Strong-coupling asymptotic expansion for Schr\"odinger operators with a singular interaction supported by a curve in R3\mathbb{R}^3

We investigate a class of generalized Schr\"{o}dinger operators in $L^2(\mathbb{R}^3)$ with a singular interaction supported by a smooth curve $\Gamma$. We find a strong-coupling asymptotic expansion of the discrete spectrum in 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 Schr\"{o}dinger operator with a potential determined by the curvature of $\Gamma$. In the same way we obtain an 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.Comment: LaTeX 2e, 30 pages; minor improvements, to appear in Rev. Math. Phy

Scattering by local deformations of a straight leaky wire

We consider a model of a leaky quantum wire with the Hamiltonian $-\Delta -\alpha \delta(x-\Gamma)$ in $L^2(\R^2)$, where $\Gamma$ is a compact deformation of a straight line. The existence of wave operators is proven and the S-matrix is found for the negative part of the spectrum. Moreover, we conjecture that the scattering at negative energies becomes asymptotically purely one-dimensional, being determined by the local geometry in the leading order, if $\Gamma$ is a smooth curve and $\alpha \to\infty$.Comment: Latex2e, 15 page

Fractional Schrödinger Operator With Delta Potential Localized On Circle

We consider a system governed by the fractional Schödinger operator with a delta potential supported by a circle in R 2. We find out the function counting the number of bound states, in particular, we give the necessary and sufficient conditions for the absence of bound state in our system. Furthermore, we reproduce the form of eigenfunctions and analyze the asymptotic behavior of eigenvalues for the strong coupling constant case. © 2012 American Institute of Physics.533Albeverio, S., Gesztesy, F., Høegh-Krohn, R., Holden, H., (2004) Solvable Models in Quantum Mechanics, , 2nd ed. (with Appendix by P. Exner), (American Mathematical Society, Providence, RI)Bandrowski, B., Karczewska, A., Rozmej, P., Numerical solutions to integral equations equivalent to differential equations with fractional time derivative (2010) Int. J. Appl. Math Comput. Sci., 20 (2), pp. 261-269. , 10.2478/v10006-010-0019-1Bandrowski, B., Rozmej, P., On fractional Schrödinger equation (2010) Comput. Methods Sci. Technol., 16 (2), pp. 191-194. , http://www.man.poznan.pl/cmst/2010/_V_16_2/14_Rozmej_G.pdfBraaksma, B.L.J., Asymptotic expansions and analytic continuations for a class of Barnes-integrals (1962) Compos. Math., 15, pp. 239-341. , http://archive.numdam.org/ARCHIVE/CM/CM_1962-1964__15_/CM_1962-1964__15__239_0/CM_1962-1964__15__239_0.pdfCapelas de Oliveira, E., Silva Costa, F., Vaz, J., The fractional Schödinger operator equation for delta potentials (2010) J. Math. Phys., 51, p. 123517. , 10.1063/1.3525976Capelas de Oliveira, E., Vaz, J., Tunneling in fractional quantum mechanics (2011) J. Phys. A: Math. Theor., 44, p. 185303. , 10.1088/1751-8113/44/18/185303Exner, P., Ichinose, T., Geometrically induced spectrum in curved leaky wires (2001) J. Phys. A, 34, pp. 1439-1450. , 10.1088/0305-4470/34/7/315Exner, P., Kondej, S., Curvature-induced bound states for a δ interaction supported by a curve in (2002) Ann. Henri Poincaré, 3, pp. 967-981. , 10.1007/s00023-002-8644-3Exner, P., Kondej, S., Bound states due to a strong delta interaction supported by a curved surface (2003) J. Phys. A, 36, pp. 443-457. , 10.1088/0305-4470/36/2/311Exner, P., Tater, M., Spectra of soft ring graphs (2004) Waves Random Complex MediaMedia, 14, pp. S47-60. , 10.1088/0959-7174/14/1/010Guo, X., Xu, M., Some physical applications of fractional Schrr̈odinger equation (2006) J. Math. Phys., 47, p. 082104. , 10.1063/1.2235026Gradshteyn, I.S., Ryzhik, I.M., (2007) Table of Integrals, Series, and Products, , 7th ed., (Academic, New York)Jeng, M., Xu, S.-L.-Y., Hawkins, E., Schwarz, J.M., On the nonlocality of the fractional Schrödinger equation (2010) J. Math. Phys., 51, p. 062102. , 10.1063/1.3430552Dong, J., Xu, M., Some solutions to the space fractional Schrödinger equation using momentum representation method (2007) J. Math. Phys., 48, p. 072105. , 10.1063/1.2749172Kilbas, A.A., Srivastava, H.M., Trujillo, J.J., (2006) Theory and Applications of Fractional Differential Equations, , (Elsevier, Amsterdam)Laskin, N., Fractional quantum mechanics and Lévy path integrals (2000) Phys. Lett. A, 268, pp. 298-305. , 10.1016/S0375-9601(00)00201-2Laskin, N., Fractional quantum mechanics (2000) Phys. Rev. E, 62, pp. 3135-3145. , 10.1103/PhysRevE.62.3135Laskin, N., Fractal and quantum mechanics (2000) Chaos, 10, pp. 780-790. , 10.1063/1.1050284Mathai, A.M., Saxena, R.K., Haubold, H.J., (2009) The H-Function, , (Springer, New York)Naber, M., Time fractional Schrödinger equation (2004) J. Math. Phys., 45, pp. 3339-3352. , 10.1063/1.1769611Oberhetting, F., (1974) Tables of Mellin Transforms, , (Springer, New York)Reed, M., Simon, B., (1975) Methods of Modern Mathematical Physics. II. Fourier Analysis, , (Academic, New York)Posilicano, A., A Krein-like formula for singular perturbations of self-adjoint operators and applications (2001) J. Funct. Anal., 183, pp. 109-147. , 10.1006/jfan.2000.3730Stollmann, P., Voigt, J., Perturbation of Dirichlet forms by measures (1996) Potential Anal., 5, pp. 109-138. , 10.1007/BF0039677

