a delta-like singularity of the potential in two dimensions [19] [20]: ...... where k = √ 2mE/ and a is a parameter characterising the strength of the potential, and γ is Euler's constant. • Diffraction on a delta-like singularity of the potential in three dimensions [19] [20]: ...Article
- Oct 1999
- J Phys Math Gen

We investigate the influence of diffraction on the statistics of energy levels in quantum systems with a chaotic classical limit. By applying the geometrical theory of diffraction we show that diffraction on singularities of the potential can lead to modifications in semiclassical approximations for spectral statistics that persist in the semiclassical limit $\hbar \to 0$. This result is obtained by deriving a classical sum rule for trajectories that connect two points in coordinate space. Comment: 14 pages, no figure, to appear in J. Phys. A - ... Numerous applications of point and contact interactions can be found in various areas of physics such as quantum physics, 1,2 solid state physics [3][4][5][6][7] or in optics. 8 The increased interest in these models last couple of decades is mainly due to the rapid progress in fabricating nanoscale quantum devices. ...Article
- Jun 2012
- MOD PHYS LETT B

The scattering properties of four-parameter one-dimensional point interactions also define time aspect of these interactions. In this paper we investigate how different families of point interactions affect the relevant tunneling times. Particular emphasis is given to various interpretations of so called δ' potential. - ... The strength of disorder is thus measured by the parameter 1/Kl. In most theoretical approaches, the randomly distributed impurities are often approximated by point-like potentials [3, 4, 5]. Although this type of potential greatly simplifies the analytical calculations, it has been found that this approximation can lead to incorrect results, in particular as far as causality violation is concerned [6] . ...The mean free path is an essential characteristic length in disordered systems. In microscopic calculations, it is usually approximated by the classical value of the elastic mean free path. It corresponds to the Boltzmann mean free path when only isotropic scattering is considered, but it is different for anisotropic scattering. In this paper, we work out the corrections to the so called Boltzmann mean free path due to multiple scattering effects on finite size scatterers, in the s-wave approximation, ie. when the elastic mean free path is equivalent to the Boltzmann mean free path. The main result is the expression for the mean free path expanded in powers of the perturbative parameter given by the scatterer density. Comment: 12 pages
- ... where D is the diffraction coefficient for the diffraction on the singularity of the potential [9, 16]. It can be parameterised in the following form ...Article
- Oct 2000
- J Phys Math Gen

We consider quantum systems with a chaotic classical limit that are perturbed by a point-like scatterer. The spectral form factor K(τ) for these systems is evaluated semiclassically in terms of periodic and diffractive orbits. It is shown for order τ2 and τ3 that off-diagonal contributions to the form factor which involve diffractive orbits cancel exactly the diagonal contributions from diffractive orbits, implying that the perturbation by the scatterer does not change the spectral statistic. We further show that parametric spectral statistics for these systems are universal for small changes of the strength of the scatterer. - ... section 3 for a detailed discussion). It is known that smooth potentials may be modeled by delta potentials in a suitable energy range, where the wavelength is much larger than the size of the support of the smooth potential (cf. for instance [10, 11]). We consider a weak disorder regime. ...ArticleFull-text available
- Apr 2016

We study a random Schroedinger operator, the Laplacian with random Dirac delta potentials on a torus T^d_L = R^d/LZ^d, in the thermodynamic limit L\to\infty, for dimension d=2. The potentials are located on a randomly distorted lattice Z^2+\omega, where the displacements are i.i.d. random variables sampled from a compactly supported probability density. We prove that, if the disorder is sufficiently weak, there exists a certain energy threshold E_0>0 above which exponential localization of the eigenfunctions must break down. In fact we can rule out any decay faster than a certain polynomial one. Our results are obtained by translating the problem of the distribution of eigenfunctions of the random Schroedinger operator into a study of the spatial distribution of two point correlation densities of certain random superpositions of Green's functions and its relation with a lattice point problem. - ... There has also been attempts to explore the scattering properties of semi-infinite periodic arrays of δ-function potentials in one dimension[18]. The study of higher dimensional generalizations of these potentials, however, is a much more difficult problem[6,19,20]. In particular, the divergences appearing in the Green's function defined by these potentials have led to intricate renormalization schemes for dealing with them[2,3,4,5]. ...Article
- Aug 2017
- J PHYS A-MATH THEOR

