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# Point Interactions in a Strip

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*in*Annals of Physics 252(1) · July 1996

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DOI: 10.1006/aphy.1996.0127 · Source: arXiv

Cite this publicationAbstract

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

- ... Mimicking the argument of [4] we also get Proposition 1.3. The on-shell S-matrix at energy z = k 2 is a 2N open ×2N open unitary matrix with elementary blocks ...... The discrete spectrum is determined by poles of the resolvent coming from the coefficients λ jk in (2.2). This yields (2.4); the eigenfunctions are obtained in the same way as in [1] [4]. The next question concerns the existence of solutions to (2.4). ...... The condition ν n < z in the above argument is crucial; the operator H(α, a) can have embedded eigenvalues with eigenfunctions in the orthogonal complement of the mentioned subspace if N > 1. Examples can be constructed as in [4] (or other similar systems – cf.[3] ...Article
- Sep 1999

We discuss discuss spectral and scattering properties of a particle confined to a straight Dirichlet tube in $\mathbb{R}^3$ with a family of point interactions. - ... Today there are many papers treating point interaction in restricted areas; a bibliography is given in the introduction of [5]. They typically put emphasis on the description of a specific model rather than a proper handling of the point interaction. ...... They typically put emphasis on the description of a specific model rather than a proper handling of the point interaction. Among few existing rigorous treatments of the problem it is the paper [5] which motivates the present study analyzing point interactions in an infinite planar strip with Dirichlet boundary conditions, together with similar systems. There are two ways in which the results can be generalized to dimension three. ...... Here the dimension of the configuration space is decisive. While for two-dimensional system the scaling amounts to logarithmic shift of the function ξ as shown in [5], in three dimension the transformation is multiplicative. We find easily that the situation is the same as for straight tubes in R 3 studied in [6], i.e. we have ξ( a σ ; zσ −2 ) = σ −1 ξ( a; z), where a σ := (aσ, bσ). ...
- ... Here, too, point-interaction Hamiltonians proved as a useful tool and yielded some unexpected results such as the existence of a chaotic behaviour in systems whose classical counterparts are integrable [4]. Today there are many papers treating point interaction in restricted areas; a bibliography is given in the introduction of [5]. They typically put emphasis on the description of a specific model rather than a proper handling of the point interaction. ...... They typically put emphasis on the description of a specific model rather than a proper handling of the point interaction. Among few existing rigorous treatments of the problem it is the paper [5] which motivates the present study analyzing point interactions in an infinite planar strip with Dirichlet boundary conditions, together with similar systems. There are two ways in which the results can be generalized to dimension three. ...... Here the dimension of the configuration space is decisive. While for two-dimensional system the scaling amounts to logarithmic shift of the function ξ as shown in [5], in three dimension the transformation is multiplicative. We find easily that the situation is the same as for straight tubes in R 3 studied in [6], i.e. we have ξ( a σ ; zσ −2 ) = σ −1 ξ( a; z), where a σ := (aσ, bσ). ...Article
- Mar 2002
- J MATH PHYS

We study spectral and scattering properties of a spinless quantum particle confined to an infinite planar layer with hard walls containing a finite number of point perturbations. A solvable character of the model follows from the explicit form of the Hamiltonian resolvent obtained by means of Krein’s formula. We prove the existence of bound states, demonstrate their properties, and find the on-shell scattering operator. Furthermore, we analyze the situation when the system is put into a homogeneous magnetic field perpendicular to the layer; in that case the point interactions generate eigenvalues of a finite multiplicity in the gaps of the free Hamiltonian essential spectrum. © 2002 American Institute of Physics. - ... Following the observation of Berezin and Faddeev [4], the problem was studied systematically in the eighties. The results are summarized in the monograph [1], references to some recent work are given, e.g., in [2] [8]. In its present form the point–interaction method represents a versatile tool for constructing solvable models whose power has not been yet, to our opinion, appreciated fully in the physical community. ...... This gives these point interactions a flavour of something exceptional, for instance, one is led to believe that they cannot be used to model well–localized repulsive potentials. That would have consequences, in particular, for recent studies of quantum wires with natural or artificial impurities [5] [6] [8] [10] [12]. ...Article
- Oct 1996
- PHYS LETT A

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. - ... The point interactions can be extended on more general class of manifolds as well [22]. In particular, they have been studied on some particular surfaces in R 3 , namely on the infinite planar strip as a natural model for quantum wires containing impurities [23] and on the torus [24]. A more heuristic approach for point interactions on Riemannian manifolds has been constructed through the heat kernel in [13,14]. ...Preprint
- Apr 2019