Schroedinger operators with singular interactions: a model of tunneling resonances

We discuss a generalized Schr\"odinger operator in $L^2(\mathbb{R}^d), d=2,3$, with an attractive singular interaction supported by a $(d-1)$-dimensional hyperplane and a finite family of points. It can be regarded as a model of a leaky quantum wire and a family of quantum dots if $d=2$, or surface waves in presence of a finite number of impurities if $d=3$. We analyze the discrete spectrum, and furthermore, we show that the resonance problem in this setting can be explicitly solved; by Birman-Schwinger method it is cast into a form similar to the Friedrichs model.Comment: LaTeX2e, 34 page

Asymptotic behaviour of the spectrum of a waveguide with distant perturbations

We consider the waveguide modelled by a $n$-dimensional infinite tube. The operator we study is the Dirichlet Laplacian perturbed by two distant perturbations. The perturbations are described by arbitrary abstract operators ''localized'' in a certain sense, and the distance between their ''supports'' tends to infinity. We study the asymptotic behaviour of the discrete spectrum of such system. The main results are a convergence theorem and the asymptotics expansions for the eigenvalues. The asymptotic behaviour of the associated eigenfunctions is described as well. We also provide some particular examples of the distant perturbations. The examples are the potential, second order differential operator, magnetic Schroedinger operator, curved and deformed waveguide, delta interaction, and integral operator

Bound states due to a strong δ\delta interaction supported by a curved surface

We study the Schr\"odinger operator $-\Delta -\alpha \delta (x-\Gamma)$ in $L^2(\R^3)$ with a $\delta$ interaction supported by an infinite non-planar surface $\Gamma$ which is smooth, admits a global normal parameterization with a uniformly elliptic metric. We show that if $\Gamma$ is asymptotically planar in a suitable sense and $\alpha>0$ is sufficiently large this operator has a non-empty discrete spectrum and derive an asymptotic expansion of the eigenvalues in terms of a ``two-dimensional'' comparison operator determined by the geometry of the surface $\Gamma$. [A revised version, to appear in J. Phys. A]Comment: LaTeX 2e, 21 page

Spectra of soft ring graphs

We discuss of a ring-shaped soft quantum wire modeled by $\delta$ interaction supported by the ring of a generally nonconstant coupling strength. We derive condition which determines the discrete spectrum of such systems, and analyze the dependence of eigenvalues and eigenfunctions on the coupling and ring geometry. In particular, we illustrate that a random component in the coupling leads to a localization. The discrete spectrum is investigated also in the situation when the ring is placed into a homogeneous magnetic field or threaded by an Aharonov-Bohm flux and the system exhibits persistent currents.Comment: LaTeX 2e, 17 pages, with 10 ps figure

Schrödinger operators with δ and δ′-potentials supported on hypersurfaces

Self-adjoint Schrödinger operators with δ and δ′-potentials supported on a smooth compact hypersurface are defined explicitly via boundary conditions. The spectral properties of these operators are investigated, regularity results on the functions in their domains are obtained, and analogues of the Birman–Schwinger principle and a variant of Krein’s formula are shown. Furthermore, Schatten–von Neumann type estimates for the differences of the powers of the resolvents of the Schrödinger operators with δ and δ′-potentials, and the Schrödinger operator without a singular interaction are proved. An immediate consequence of these estimates is the existence and completeness of the wave operators of the corresponding scattering systems, as well as the unitary equivalence of the absolutely continuous parts of the singularly perturbed and unperturbed Schrödinger operators. In the proofs of our main theorems we make use of abstract methods from extension theory of symmetric operators, some algebraic considerations and results on elliptic regularity

Generalized eigenfunctions and spectral theory for strongly local Dirichlet forms

We present an introduction to the framework of strongly local Dirichlet forms and discuss connections between the existence of certain generalized eigenfunctions and spectral properties within this framework. The range of applications is illustrated by a list of examples