We use the transfer matrix formulation of scattering theory in two-dimensions to treat the scattering problem for a potential of the form $v(x,y)=\zeta\,\delta(ax+by)g(bx-ay)$ where $\zeta,a$, and $b$ are constants, $\delta(x)$ is the Dirac $\delta$ function, and $g$ is a real- or complex-valued function. We map this problem to that of $v(x,y)=\zeta\,\delta(x)g(y)$ and give its exact and analytic solution for the following choices of $g(y)$: i) A linear combination of $\delta$-functions, in which case $v(x,y)$ is a finite linear array of two-dimensional $\delta$-functions; ii) A linear combination of $e^{i\alpha_n y}$ with $\alpha_n$ real; iii) A general periodic function that has the form of a complex Fourier series. In particular we solve the scattering problem for a potential consisting of an infinite linear periodic array of two-dimensional $\delta$-functions. We also prove a general theorem that gives a sufficient condition for different choices of $g(y)$ to produce the same scattering amplitude within specific ranges of values of the wavelength $\lambda$. For example, we show that for arbitrary real and complex parameters, $a$ and $\mathfrak{z}$, the potentials $ \mathfrak{z} \sum_{n=-\infty}^\infty\delta(x)\delta(y-an)$ and $a^{-1}\mathfrak{z}\delta(x)[1+2\cos(2\pi y/a)]$ have the same scattering amplitude for $a< \lambda\leq 2a$. - ... In the Quantum Chaos literature the Seba billiard [15], a delta potential placed inside an irrational rectangular billiard, has attracted considerable attention [17, 18, 19, 20, 16, 21, 4]. Seba introduced the model to investigate the transition between integrability and chaos in quantum systems and numerical experiments revealed features characteristic of chaotic systems: level repulsion and a Gaussian value distribution of the wave functions — in agreement with Berry's random wave conjecture [1]. ...We consider the Laplacian with a delta potential (a "point scatterer") on an irrational torus, where the square of the side ratio is diophantine. The eigenfunctions fall into two classes ---"old" eigenfunctions (75%) of the Laplacian which vanish at the support of the delta potential, and therefore are not affected, and "new" eigenfunctions (25%) which are affected, and as a result feature a logarithmic singularity at the location of the delta potential. Within a full density subsequence of the new eigenfunctions we determine all semiclassical measures in the weak coupling regime and show that they are localized along 4 wave vectors in momentum space --- we therefore prove the existence of so-called "superscars" as predicted by Bogomolny and Schmit. This result contrasts the phase space equidistribution which is observed for a full density subset of the new eigenfunctions of a point scatterer on a rational torus. Further, in the strong coupling limit we show that a weaker form of localization holds for a positive proportion of the new eigenvalues; in particular quantum ergodicity does not hold. We also explain how our results can be modified for rectangles with Dirichlet boundary conditions with a point scatterer in the interior. In this case our results extend previous work of Keating, Marklof and Winn who proved the existence of localized semiclassical measures under a non-clustering condition on the spectrum of the Laplacian.
- ... The perturbation series solution of the Lippmann-Schwinger equation [18] for the delta-function potential in two dimensions is plagued with the emergence of infinities. The same problem arises in the study of the corresponding spectral problem and has led to the development of intricate renormalization schemes that amount to the introduction of a length scale for the problem [29][30][31][32][33][34][35][36][37][38]. The problem of divergences also arises in the standard treatment of the delta-function potential in 3D. ...Article
- Nov 2015

Transfer matrix is an indispensable tool in the study of scattering phenomena in (effectively) one-dimensional systems. We introduce a genuine multidimensional notion of transfer matrix and use it to develop a powerful alternative to the standard S-matrix formulation of scattering theory. Because this transfer matrix shares the composition property of its one-dimensional analog, our formulation allows for the determination of the scattering properties of any scattering potential $v$ from that of any collection of its truncations $v_j$ along the scattering axis as long as $v=\sum_jv_j$. This is the main advantage of our approach particularly with regard to the numerical solution of multidimensional scattering problems. We demonstrate its application in solving the scattering problem for delta-function potentials in two and three-dimensions and provide an analytic treatment of the scattering of electromagnetic waves from an infinite slab of optically active material with a surface line defect. In particular, we determine the laser threshold condition for the latter system and show that the presence of the defect makes the slab begin lasing for arbitrarily small gain coefficients. - Article
- Feb 1997
- PHYS LETT A