The double-well potential is a good example, where we can compute the splitting in the bound state energy of the system due to the tunneling effect with various methods, namely WKB or instanton calculations. All these methods are non-perturbative and there is a common belief that it is difficult to find the splitting in the energy due to the barrier penetration from a perturbative analysis. However, we will illustrate by explicit examples containing singular potentials (e.g., Dirac delta potentials supported by points and curves and their relativistic extensions)that it is possible to find the splitting in the bound state energies by developing some kind of perturbation method. - ... Furthermore, renormalization group equations and the β function have been discussed in [73, 84] and the model has been shown to asymptotically free in two dimensions. Apart from point interactions in flat two and three dimensions, they can also be defined on the infinite planar strips as a natural model for quantum wires containing impurities and onto torus (see [86, 87]) and on the two and three -dimensional Riemannian manifolds, where the renormalization is required, and this subject has been studied in [88, 89, 90]. One can also construct many-body models, where the particles are interacting through the two-body onedimensional Dirac delta potentials, known as the Lieb-Liniger model [91] and they have been studied in great detail in the literature [92, 93, 94, 95] (also see the references in [16]). ...ArticleFull-text available
- Mar 2017

We give a brief exposition of the formulation of the bound states and scattering problems for the one-dimensional system of $N$ attractive Dirac delta potentials, as an $N \times N$ matrix eigenvalue problem ($\Phi A =\omega A$) and discuss various physical results of the spectrum through the tools of linear algebra. In particular, we illustrate that the non-degeneracy theorem given in one-dimensional bound states breaks down for the periodically distributed Dirac delta potential, where the matrix $\Phi$ becomes a special form of the circulant matrix. We then show that the ground state is always non-degenerate and the associated wave function may be chosen to be positive by using the Perron-Frobenius theorem. We also prove that removing a single center from the system of $N$ delta centers shift all the bound state energy levels upward as a simple consequence of the Cauchy interlacing theorem. In addition, we show from the Feynman-Hellmann theorem that there are least one, at most $N$ bound states, and give a criterion for the existence of exactly $N$ bound states using the Gershgorin theorem. Moreover, we give an explicit solution of the Lippmann-Schwinger equation in terms of the matrix $\Phi$ and show that band gaps in the energy spectrum appear as the number of periodically located Dirac delta centers increases. We also discuss the so-called threshold anomaly and show the presence of scattering resonances, for which we obtain their Gamow vectors. Finally, we investigate the motion of the poles of the transmission coefficient for two center problem. - ... Here the problem is interesting even for straight strips and much less studied in the literature. Apart from the curved quantum waveguides, the discrete spectrum can be also generated by a local deformation of the boundary ∂Ω of straight tubes and layers [9, 8, 47], via introducing an obstacle [23, 14, 2] or impurities modelled by a Dirac interaction [26, 37, 39, 40], coupling several waveguides by a window [44, 45, 46, 6] , etc. The spectrum of periodically and randomly curved waveguides was investigated in [76, 74] and [55], respectively. ...Article
- Sep 2005
- PUBL RES I MATH SCI

The spectrum of the Laplace operator in a curved strip of constant width built along an infinite plane curve, subject to three different types of boundary conditions (Dirichlet, Neumann and a combination of these ones, respectively), is investigated. We prove that the essential spectrum as a set is stable under any curvature of the reference curve which vanishes at infinity and find various sufficient conditions which guarantee the existence of geometrically induced discrete spectrum. Furthermore, we derive a lower bound to the distance between the essential spectrum and the spectral threshold for locally curved strips. The paper is also intended as an overview of some new and old results on spectral properties of curved quantum waveguides. - ... Apart from the self-adjoint extension approaches developed by von Neumann and Krein, and the above two approximation procedures, there are other approaches to point interactions, namely non-standard analysis and the theory of quadratic forms [1]. Point interactions have been generalized onto some particular surfaces in R 3 (onto infinite planar strip as a natural model for quantum wires containing impurities and onto torus using the Von Neumann's and Krein's theory of self-adjoint extensions (see [5,6] and references therein)) and their spectral properties have been studied in great detail. Moreover, they have been constructed rigorously on even more general spaces, e.g. ...We study the bound state problem for $N$ attractive point Dirac $\delta$-interactions in two and three dimensional Riemannian manifolds. We give a sufficient condition for the Hamiltonian to have $N$ bound states and give an explicit criterion for it in hyperbolic manifolds $\mathbb{H}^2$ and $\mathbb{H}^3$. Furthermore, we study the same spectral problem for a relativistic extension of the model on $\mathbb{R}^2$ and $\mathbb{H}^2$.
- ... Many methodologies have been developed to analyze this problem, for instance, multiple scattering theory [2,3,5], the T-matrix method [6][7][8][9][10], and many hybrid numerical methods (see, for example, Ref. [9]). The basic result of such multiple scattering is that the wave nature of light leads to a reduction in transmittance due to destructive interference effects [10][11][12][13]. ...Article
- Jul 2016
- J OPT SOC AM A

