Schrödinger operators with nonlocal point interactions

Albeverio, Sergio; Nizhnik, L. P.

doi:10.1016/j.jmaa.2006.10.070

Cited by 44 publications

(63 citation statements)

References 17 publications

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“…This problem is the combination of the interacting system studied in [1,7] and the free problem in the presence of a point-like potential, see e.g. [23] and [24]. Each of these problems has a well-defined translation in terms of a partial differential equation problem together with boundary conditions for the wavefunction.…”

Section: Combining Two Systemsmentioning

confidence: 99%

Exact Results for the One-Dimensional Many-Body Problem With Contact Interaction: Including a Tunable Impurity

Caudrelier

Crampé

2007

Rev. Math. Phys.

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The one-dimensional problem of N particles with contact interaction in the presence of a tunable transmitting and reflecting impurity is investigated along the lines of the coordinate Bethe ansatz. As a result, the system is shown to be exactly solvable by determining the eigenfunctions and the energy spectrum. The latter is given by the solutions of the Bethe ansatz equations which we establish for different boundary conditions in the presence of the impurity. These impurity Bethe equations contain as special cases well-known Bethe equations for systems on the half-line. We briefly study them on their own through the toy-examples of one and two particles. It turns out that the impurity can be tuned to lift degeneracies in the energies and can create bound states when it is sufficiently attractive. The example of an impurity sitting at the center of a box and breaking parity invariance shows that such an impurity can be used to confine asymmetrically a stationary state. This could have interesting applications in condensed matter physics.

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Section: Combining Two Systemsmentioning

confidence: 99%

Exact Results for the One-Dimensional Many-Body Problem With Contact Interaction: Including a Tunable Impurity

Caudrelier

Crampé

2007

Rev. Math. Phys.

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“…In this work, we wish to further extend our previous research on various types of point perturbations of Schrödinger Hamiltonians with or without harmonic confinement (see, e.g., [2][3][4][5][6][7][8][9][10][11]) to the physical models utilized to describe three-dimensional quantum dots.…”

Section: Introductionmentioning

confidence: 99%

Spectral properties of a symmetric three-dimensional quantum dot with a pair of identical attractive δ-impurities symmetrically situated around the origin

Albeverio¹,

Fassari²,

Rinaldi³

2016

Nanosystems: Phys. Chem. Math.

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In this presentation, we wish to provide an overview of the spectral features for the self-adjoint Hamiltonian of the three-dimensional isotropic harmonic oscillator perturbed by either a single attractive δ-interaction centered at the origin or by a pair of identical attractive δ-interactions symmetrically situated with respect to the origin. Given that such Hamiltonians represent the mathematical model for quantum dots with sharply localized impurities, we cannot help having the renowned article by Brüning, Geyler and Lobanov [1] as our key reference. We shall also compare the spectral features of the aforementioned three-dimensional models with those of the self-adjoint Hamiltonian of the harmonic oscillator perturbed by an attractive δ -interaction in one dimension, fully investigated in [2,3], given the existence in both models of the remarkable spectral phenomenon called "level crossing". The rigorous definition of the self-adjoint Hamiltonian for the singular double well model will be provided through the explicit formula for its resolvent (Green's function). Furthermore, by studying in detail the equation determining the ground state energy for the double well model, it will be shown that the concept of "positional disorder", introduced in [1] in the case of a quantum dot with a single δ-impurity, can also be extended to the model with the twin impurities in the sense that the greater the distance between the two impurities is, the less localized the ground state will be. Another noteworthy spectral phenomenon will also be determined; for each value of the distance between the two centers below a certain threshold value, there exists a range of values of the strength of the twin point interactions for which the first excited symmetric bound state is more tightly bound than the lowest antisymmetric bound state. Furthermore, it will be shown that, as the distance between the two impurities shrinks to zero, the 3D-Hamiltonian with the singular double well (requiring renormalization to be defined) does not converge to the one with a single δ-interaction centered at the origin having twice the strength, in contrast to its one-dimensional analog for which no renormalization is required. It is worth stressing that this phenomenon has also been recently observed in the case of another model requiring the renormalization of the coupling constant, namely the one-dimensional Salpeter Hamiltonian perturbed by two twin attractive δ-interactions symmetrically situated at the same distance from the origin.Spectral properties of a symmetric three-dimensional quantum dot . . . 269

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“…In a way inspired by the older developments in self-adjoint context [8] we found a key to the technical new results in the use of the language of the formalism of boundary triplets. We managed to demonstrate that even after a restriction of our attention to the first nontrivial class of one point nonlocal interactions the wealth of the spectral properties of the models remains satisfactorily rich involving not only the usual regularities/anomalies in the discrete spectra but, equally well, also the advanced (and, in the finite-dimensional models, inaccessible) features of the presence of the exceptional points and of the spectral singularities.…”

Section: Discussionmentioning

confidence: 99%

“…Such kind of new solvable models with point interactions has recently been proposed and studied (for self-adjoint case) by S. Albeverio and L. Nizhnik [8] (see also [9] - [13]). Our interest to the non-self-adjoint case was inspirited in part by an intensive development of Pseudo-Hermitian (PT -Symmetric) Quantum Mechanics PHQM (PTQM) during last decades [14]- [16].…”

Section: Introductionmentioning

confidence: 99%

Non-self-adjoint Schrödinger operators with nonlocal one-point interactions

Kuzhel¹,

Znojil²

2017

Banach J. Math. Anal.

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Abstract. Within the framework of quantum mechanics working with one-dimensional, manifestly non-Hermitian Hamiltonians H = −d 2 /dx 2 + V the traditional class of the exactly solvable models with local point interactions V = V (x) is generalized and studied. The consequences of the use of the nonlocal point interactions such that (V f )(x) = K(x, s)f (s)ds are discussed using the suitably adapted formalism of boundary triplets.

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Schrödinger operators with nonlocal point interactions

Abstract: Schrödinger operators with nonlocal point interactions are considered as new solvable models with point interactions. Examples in one and three dimensions are discussed. Corresponding direct and inverse scattering problems in one dimension are also discussed.

Cited by 44 publications

References 17 publications

Exact Results for the One-Dimensional Many-Body Problem With Contact Interaction: Including a Tunable Impurity

Exact Results for the One-Dimensional Many-Body Problem With Contact Interaction: Including a Tunable Impurity

Spectral properties of a symmetric three-dimensional quantum dot with a pair of identical attractive δ-impurities symmetrically situated around the origin

Non-self-adjoint Schrödinger operators with nonlocal one-point interactions

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