We propose a model for scattering in a flat resonator with a thin antenna. The results are applied to rectangular microwave cavities. We compute the resonance spacing distribution and show that it agrees well with experimental data provided the antenna radius is much smaller than wavelengths of the resonance wavefunctions. - Article
- Jun 2007
- NONLINEARITY

We derive semiclassical approximations for wavefunctions, Green's functions and expectation values for classically chaotic quantum systems. Our method consists of applying singular and regular perturbations to quantum Hamiltonians. The wavefunctions, Green's functions and expectation values of the unperturbed Hamiltonian are expressed in terms of the spectral determinant of the perturbed Hamiltonian. Semiclassical resummation methods for spectral determinants are applied and yield approximations in terms of a finite number of classical trajectories. The final formulas have a simple form. In contrast to Poincare surface of section methods, the resummation is done in terms of the periods of the trajectories. Comment: 18 pages, no figures - ArticleFull-text available
- Nov 2007

We investigate relations between spectral properties of a single-centre point-interaction Hamiltonian describing a particle confined to a bounded domain $\Omega\subset\mathbb{R}^{d},\: d=2,3$, with Dirichlet boundary, and the geometry of $\Omega$. For this class of operators Krein's formula yields an explicit representation of the resolvent in terms of the integral kernel of the unperturbed one, $(-\Delta_{\Omega}^{D}+z) ^{-1}$. We use a moving plane analysis to characterize the behaviour of the ground-state energy of the Hamiltonian with respect to the point-interaction position and the shape of $\Omega$, in particular, we establish some conditions showing how to place the interaction to optimize the principal eigenvalue. Comment: LaTeX, 15 pages - Article
- Mar 2002
- PHYS LETT A

We discuss properties of the two-dimensional Landau Hamiltonian perturbed by a family of identical $\delta$ potentials arranged equidistantly along a closed loop. It is demonstrated that for the loop size exceeding the effective size of the point obstacles and the cyclotronic radius such a system exhibits persistent currents at the bottom of the spectrum. We also show that the effect is sensitive to a small disorder. - Article
- Nov 1999

Scattering theory provides a convenient framework for the solution of a variety of problems. In this thesis we focus on the combination of boundary conditions and scattering potentials and the combination of non-overlapping scattering potentials within the context of scattering theory. Using a scattering t-matrix approach, we derive a useful relationship between the scattering t-matrix of the scattering potential and the Green function of the boundary, and the t-matrix of the combined system, effectively renormalizing the scattering t-matrix to account for the boundaries. In the case of the combination of scattering potentials, the combination of t-matrix operators is achieved via multiple scattering theory. We also derive methods, primarily for numerical use, for finding the Green function of arbitrarily shaped boundaries of various sorts. These methods can be applied to both open and closed systems. In this thesis, we consider single and multiple scatterers in two dimensional strips ... - BookFull-text available
- Jan 2010

This chapter is an introduction to the semiclassical approach for the Helmholtz equation in complex systems originating in the field of quantum chaos. A particular emphasis will be made on the applications of trace formulae in paradigmatic wave cavities known as wave billiards. Its connection with random matrix theory (RMT) and disordered scattering systems will be illustrated through spectral statistics. Introduction The study of wave propagation in complicated structures can be achieved in the high-frequency (or small-wavelength) limit by considering the dynamics of rays. The complexity of wave media can be due either to the presence of inhomogeneities (scattering centers) of the wave velocity or to the geometry of boundaries enclosing a homogeneous medium. It is the latter case that was originally addressed by the field of quantum chaos to describe solutions of the Schrödinger equation when the classical limit displays chaos. The Helmholtz equation is the strict formal analog of the Schrödinger equation for electromagnetic or acoustic waves, the geometrical limit of rays being equivalent to the classical limit of particle motion. To qualify this context, the new expression wave chaos has naturally emerged. Accordingly, billiards have become geometrical paradigms of wave cavities. In this chapter we will particularly discuss how the global knowledge about ray dynamics in a chaotic billiard may be used to explain universal statistical features of the corresponding wave cavity, concerning spatial wave patterns of modes, as well as frequency spectra. - In a two-dimensional rectangular microwave cavity dressed with one pointlike scatterer, a semiclassical approach is used to analyze the spectrum in terms of periodic orbits and diffractive orbits. We show, both numerically and experimentally, how the latter can be accounted for in the so-called length spectrum, which is retrieved from two-point correlations of a finite-range frequency spectrum. Beyond its fundamental interest, this first experimental evidence of the role played by diffractive orbits in the spectrum of an actual cavity, can be the first step towards a technique to detect and track small defects in wave cavities.
- Article
- Feb 2006
- PHYS REV E