We study wave transmission through a Fabry-Perot resonator (FPR) loaded with point-like impurities. We show both analytically in the framework of the coupled mode theory and numerically that there are two different regimes for transmission dependent on the quality of the FPR mirrors. For low quality, we obtain transmittance very similar to the clean FPR with slightly shifted Lorentz peaks. However, for good quality, the transmittance peaks are strongly reduced and substituted with Gaussian peaks because of multiple scattering of waves by each impurity. As a side effect, we observe the angular (channel) conversion in the disordered FPR. We demonstrate that the resonant peaks are dependent on the concentration of impurities to pave a way for resonant measurement of the concentration. - ... Apart from the self-adjoint extension approaches developed by von Neumann and Krein, and the above two approximation procedures in the resolvent sense, there are other approaches to point interactions, namely nonstandard analysis and the theory of quadratic forms [2]. Point interactions have been also generalized onto particular surfaces in R 3 , e.g., onto the infinite planar strip as a natural model for quantum wires [6] and onto the torus [7] using the theory of self-adjoint extensions. Our approach in this work is not to construct the Hamiltonian operator for point interactions on Riemannian manifolds through the self-adjoint extension theory. ...Article
- Nov 2015

We first review the construction of the problem of a quantum particle interacting with $N$ attractive point $\delta$-interactions in two and three dimensional Riemannian manifolds and improve it in a more rigorous way so that our somewhat heuristic arguments, which were used in our earlier works, are justified. We prove that the principal matrix is a matrix-valued holomorphic function on the region $\mathcal{R}=\{ z \in \mathbb{C}| \Re(z) <0 \}$ for two particular classes of Riemannian manifolds. We show that the essential spectrum of the Hamiltonian and that of the free Hamiltonian coincides. We finally give a sufficient condition for the Hamiltonian to have $N$ bound states and give an explicit criterion for it in hyperbolic manifolds. - ... Proof: Since the assertions (ii)–(iv) are easy consequences of the first claim and the above corollaries, it is sufficient to check (i). The free Green's function for the strip Ω was written down in [7]. In particular, we have ...Article
- Mar 2001
- REV MATH PHYS

The paper is devoted to a model of a mesoscopic system consisting of a pair of parallel planar waveguides separated by an infinitely thin semitransparent boundary modeled by a transverse delta interaction. We develop the Birman-Schwinger theory for the corresponding generalized Schrödinger operator. The spectral properties become nontrivial if the barrier coupling is not invariant with respect to longitudinal translations, in particular, there are bound states if the barrier is locally more transparent in the mean and the coupling parameter reaches the same asymptotic value in both directions along the guide axis. We derive the weak-coupling expansion of the ground-state eigenvalue for the cases when the perturbation is small in the supremum and the L1-norms. The last named result applies to the situation when the support of the leaky part shrinks: the obtained asymptotics differs from that of a double guide divided by a pierced Dirichlet barrier. We also derive an upper bound on the number of bound states. - ... Here the problem is interesting even for straight strips and much less studied in the literature. Apart from the curved quantum waveguides, the discrete spectrum can be also generated by a local deformation of the boundary ∂Ω of straight tubes and layers [9, 8, 47], via introducing an obstacle [23, 14, 2] or impurities modelled by a Dirac interaction [26, 37, 39, 40], coupling several waveguides by a window [44, 45, 46, 6], etc. The spectrum of periodically and randomly curved waveguides was investigated in [76, 74] and [55], respectively. ...Article
- Jul 2003

