The Birman-Schwinger Operator for a Parabolic Quantum Well in a Zero-Thickness Layer in the Presence of a Two-Dimensional Attractive Gaussian Impurity

In this note we are concerned with the limiting case of a zero-thickness layer with harmonic confinement along one of the two available dimensions. We investigate the Birman-Schwinger operator for such a model assuming the presence of a Gaussian impurity inside the layer and prove that such an integral operator is Hilbert-Schmidt, which allows the use of the modified Fredholm determinant in order to compute the impurity bound states. Furthermore, we consider the Hamiltonian H0 −λ√πδ(x)e−y2, that is to say the energy operator with the interaction term having a point interaction in place of the Gaussian along the x-direction, and prove that such an operator is self-adjoint as well as that it is the limit in the norm resolvent sense of the sequence H0 −λne−(n2x2+y2) as n →∞.

[1]  M. Glasser,et al.  Spectral properties of the two-dimensional Schrödinger Hamiltonian with various solvable confinements in the presence of a central point perturbation , 2019, Physica Scripta.

[2]  D. Panchenko,et al.  Point-Like Rashba Interactions as Singular Self-Adjoint Extensions of the Schrödinger Operator in One Dimension , 2018, Front. Phys..

[3]  D. Panchenko,et al.  Mass-jump and mass-bump boundary conditions for singular self-adjoint extensions of the Schrödinger operator in one dimension , 2018, Annals of Physics.

[4]  M. Glasser,et al.  LEVEL CROSSINGS OF EIGENVALUES OF THE SCHRöDINGER HAMILTONIAN OF THE ISOTROPIC HARMONIC OSCILLATOR PERTURBED BY A CENTRAL POINT INTERACTION IN DIFFERENT DIMENSIONS , 2018 .

[5]  J. M. M. Castañeda.  δ − δ ′ generalized Robin boundary conditions and quantum vacuum fluctuations , 2018 .

[6]  ON THE SPECTRUM OF THE ONE-DIMENSIONAL SCHRÖDINGER HAMILTONIAN PERTURBED BY AN ATTRACTIVE GAUSSIAN POTENTIAL , 2017 .

[7]  Hyunjoong Kim,et al.  Functional Analysis I , 2017 .

[8]  J. Mushanyu,et al.  A Note on the Discrete Spectrum of Gaussian Wells (I): The Ground State Energy in One Dimension , 2016 .

[9]  Anirban Bhaduri,et al.  Quantitative Assessment of Heart Rate Dynamics during Meditation: An ECG Based Study with Multi-Fractality and Visibility Graph , 2016, Front. Physiol..

[10]  B. Mityagin,et al.  Root System of Singular Perturbations of the Harmonic Oscillator Type Operators , 2013, 1307.6245.

[11]  Physical structure of point-like interactions for one-dimensional Schrödinger operator and the gauge symmetry , 2015 .

[12]  M. Gadella,et al.  Two-point one-dimensional δ- δ ′ ?> interactions: non-abelian addition law and decoupling limit , 2015, 1505.04359.

[13]  A. Zolotaryuk,et al.  Controllable resonant tunnelling through single-point potentials: A point triode , 2015 .

[14]  J. M. Guilarte,et al.  δ -δ ' generalized Robin boundary conditions and quantum vacuum fluctuations , 2014, 1407.4212.

[15]  M. Bordag,et al.  QFT Over the Finite Line. Heat Kernel Coefficients, Spectral Zeta Functions and Selfadjoint Extensions , 2014, Letters in Mathematical Physics.

[16]  A zero-thickness limit of multilayer structures: a resonant-tunnelling δ′-potential , 2015 .

[17]  M. Bordag.  Monoatomically thin polarizable sheets , 2014, 1404.4842.

[18]  M. Asorey,et al.  Attractive and repulsive Casimir vacuum energy with general boundary conditions , 2013, 1306.4370.

[19]  R. Seiringer,et al.  On contact interactions as limits of short-range potentials , 2013, 1305.3805.

[20]  J. M. Guilarte,et al.  Quantum vacuum energies and Casimir forces between partially transparent $\delta$-function plates , 2013, 1305.2054.

[21]  F. Fernández.  Quantum Gaussian wells and barriers , 2011 .

[22]  S. Nandi.  The quantum Gaussian well , 2010, 1011.3637.

[23]  P. Šťovíček,et al.  On the two-dimensional Coulomb-like potential with a central point interaction , 2010, 1006.5952.

[24]  On the spectrum of the Schrödinger Hamiltonian with a particular configuration of three one-dimensional point interactions , 2009 .

[25]  A delta well with a mass jump , 2009 .

[26]  G. Dell'Antonio,et al.  Spectral analysis of a two body problem with zero-range perturbation , 2007, 0708.0371.

[27]  W.-M. Wang,et al.  Pure Point Spectrum of the Floquet Hamiltonian for the Quantum Harmonic Oscillator Under Time Quasi-Periodic Perturbations , 2007, 0805.3764.

[28]  E. Demiralp,et al.  Bose-Einstein condensate in a harmonic trap decorated with Dirac delta functions , 2007, cond-mat/0701668.

[29]  Quantitative analysis of photon density of states for a realistic superlattice with omnidirectional light propagation. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[30]  D. García-Álvarez,et al.  Casimir effect and global theory of boundary conditions , 2006, hep-th/0604089.

[31]  V. Rostiashvili,et al.  Directed polymers with constrained winding angle. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[32]  Halevi,et al.  Density of states for a dielectric superlattice. II. TM polarization , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[33]  Coupling constant thresholds of perturbed periodic Hamiltonians , 1998 .

[34]  S. Fassari.  On the bound states of non-relativistic Krönig-Penney Hazmiltonians with short range impurities , 1990 .

[35]  S. Fassari.  On the bound states of relativistic Kröning-Penney Hamiltonians with short range impurities , 1990 .

[36]  Spectral properties of the Kronig–Penney Hamiltonian with a localized impurity , 1989 .

[37]  P. Seba.  The generalized point interaction in one dimension , 1986 .

[38]  S. Fassari.  On the Schrödinger operator with periodic point interactions in the three‐dimensional case , 1984 .

[39]  B. Simon.  Trace ideals and their applications , 1979 .

[40]  M. Reed,et al.  Fourier Analysis, Self-Adjointness , 1975 .

[41]  M. Glasser,et al.  Relativistic theory of surface states , 1973 .