We study the diffusion of monochromatic classical waves in a disordered acoustic medium by scattering theory. In order to avoid artifacts associated with mathematical point scatterers, we model the randomness by small but finite insertions. We derive expressions for the configuration-averaged energy flux, energy density, and intensity for one-, two-, and three-dimensional (3D) systems with an embedded monochromatic source using the ladder approximation to the Bethe-Salpeter equation. We study the transition from ballistic to diffusive wave propagation and obtain results for the frequency dependence of the medium properties such as mean free path and diffusion coefficient as a function of the scattering parameters. We discover characteristic differences of the diffusion in 2D as compared to the conventional 3D case, such as an explicit dependence of the energy flux on the mean free path and quite different expressions for the effective transport velocity. - ArticleOrientador: Prof. Dr. Marcos Gomes Eleuterio da Luz Tese (doutorado) - Universidade Federal do Paraná, Setor de Ciências Exatas, Curso de Pós-Graduaçao em Física. Defesa: Curitiba, 2008 Inclui bibliografia
- Article
- Aug 2010

The effective mass density of an inhomogeneous medium is discussed. Random configurations of circular cylindrical scatterers are considered, in various physical contexts: fluid cylinders in another fluid, elastic cylinders in a fluid or in another solid, and movable rigid cylinders in a fluid. In each case, time-harmonic waves are scattered, and an expression for the effective wavenumber due to Linton and Martin [J. Acoust. Soc. Am. 117, 3413-3423 (2005)] is used to derive the effective density in the low frequency limit, correct to second order in the area fraction occupied by the scatterers. Expressions are recovered that agree with either the Ament formula or the effective static mass density, depending upon the physical context. - Article
- May 2009
- PHYS LETT A

The one-dimensional Schr\"odinger equation with the point potential in the form of the derivative of Dirac's delta function, $\lambda \delta'(x)$ with $\lambda$ being a coupling constant, is investigated. This equation is known to require an extension to the space of wave functions $\psi(x)$ discontinuous at the origin under the two-sided (at $x=\pm 0$) boundary conditions given through the transfer matrix ${cc} {\cal A} 0 0 {\cal A}^{-1})$ where ${\cal A} = {2+\lambda \over 2-\lambda}$. However, the recent studies, where a resonant non-zero transmission across this potential has been established to occur on discrete sets $\{\lambda_n \}_{n=1}^\infty$ in the $\lambda$-space, contradict to these boundary conditions used widely by many authors. The present communication aims at solving this discrepancy using a more general form of boundary conditions. Comment: Submitted Phys. Lett. A. Essentially revised and extended version, 1 figure added. 12 pages - ArticleFull-text available
- Feb 2011

Two-dimensional dielectric microcavities are of widespread use in microoptics applications. Recently, a trace formula has been established for dielectric cavities which relates their resonance spectrum to the periodic rays inside the cavity. In the present paper we extend this trace formula to a dielectric disk with a small scatterer. This system has been introduced for microlaser applications, because it has long-lived resonances with strongly directional far field. We show that its resonance spectrum contains signatures not only of periodic rays, but also of diffractive rays that occur in Keller's geometrical theory of diffraction. We compare our results with those for a closed cavity with Dirichlet boundary conditions. - Article
- Mar 2006
- J Phys Math Gen

We propose a classification scheme for the complete family of 1D point interactions. To do so, we first review the solutions of the wavefunctions and Green functions of the problem. Second, we derive the exact time-dependent propagators in such a way that we can write the expressions for the K's in a very compact form. As they should, such expressions do reduce to the known results in the literature according to the potential parameter values. Then, we analyse in general terms how the different point interactions scatter off arbitrary initially localized wave packets. Finally, we show that the physical features associated with the scattering process can be used to establish a classification procedure. Moreover, these physical characteristics are directly related to the potential parameters leading to the many formulae for the K's. As an application, we present numerical calculations for Gaussian wave packets. - ArticleFull-text available
- Feb 2012