The spectrum of the Laplace operator in a curved strip of constant width built along an infinite plane curve, subject to three different types of boundary conditions (Dirichlet, Neumann and a combination of these ones, respectively), is investigated. We prove that the essential spectrum as a set is stable under any curvature of the reference curve which vanishes at infinity and find various sufficient conditions which guarantee the existence of geometrically induced discrete spectrum. Furthermore, we derive a lower bound on the distance between the essential spectrum and the spectral threshold for locally curved strips. The paper is also intended as an overview of some new and old results on spectral properties of curved quantum waveguides. - We investigate the scattering phenomena produced by a general finite-range nonseparable potential in a multi-channel two-probe cylindrical nanowire heterostructure. The multi-channel current scattering matrix is efficiently computed using the R-matrix formalism extended for cylindrical coordinates. Considering the contribution of the evanescent channels to the scattering matrix, we are able to put in evidence the specific dips in the tunneling coefficient in the case of an attractive potential. The cylindrical symmetry cancels the "selection rules" known for Cartesian coordinates. If the attractive potential is superposed over a non-uniform potential along the nanowire, then resonant transmission peaks appear. We can characterize them quantitatively through the poles of the current scattering matrix. Detailed maps of the localization probability density sustain the physical interpretation of the resonances (dips and peaks). Our formalism is applied to a variety of model systems like a quantum dot, a core/shell quantum ring or a double-barrier, embedded into the nanocylinder.
- The effect of spin-involved interaction on the transport properties of disordered two-dimensional electron systems with ferromagnetic contacts is described using a two-component model. Components representing spin-up and spin-down states are supposed to be coupled at a discrete set of points. We have found that due to the additional interference arising in two-component systems the difference between conductances for the parallel and antiparallel orientations of the contact magnetization changes its sign as a function of the length of the conducting channel.
- Article
- Aug 2004
- PHYS REV B

A nanodevice consisting of a conductive cylinder in an axial magnetic field with one-dimensional wires attached to its lateral surface is considered. An explicit form for transmission and reflection coefficients of the system as a function of electron energy is found from the first principles. The form and the position of transmission resonances and zeros are studied. It is found that, in the case of one wire being attached to the cylinder, reflection peaks occur at energies coinciding with the discrete part of the electronic spectrum of the cylinder. These peaks are split in a magnetic field. In the case of two wires the asymmetric Fano-type resonances are detected in the transmission between the wires for integer and half-integer values of the magnetic flux. The collapse of the resonances appears for certain position of contacts. Magnetic field splits transmission peaks and leads to spin polarization of transmitted electrons. - Article
- Jan 2003
- REP MATH PHYS

We study spectral properties of a spinless quantum particle confined to an infinite planar layer with hard walls which interacts with a periodic lattice of point perturbations and a homogeneous magnetic field perpendicular to the layer. It is supposed that the lattice cell contains a finite number of impurities and the flux through the cell is rational. Using the Landau-Zak transformation, we convert the problem into investigation of the corresponding fiber operators which is performed by means of Krein's formula. This yields an explicit description of the spectral bands which may be absolutely continuous or degenerate, depending on the parameters of the model. - Article
- Apr 2001
- REP MATH PHYS

We discuss the Nöckel model of an open quantum dot, i.e. a straight hard-wall channel with a potential well. If this potential depends on the longitudinal variable only, there are embedded eigenvalues which turn into resonances if the symmetry is violated, either by applying a magnetic field or by deformation of the well. For a weak symmetry breaking we derive the perturbative expansion of these resonances. We also deduce a sufficient condition under which the discrete spectrum of such a system (without any symmetry) survives the presence of a strong magnetic field. It is satisfied, in particular, if the dot potential is purely attractive. - 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 - ArticleFull-text available
- Nov 2010

Bound states of the Hamiltonian describing a quantum particle living on three dimensional straight strip of width $d$ are investigated. We impose the Neumann boundary condition on the two concentric windows of the radii $a$ and $ b$ located on the opposite walls and the Dirichlet boundary condition on the remaining part of the boundary of the strip. We prove that such a system exhibits discrete eigenvalues below the essential spectrum for any $a,b>0$. When $a$ and $b$ tend to the infinity, the asymptotic of the eigenvalue is derived. A comparative analysis with the one-window case reveals that due to the additional possibility of the regulating energy spectrum the anticrossing structure builds up as a function of the inner radius with its sharpness increasing for the larger outer radius. Mathematical and physical interpretation of the obtained results is presented; namely, it is derived that the anticrossings are accompanied by the drastic changes of the wave function localization. Parallels are drawn to the other structures exhibiting similar phenomena; in particular, it is proved that, contrary to the two-dimensional geometry, at the critical Neumann radii true bound states exist. - A single impurity in a quantum wire exhibits a conductance reduction. In this paper we prove (without any approximations) that for any point impurity this conduction reduction in all the subbands is exactly 2e2/h. It is also shown that in the case of a surface defect, not only are the conductance minima independent of the defect characteristics, but also the transmission matrix converges to universal values. We use these calculations to discuss particle confinement between two arbitrarily weak point defects.
- ArticleFull-text available
- Apr 2012

A non-perturbative renormalization of a many-body problem, where non-relativistic bosons living on a two dimensional Riemannian manifold interact with each other via the two-body Dirac delta potential, is given by the help of the heat kernel defined on the manifold. After this renormalization procedure, the resolvent becomes a well-defined operator expressed in terms of an operator (called principal operator) which includes all the information about the spectrum. Then, the ground state energy is found in the mean field approximation and we prove that it grows exponentially with the number of bosons. The renormalization group equation (or Callan-Symanzik equation) for the principal operator of the model is derived and the $\beta$ function is exactly calculated for the general case, which includes all particle numbers. - 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.
- Article
- May 1998
- Czech J Phys

We discuss differences between the exactS-matrix for scattering on serial structures and a known factorized expression constructed of single-elementS-matrices. As an illustration, we use an exactly solvable model of a quantum wire with two point impurities. - Article
- Feb 2005
- REP MATH PHYS

We show how the von Neumann theory of self-adjoint extensions can be used to investigatequantum systems the configuration space of which can be decomposed into parts of different dimensionalities. The method can be applied in many situations; we illustrate it on examples including point contact spectroscopy, nanotube systems, microwave resonators, or spin conductance oscillations. - We consider the resonances of the self-adjoint three-dimensional Schr\"odinger operator with point interactions of constant strength supported on the set $X = \{ x_n \}_{n=1}^N$. The size of $X$ is defined by $V_X = \max_{\pi\in\Pi_N} \sum_{n=1}^N |x_n - x_{\pi(n)}|$, where $\Pi_N$ is the family of all the permutations of the set $\{1,2,\dots,N\}$. We prove that the number of resonances counted with multiplicities and lying inside the disc of radius $R$ asymptotically behaves as $\frac{W_X}{\pi} R + O(1)$ as $R \to \infty$, where $W_X \in [0,V_X]$ is the effective size of $X$. Moreover, we show that there exist configurations of any number of points such that $W_X = V_X$. Finally, we construct an example for $N = 4$ with $W_X < V_X$, which can be viewed as an analogue of a non-Weyl quantum graph.
- Article
- Sep 2001
- J Phys Math Gen

We discuss the discrete spectrum of the Hamiltonian describing a two-dimensional quantum particle interacting with an infinite family of point interactions. We suppose that the latter are arranged into a star-shaped graph with N arms and a fixed spacing between the interaction sites. We prove that the essential spectrum of this system is the same as that of the infinite straight `polymer', but in addition there are isolated eigenvalues unless N = 2 and the graph is a straight line. We also show that the system has many strongly bound states if at least one of the angles between the star arms is small enough. Examples of eigenfunctions and eigenvalues are computed numerically. - ArticleFull-text available
- Apr 2007

A straight quasi-one-dimensional Dirichlet wave guide with a Neumann window of length L on one or two confining surfaces is considered theoretically with and without perpendicular homogeneous magnetic field . It is shown that for the field-free case, a bound state in the continuum (BIC) for one Neumann window exists for some critical lengths only, while for the two Neumann segments symmetrically located on the opposite walls, due to the restored transverse symmetry of the system, BICs exist for the arbitrary L. Bound states lying below the fundamental propagation threshold of the Dirichlet strip survive any strength of the uniform magnetic field and do not depend on its direction. Moreover, an increasing field induces new bound states regularly arranged with the levels present at B = 0. For two Neumann windows, strong magnetic fields lead to the degeneracy of the adjacent odd and even bound states with their energies almost equal to each other and to their corresponding counterpart for one Neumann segment, which is explained by mapping the problem onto the field-free one or two purely attractive one-dimensional quantum wells with field-dependent depth. Miscellaneous magnetotransport characteristics of the structures are also considered; in particular, it is demonstrated that small fields applied to the channel with two Neumann windows destroy BICs by coupling them to the continuum states. This is manifested in the conductance-Fermi energy dependence by Fano resonances. Currents flowing in the wave guide are investigated too, and it is shown that current density patterns near the resonances form vortices which change their chirality as energy sweeps through the resonant region. Generalizations to any other arbitrary combination of the boundary conditions are provided. Comparison with other structures such as window-coupled Dirichlet wave guides, a bent strip or straight Dirichlet channel with electrostatic impurity inside, is performed. - We investigate a class of generalized Schrödinger operators in L2(R3) with a singular interaction supported by a smooth curve Gamma. We find a strong-coupling asymptotic expansion of the discrete spectrum in the case when Gamma is a loop or an infinite bent curve which is asymptotically straight. It is given in terms of an auxiliary one-dimensional Schrödinger operator with a potential determined by the curvature of Gamma. In the same way, we obtain asymptotics of spectral bands for a periodic curve. In particular, the spectrum is shown to have open gaps in this case if Gamma is not a straight line and the singular interaction is strong enough.
- Article
- Sep 2003
- AM J PHYS