Recently, the non-zero transmission of a quantum particle through the one-dimensional singular potential given in the form of the derivative of Dirac's delta function, $\lambda \delta'(x) $, with $\lambda \in \R$, being a potential strength constant, has been discussed by several authors. The transmission occurs at certain discrete values of $\lambda$ forming a resonance set ${\lambda_n}_{n=1}^\infty$. For $\lambda \notin {\lambda_n}_{n=1}^\infty$ this potential has been shown to be a perfectly reflecting wall. However, this resonant transmission takes place only in the case when the regularization of the distribution $\delta'(x) $ is constructed in a specific way. Otherwise, the $\delta'$-potential is fully non-transparent. Moreover, when the transmission is non-zero, the structure of a resonant set depends on a regularizing sequence $\Delta'_\varepsilon(x)$ that tends to $\delta'(x)$ in the sense of distributions as $\varepsilon \to 0$. Therefore, from a practical point of view, it would be interesting to have an inverse solution, i.e. for a given $\bar{\lambda} \in \R$ to construct such a regularizing sequence $\Delta'_\varepsilon(x)$ that the $\delta'$-potential at this value is transparent. If such a procedure is possible, then this value $\bar{\lambda}$ has to belong to a corresponding resonance set. The present paper is devoted to solving this problem and, as a result, the family of regularizing sequences is constructed by tuning adjustable parameters in the equations that provide a resonance transmission across the $\delta'$-potential. - Article
- Jun 2015
- J MATH PHYS

Consider the 3-dimensional Laplacian with a potential described by point scatterers placed on the integer lattice. We prove that for Floquet-Bloch modes with fixed quasi-momentum satisfying a certain Diophantine condition, there is a subsequence of eigenvalues of positive density whose eigenfunctions exhibit equidistribution in position space and localisation in momentum space. This result complements the result of Uebersch\"{a}r and Kurlberg who show momentum localisation for zero quasi-momentum in 2-dimensions, and is the first result in this direction in 3-dimensions. - Article
- Apr 2007

A three-parameter family of point interactions constructed from sequences of symmetric barrier–well–barrier and well–barrier–well rectangles is studied in the limit, when the rectangles are squeezed to zero width but the barrier height and the well depth become infinite (the zero-range limit). The limiting generalized potentials are referred to as the second derivative of Dirac's delta function ±λδ''(x) with a renormalized coupling constant λ > 0 or simply as ±δ''-like point interactions. As a result, a whole family of self-adjoint extensions of the one-dimensional Schrödinger operator is shown to exist, which results in full and partial resonant tunnelling through this class of singular potentials. The resonant tunnelling occurs for countable sets of interaction strength values in the λ-space which are the roots of several transcendental equations. The comparison with the previous results for δ'-like point interactions is also discussed. - Article
- Jul 2006
- J Phys Math Gen

The scattering properties of a three-parameter family of point dipole-like interactions constructed from a sequence of barrier-well rectangles are studied in the zero-range limit. Besides the real (unrenormalized) δ'-interaction, the derivative of Dirac's delta function, a whole family of point dipole interactions with a renormalized coupling constant are analysed. Depending on the parameter values, all these interactions being self-adjoint extensions of the one-dimensional Schrödinger operator are shown to be divided into four types: (i) interactions will full transparency, (ii) non-transparent interactions, (iii) partially transparent interactions acting effectively as a δ-interaction and (iv) interactions with partial transparency at discrete resonant values of the coupling constant. - The scattering properties of regularizing finite-range potentials constructed in the form of squeezed rectangles, which approximate the first and second derivatives of the Dirac delta function δ(x), are studied in the zero-range limit. Particularly, for a countable set of interaction strength values, a non-zero transmission through the point potential δ'(x), defined as the weak limit (in the standard sense of distributions) of a special dipole-like sequence of rectangles, is shown to exist when the rectangles are squeezed to zero width. A tripole sequence of rectangles, which gives in the weak limit the distribution δ''(x), is demonstrated to exhibit the total transmission for a countable sequence of the rectangle's width that tends to zero. However, this tripole sequence does not admit a well-defined point interaction in the zero-range limit, making sense only for a finite range of the regularizing rectangular-like potentials.
- Article
- Feb 2010