An integral equation approach is developed for the propagation of electrons in two-dimensional quantum waveguides. The original two-dimensional problem is transformed into a set of one-dimensional coupled equations by expanding the full wave function in terms of simple transverse basis functions. The equivalence of the Schrödinger equation with suitable boundary conditions in configuration space to an integral equation approach in momentum space can thus be illustrated in a coupled channel situation with a minimum of geometrical complications. The application to scattering from a point defect embedded in a waveguide is considered. In this case the scattering integral equations reduce to a set of algebraic equations, and typical coupled channel phenomena can be discussed through straightforward mathematical techniques. The convergence problems due to a singular perturbation are briefly considered, and the differences between genuine one-dimensional problems and the present two-dimensional case are discussed. - Making use of recent techniques in the theory of selfadjoint extensions of symmetric operators, we characterize the class of point interaction Hamiltonians in a 3-D bounded domain with regular boundary. In the particular case of one point interaction acting in the center of a ball, we obtain an explicit representation of the point spectrum of the operator togheter with the corresponding related eigenfunctions. These operators are used to build up a model-system where the dynamics of a quantum particle depends on the state of a quantum bit. Comment: 18 pages, 3 figures, latex

- Article
- May 1998
- Czech J Phys

We discuss differences between the exactS-matrix for scattering on serial structures and a known factorized expression constructed of single-elementS-matrices. As an illustration, we use an exactly solvable model of a quantum wire with two point impurities. - Schrödinger operators with singular interactions Cahay: Spatial distribution of current and Fermi carriers around localized elastic scatterers in quantum transport
- Jan 1992
- 112-139

- J Brasche
- P Exner
- Yu A Kuperin
- Seba

J. Brasche, P. Exner, Yu.A. Kuperin, P Seba: Schrödinger operators with singular interactions, J.Math.Anal.Appl. 184 (1994), 112–139. [9] S. Chaudhury, S. Bandyopadhyay, M. Cahay: Spatial distribution of current and Fermi carriers around localized elastic scatterers in quantum transport, Phys.Rev. B45 (1992), 11126–11135. - 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.- Skriganov: Proof of the Bethe–Sommerfeld conjecture in dimension two
- Jan 1979
- 956-959

M.M. Skriganov: Proof of the Bethe–Sommerfeld conjecture in dimension two, Sov.Math. Doklady 20 (1979), 956–959. - Article
- Jan 1993
- J MATH PHYS

Using two simple tricks, the results of our previous papers concerning bound states of Dirichlet Laplacians in curved tubes in Rν, ν=2, 3 are improved. If the tube is smooth and thin enough, the bound states are shown to exist if the tube curvature (and torsion for ν=3) decays with respect to the arclength parameter s as ‖s‖−1/2−&egr;. A sufficient condition is found and it is proved that it is necessary at the same time, provided the curvature (torsion) decays as ‖s‖−1−&egr; and the tube is only mildly bent. - 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. - Article
- Apr 1980
- J MATH PHYS

The existence of a family of self-adjoint Hamiltonians Htheta, theta ∈ [0, 2pi), corresponding to the formal expression H0+nudelta (x) is shown for a general class of self-adjoint operators H0. Expressions for the Green's function and wavefunction corresponding to Htheta are obtained in terms of the Green's function and wavefunction corresponding to H0. Similar results are shown for the perturbation of H0 by a finite sum of Dirac distributions. A prescription is given for obtaining Htheta as the strong resolvent limit of a family of momentum cutoff Hamiltonians HN. The relationship between the scattering theories corresponding to HN and Htheta is examined. - Article
- May 1981
- Phys Rev B

An elementary, exact calculation of two-dimensional electrons in crossed electric and magnetic fields with a δ-function impurity is carried out in the quantum limit. A state localized on the impurity exists and carries no current. However, the remaining mobile electrons passing near the impurity carry an extra dissipationless Hall current exactly compensating the loss of current by the localized electron. The Hall resistance should thus be precisely h/e2, as found experimentally by Klitzing et al. Other possible sources of deviation from this result are briefly examined. - Article
- Feb 1994
- J Fluid Mech