A family of point interactions of the dipole type is studied in one dimension using a regularization by rectangles in the form of a barrier and a well separated by a finite distance. The rectangles and the distance are parametrized by a squeezing parameter ε → 0 with three powers μ, ν and τ describing the squeezing rates for the barrier, the well and the distance, respectively. This parametrization allows us to construct a whole family of point potentials of the dipole type including some other point interactions, such as e.g. δ-potentials. Varying the power τ, it is possible to obtain in the zero-range limit the following two cases: (i) the limiting δ'-potential is opaque (the conventional result obtained earlier by some authors) or (ii) this potential admits a resonant tunneling (the opposite result obtained recently by other authors). The structure of resonances (if any) also depends on a regularizing sequence. The sets of the {μ, ν, τ}-space where a non-zero (resonant or non-resonant) transmission occurs are found. For all these cases in the zero-range limit the transfer matrix is shown to be of the form with real parameters χ and g depending on a regularizing sequence. Those cases when χ ≠ 1 and g ≠ 0 mean that the corresponding δ'-potential is accompanied by an effective δ-potential. - This survey article deals with a delta potential - also known as a point scatterer - on flat 2D and 3D tori. We introduce the main conjectures regarding the spectral and wave function statistics of this model in the so- called weak and strong coupling regimes. We report on recent progress as well as a number of open problems in this field.
- ArticleFull-text available
- Mar 2005

We consider a very straightforward extension of the usual self-adjoint techniques to solve the quantum mechanical problem of contact interaction potentials (i.e., infinite walls with different boundary conditions and point interactions) moving with constant velocity. From these calculations we obtain general expressions that describe the scattering of wavepackets. Furthermore, we also discuss the cases where the potentials oscillate with constant velocity or have time dependent strengths. Finally, we show, with an explicit example, that the prescription used is not the only possibility to implement self-adjoint extensions for contact interactions. Thus, in principle different models can be realized from such moving systems. Throughout the work we propose some simple physical applications of our results. - We present a new design of dielectric microcavities supporting modes with large quality factors and highly directional light emission. The key idea is to place a point scatterer inside a dielectric circular microdisk. We show that, depending on the position and strength of the scatterer, this leads to strongly directional modes in various frequency regions while preserving the high Q-factors reminiscent of the whispering gallery modes of the microdisk without scatterer. The design is very appealing due to its simplicity, promising a cleaner experimental realisation than previously studied microcavity designs on the one hand and analytic tractability based on Green's function techniques and self-adjoint extension theory on the other.
- Article
- Aug 1998
- PHYS LETT A

We consider scattering of a three-dimensional particle on a finite family of δ potentials. For some parameter values the scattering wavefunctions exhibit nodal lines in the form of closed loops, which may touch but do not entangle. The corresponding probability current forms vortical singularities around these lines; if the scattered particle is charged, this gives rise to magnetic flux loops in the vicinity of the nodal lines. The conclusions extend to scattering on hard obstacles or smooth potentials. - Quantum scattering is studied in a system consisting of randomly distributed point scatterers in the strip. The model is continuous yet exactly solvable. Varying the number of scatterers (the sample length) we investigate a transition between the ballistic and the localized regimes. By considering the cylinder geometry and introducing the magnetic flux we are able to study time reversal symmetry breaking in the system. Both macroscopic (conductance) and microscopic (eigenphases distribution, statistics of S-matrix elements) characteristics of the system are examined.

- Wave propagation through waveguides, quantum wires, or films with a modest amount of disorder is in the semiballistic regime when in the transversal direction(s) almost no scattering occurs, while in the long direction(s) there is so much scattering that the transport is diffusive. For such systems, randomness is modeled by an inhomogeneous density of pointlike scatterers. These are first considered in the second order Born approximation and then beyond that approximation. In the latter case, it is found that attractive point scatterers in a cavity always have geometric resonances, even for Schrödinger wave scattering. In the long sample limit, the transport equation is solved analytically. Various geometries are considered: waveguides, films, and tunneling geometries such as Fabry-Perot interferometers and double-barrier quantum wells. The predictions are compared with new and existing numerical data and with experiment. The agreement is quite satisfactory
- Wirzba: Small disks and semiclassical resonances, chao–dyn/9602017 [14] P. ˇ Seba: Some remarks on the δ ′ –interaction in one dimension
- P Rosenquist
- N D Whelan