A two-dimensional acoustic waveguide of infinite extent described by two parallel lines contains an obstruction of fairly general shape which is symmetric about the centreline of the waveguide. It is proved that there exists at least one mode of oscillation, antisymmetric about the centreline, that corresponds to a local oscillation at a particular frequency, in the absence of excitation, which decays with distance down the waveguide away from the obstruction. Mathematically, this trapped mode is related to an eigenvalue of the Laplace operator in the waveguide. The proof makes use of an extension of the idea of the Rayleight quotient to characterize the lowest eigenvalue of a differential operator on an infinite domain. - Methods of Modern Mathematical Physics, II. Fourier Analysis Sel–Adjointness, III. Scattering Theory, IV Analysis of Operators [26] B. Simon: The bound state of weakly coupled Schrödinger operators in one and two dimensions
- Jan 1975
- 279-288

- M Reed
- B Simon

M. Reed, B. Simon: Methods of Modern Mathematical Physics, II. Fourier Analysis. Sel–Adjointness, III. Scattering Theory, IV. Analysis of Operators, Academic Press, New York 1975–1979. [26] B. Simon: The bound state of weakly coupled Schrödinger operators in one and two dimensions, Ann.Phys. 97 (1976), 279–288. - Article
- Feb 1985
- INVENT MATH

On montre que les bandes de spectre de l'operateur de Schrodinger a 3 dimensions recouvrent tous les λ reels assez grands et que la multiplicite du recouvrement tend vers l'infini quand λ→∞. On obtient la forme finale a la justification rigoureuse de l'hypothese de Bethe-Sommerfeld - Article
- Dec 1993
- PHYS LETT A

Viewing electron transmission as a probability flow problem, we study the stream lines of the flow. Velocity nodes are found at the points of wavefunction stationary phase, while vortices occur at wavefunction nodes. Analytical results are given for stream lines around such points. The sum of the action along a closed path and the charge times the enclosed magnetic flux is quantized in terms of the number of “quantum vortices”. - Article
- Apr 1976
- ANN PHYS-NEW YORK

We study the unique bound state which and −Δ + λV (in two dimensions) have when λ is small and V is suitable. Our main results give necessary and sufficient conditions for there to be a bound state when λ is small and we prove analyticity (resp. nonanalyticity) of the energy eigenvalue at λ = 0 in one (resp. two) dimensions. - Article
- May 1990
- PHYS REV B

We calculate the current transmission amplitudes and electrical conductance as a function of Fermi energy for electrons scattering from a single defect in a quasi-one-dimensional wire. In a confined geometry the scattering boundary conditions couple propagating modes in the wire to nonpropagating or evanescent modes. Therefore, the applied steady current causes localized or evanescent modes to build up around any defects in the wire. These extra stored electrons strongly affect the scattering boundary conditions for the propagating modes whenever the Fermi energy approaches either a new quasi-one-dimensional subband or a quasi-bound-state splitting off of the higher confinement subbands. We show that the presence of evanescent modes can lead to either perfect transparency or perfect opaqueness for the scattering modes, even in the presence of scattering defects. For the special case of a delta-function scatterer in the wire we analytically obtain the scattering amplitudes. We also numerically examine a finite-range scatterer. - We consider eigenvaluesEλ of the HamiltonianHλ=−Δ+V+λW,W compactly supported, in the λ→∞ limit. ForW≧0 we find monotonic convergence ofEλ to the eigenvalues of a limiting operatorH∞ (associated with an exterior Dirichlet problem), and we estimate the rate of convergence for 1-dimensional systems. In 1-dimensional systems withW≦0, or withW changing sign, we do not find convergence. Instead, we find a cascade phenomenon, in which, as λ→∞, each eigenvalueEλ stays near a Dirichlet eigenvalue for a long interval (of lengthO( )) of the scaling range, quickly drops to the next lower Dirichlet eigenvalue, stays there for a long interval, drops again, and so on. As a result, for most large values of λ the discrete spectrum ofHλ is close to that ofE∞, but when λ reaches a transition region, the entire spectrum quickly shifts down by one. We also explore the behavior of several explicit models, as λ→∞.
- Article
- Apr 1996
- Phys Rev B