P. Rosenquist, N.D. Whelan, A. Wirzba: Small disks and semiclassical resonances, chao–dyn/9602017 [14] P. ˇ Seba: Some remarks on the δ ′ –interaction in one dimension, Rep.Math. - Article
- Aug 1986
- REP MATH PHYS

We discuss the existence and the physical properties of the one-dimensional δ′- interaction Hamiltonian. We show that the so called δ′-interaction Hamiltonian which appears in the literature represents a self-adjoint realization of the heuristic operator - d2 dx2+β|δ′(x)〉 〈δ′(x)| with a renormalized coupling constant β. We investigate also a possible self-adjoint realization of the formal Hamiltonian - d2 dx2+βδ′ (x). We show that this realization coincides with the ordinary one-dimensional point interaction Hamiltonian HA = - d2 dx2+Aδ(x). - Article
- Jul 1994
- ANN PHYS-NEW YORK

A simple solvable mathematical model is proposed for the mesonic decays of heavy quarkonia. It consists in coupling the bound quark channel and the free decaying meson channel through a contact interaction. The full Hamiltonian is obtained as a self-adjoint extension of the free Hamiltonian, specified by suitable boundary conditions. One obtains a Weisskopf-Van Royen-type formula for the decay width at the lowest order in the interaction strength. Finally the model is shown to remain solvable if a Coulomb interaction is added in the quark channel. 53 refs. - Article
- Jun 1994
- ANN PHYS-NEW YORK

Scattering of non-relativistic particles by an ultralocal (delta-) potential is considered in two-dimensional manifolds with various topology (cylinder, torus, sphere, and Lobachevski plane). The behavior of the bound state energy as a function of the geometrical and topological characteristics of the space is studied. It is shown that for the compact non-simply connected manifolds of small radius the variation of the twisting angles (Aharonov-Bohm fluxes) may lead to delocalization of the bound state. For a simply connected geometry the influence of curvature on the bound state is considered and the possibility of "geometric delocalization" of the impurity levels is demonstrated explicitly for the spaces of constant curvature. We also consider the Aharonov-Bohm effect for the anyons on a cylinder. It is shown that a local regular potential can induce the Aharonov-Bohm oscillations in the anyon gas with anomalous (non-mesoscopic) dependence of oscillation amplitude on the geometrical sizes of the system. - Article
- Sep 1980
- J MATH PHYS

We derive an explicit formula for the resolvent of a class of one-particle, many-center, local Hamiltonians. This formula gives, in particular, a full description of a model molecule given by point interactions at n arbitrarily placed fixed centers in three dimensions. It also gives a three−dimensional analog of the Kronig–Penney model. - Jan 1988

- S Albeverio
- F Gesztesy
- R Høegh-Krohn
- H Holden

S. Albeverio, F. Gesztesy, R. Høegh-Krohn, H. Holden: Solvable Models in Quantum Mechanics, Springer, Heidelberg 1988.- The two-dimensional spatial distribution of the current and Fermi carriers around localized elastic scatterers in phase-coherent electron transport has been calculated using a generalized scattering-matrix approach. The distributions show dramatic differences depending on whether the scatterers are attractive (donorlike) or repulsive (acceptorlike). We find that attractive scatterers can produce strong vortices in the current, resulting in localized magnetic moments, while repulsive scatterers produce much weaker vortices and may not produce any at all in quasiballistic transport (few impurities). This is a significant difference between majority-carrier transport (when the scatterers are attractive) and minority-carrier transport (scatterers are repulsive). The vortices are caused by quantum-mechanical interference between scatterers and are accentuated by evanescent modes which have a stronger effect in the case of attractive scatterers owing to the formation of quasidonor states. We also examine the influence of the impurity configuration (positions of the scatterers) on the nature of the vortices.
- Article
- Jun 1993
- Phys Rev B