We consider rectangular graph superlattices of sides l1, l2 with the wavefunction coupling at the junctions either of the delta type, when they are continuous and the sum of their derivatives is proportional to the common value at the junction with a coupling constant alpha, or the "delta-prime-S" type with the roles of functions and derivatives reversed; the latter corresponds to the situations where the junctions are realized by complicated geometric scatterers. We show that the band spectra have a hidden fractal structure with respect to the ratio theta := l1/l2. If the latter is an irrational badly approximable by rationals, delta lattices have no gaps in the weak-coupling case. We show that there is a quantization for the asymptotic critical values of alpha at which new gap series open, and explain it in terms of number-theoretic properties of theta. We also show how the irregularity is manifested in terms of Fermi-surface dependence on energy, and possible localization properties under influence of an external electric field. KEYWORDS: Schroedinger operators, graphs, band spectra, fractals, quasiperiodic systems, number-theoretic properties, contact interactions, delta coupling, delta-prime coupling. Comment: 16 pages, LaTex - 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
- Jul 1992
- Phys Rev B

The wave equation subject to Dirichlet boundary conditions has a bound state in an infinite tube of constant cross section in any number of dimensions, provided that the tube is not exactly straight. We prove this result, develop Green's-function methods to find the energy eigenvalue, and solve some simple cases. We discuss the implications for quantum systems and electromagnetic waveguides. - Article
- Apr 1993
- Phys Rev B

We have analyzed the spatial flow pattern in two-dimensional ballistic electron waveguides with circular bends. For a double bend the conductance may still be perfectly quantized in spite of the strong mixing of modes. In narrow energy regions just below the subband thresholds there is strong interference between localized and propagating solutions, causing an interference blockade. Within these regions the current flow becomes vortical. Minor changes in, e.g., energy cause drastic changes in the flow pattern. - Article
- Mar 1993
- Phys Rev B

A theoretical study of magnetotransport through two-dimensional quantum wires with double circular bends is made within the ballistic approximation. By means of the mode-matching method, the S matrix of the single bend with a hard-wall confining potential is calculated as a function of the magnetic field or the Fermi energy for the bending radius and the bending angle as parameters. The combination of S matrices is performed for bends in series. The technique is applied to systems consisting of double bends. The symmetry properties of the S matrix are clarified for the two-terminal systems with rotational symmetry or mirror-plane symmetry. The transmission matrix is evaluated in detail for the case of three propagating channels. The strong dependence of the interchannel scattering on the magnetic field is found. The structure of dips (antiresonances) appearing in the conductance is studied as a function of the distance between bends. - Article
- Mar 1988
- Phys Rev Lett

We analyze the effect of irregular classical scattering on the corresponding quantum-mechanical scattering matrix. Using semiclassical arguments, we show that the fluctuations in the S matrix and the cross sections are consistent with a random-matrix description (Ericson fluctuations). The results are illustrated by a numerical solution of a simple quantum problem, whose classical counterpart displays irregular scattering. - Article
- Dec 1990
- Phys Rev Lett

We demonstrate the existence of resistance fluctuations in experimentally realizable ballistic conductors due to scattering from geometric features. The magnetic-field and energy correlation functions are calculated both semiclassically and exactly numerically, and are found to have a scale determined by the underlying chaotic classical scattering. These systems provide a test of the ``random'' quantum behavior of classically chaotic systems. - 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. - Article
- Aug 1994
- Phys Rev Lett

We deduce the effects of quantum interference on the conductance of chaotic cavities by using a statistical ansatz for the S matrix. Assuming that the circular ensembles describe the S matrix, we find that the conductance fluctuation and weak-localization magnitudes are universal: they are independent of the size and shape of the cavity if the number of incoming modes, N, is large. The limit of small N is more relevant experimentally; here we calculate the full distribution of the conductance and find striking differences as N changes or a magnetic field is applied. - Article
- Jun 1995
- Phys Rev Lett

We discuss periodic Schrödinger operators for a particle on a rectangular lattice of sides l1, l2. In addition to the standard ( delta-type) coupling with continuous wave functions at lattice nodes, we introduce two other boundary conditions which generalize naturally the one-dimensional delta' interaction and its symmetrized version; both of them can be used as models for geometric scatterers. We show that the band spectrum of these models depends on number-theoretic properties of the parameters. In particular, the delta lattice has no gaps above the threshold if l2/l1 is badly approximable by rationals and the coupling constant is small enough. - Article
- Oct 1997

We consider curvature-induced resonances in a planar two-dimensional Dirichlet tube of a width $ d $. It is shown that the distances of the corresponding resonance poles from the real axis are exponentially small as $ d\to 0+ $, provided the curvature of the strip axis satisfies certain analyticity and decay requirements.