The spatial distributions of the current, Fermi carriers, electric field (due to space charges) and electrostatic potential in a disordered mesoscopic structure are calculated in the presence of arbitrary magnetic fields. These distributions are useful in elucidating and visulaizing many features of quantum magnetotransport, such as the formation of edge states at high magnetic fields and their near-perfect transmittivity, the evolution of the integer quantum Hall effect, the creation of magnetic bound states around an impurity, the magnetic response of current vortices that form as a result of quantum interference between scatterers and the walls of a quantum wire, the dependence of the quantized-conductance steps in a backgated quantum wire on an applied magnetic field, the behavior of residual-resistivity dipoles and the electrostatic space-charge potential in a magnetic field, the dependence of the sign of the magnetoresistance on the impurity configuration, etc. We examine the current, Fermi carrier concentrations, electric field, and both chemical- and electrostatic-potential profiles associated with each of these phenomena, and relate them to the observed terminal characteristics in each case. - Article
- Apr 1994
- Phys Rev Lett

We have introduced a controllable artificial impurity or ``antidot'' into a quantum wire and report on the novel phenomena observed as this system evolves from classical behavior at low magnetic fields to the quantum Hall regime. In the transition, conductance resonances due to magnetically bound impurity states are detected. The resonant oscillations exhibit beating and sharp period changes. A theoretical model based on an interedge state coupling mechanism and a new nonlocal effect of edge state formation at local potential energy maxima account for the principal experimental features. - We describe periodic, one dimensional Schrödinger operators, with the property that the widths of the forbidden gaps increase at large energies and the gap to band ratio is not small. Such systems can be realized by periodic arrays of geometric scatterers, e.g., a necklace of rings. Small, multichannel scatterers lead (for low energies) to the same band spectrum as that of a periodic array of (singular) point interactions known as delta'. We consider the Wannier-Stark ladder of delta' and show that the corresponding Schrödinger operator has no absolutely continuous spectrum.
- Article
- May 1997
- J Phys Math Gen

We consider a quantum mechanical particle living on a graph and discuss the behaviour of its wavefunction at graph vertices. In addition to the standard (or -type) boundary conditions with continuous wavefunctions, we investigate two types of a singular coupling which are analogous to the interaction and its symmetrized version for a particle on a line. We show that these couplings can be used to model graph superlattices in which point junctions are replaced by complicated geometric scatterers. We also discuss the band spectra for rectangular lattices with the mentioned couplings. We show that they roughly correspond to their Kronig - Penney analogues: the lattices have bands whose widths are asymptotically bounded and do not approach zero, while the lattice gap widths are bounded. However, if the lattice-spacing ratio is an irrational number badly approximable by rationals, and the coupling constant is small enough, the lattice has no gaps above the threshold of the spectrum. On the other hand, infinitely many gaps emerge above a critical value of the coupling constant; for almost all ratios this value is zero. - Article
- Jul 1996
- ANN PHYS-NEW YORK

We study the behavior of a quantum particle confined to a hard--wall strip of a constant width in which there is a finite number $ N $ of point perturbations. Constructing the resolvent of the corresponding Hamiltonian by means of Krein's formula, we analyze its spectral and scattering properties. The bound state--problem is analogous to that of point interactions in the plane: since a two--dimensional point interaction is never repulsive, there are $ m $ discrete eigenvalues, $ 1\le m\le N $, the lowest of which is nondegenerate. On the other hand, due to the presence of the boundary the point interactions give rise to infinite series of resonances; if the coupling is weak they approach the thresholds of higher transverse modes. We derive also spectral and scattering properties for point perturbations in several related models: a cylindrical surface, both of a finite and infinite heigth, threaded by a magnetic flux, and a straight strip which supports a potential independent of the transverse coordinate. As for strips with an infinite number of point perturbations, we restrict ourselves to the situation when the latter are arranged periodically; we show that in distinction to the case of a point--perturbation array in the plane, the spectrum may exhibit any finite number of gaps. Finally, we study numerically conductance fluctuations in case of random point perturbations. Comment: a LaTeX file, 38 pages, to appear in Ann. Phys.; 12 figures available at request from tater@ujf.cas.cz - We study the effect on quantum spectra of the existence of small circular disks in a billiard system. In the limit where the disk radii vanish there is no effect, however this limit is approached very slowly so that even very small radii have comparatively large effects. We include diffractive orbits which scatter off the small disks in the periodic orbit expansion. This situation is formally similar to edge diffraction except that the disk radii introduce a length scale in the problem such that for wave lengths smaller than the order of the disk radius we recover the usual semi-classical approximation; however, for wave lengths larger than the order of the disk radius there is a qualitatively different behaviour. We test the theory by successfully estimating the positions of scattering resonances in geometries consisting of three and four small disks.