The spectrum of the Schrödinger–Hamiltonian for trapped particles in a cylinder with a topological defect perturbed by two attractive delta interactions (original) (raw)
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Reports on Mathematical Physics, 2012
In this paper the self-adjoint Hamiltonian of the one-dimensional harmonic oscillator perturbed by two identical attractive point interactions (delta distributions) situated symmetrically with respect to the equilibrium position of the oscillator is rigorously defined by means of its resolvent (Green's function). The equations determining the even and odd eigenvalues of the Hamiltonian are explicitly provided in order to shed light on the behaviour of such energy levels both with respect to the separation distance between the point interaction centres and to the coupling constant.
Physica Scripta, 2019
We study three solvable two-dimensional systems perturbed by a point interaction centered at the origin. The unperturbed systems are the isotropic harmonic oscillator, a square pyramidal potential and a combination thereof. We study the spectrum of the perturbed systems. We show that, while most eigenvalues are not affected by the point perturbation, a few of them are strongly perturbed. We show that for some values of one parameter, these perturbed eigenvalues may take lower values than the immediately lower eigenvalue, so that level crossings occur. These level crossings are studied in some detail.
Journal of Physics A: Mathematical and Theoretical 46 (38 (5305)), 2013
We rigorously define the self-adjoint Hamiltonian of the harmonic oscillator perturbed by an attractive δ -interaction, of strength β, centred at 0 (the bottom of the confining parabolic potential), by explicitly providing its resolvent. Our approach is based on a 'coupling constant renormalization', related to a technique originated in quantum field theory and implemented in the rigorous mathematical construction of the self-adjoint operator representing the negative Laplacian perturbed by the δ-interaction in two and three dimensions. The way the δ -interaction enters in our Hamiltonian corresponds to the one originally discussed for the free Hamiltonian (instead of the harmonic oscillator one) by P Sěba. It should not be confused with the δ -potential perturbation of the harmonic oscillator discussed, e.g., in a recent paper by Gadella, Glasser and Nieto (also introduced by P Sěba as a perturbation of the one-dimensional free Laplacian and recently investigated in that context by Golovaty, Hryniv and Zolotaryuk). We investigate in detail the spectrum of our perturbed harmonic oscillator. The spectral structure differs from that of the one-dimensional harmonic oscillator perturbed by an attractive δ-interaction centred at the origin: the even eigenvalues are not modified at all by the δ -interaction. Moreover, all the odd eigenvalues, regarded as functions of β, exhibit the rather remarkable phenomenon called 'level crossing' after first producing the double degeneracy of all the even eigenvalues for the value β = β 0 = 2 √ π B( 3 4 , 1 2 ) ∼ = 1.47934(B(·, ·)
Reports on Mathematical Physics, 2009
In this note we investigate in detail the spectrum of the Schrodinger Hamiltonian with a configuration of three equally spaced one-dimensional point interactions (Dirac distributions), with the external ones having the same negative coupling constant. It will be seen that despite its simplicity, such a toy model exhibits a fairly rich variety of spectral combinations when the two coupling constants and the separation distance are manipulated. By analysing the equation determining the square root of the absolute value of the ground state energy and those determining the same quantity for the two possible excited states, we explicitly calculate the eigenvalues for all possible values of the separation distance and the two coupling constants. As a result of our analysis, we provide the conditions in terms of the three parameters in order to have the emergence of such excited states. Furthermore, we use our findings in order to get the confirmation of the fact that the Hamiltonian with such a configuration of three simple point interactions whose coupling constants undergo a special scaling in terms of the vanishing separation distance, converges in the norm resolvent sense to the Hamiltonian with an attractive δ′-interaction centred at the origin, as was shown by Exner and collaborators making the result previously obtained by Cheon et al. mathematically rigorous.
arXiv (Cornell University), 2017
We decorate the one-dimensional conic oscillator 1 2 − d 2 dx 2 + |x| with a point impurity of either δ-type, or local δ-type or even nonlocal δ-type. All the three cases are exactly solvable models, which are explicitly solved and analysed, as a first step towards higher dimensional models of physical relevance. We analyse the behaviour of the change in the energy levels when an interaction of the type −λ δ(x) or −λ δ(x − x 0) is switched on. In the first case, even energy levels (pertaining to antisymmetric bound states) remain invariant with λ although odd energy levels (pertaining to symmetric bound states) decrease as λ increases. In the second, all energy levels decrease when the form factor λ increases. A similar study has been performed for the so called nonlocal δ interaction, requiring a coupling constant renormalization, which implies the replacement of the form factor λ by a renormalized form factor β. In terms of β, even energy levels are unchanged. However, we show the existence of level crossings: after a fixed value of β the energy of each odd level, with the natural exception of the first one, becomes lower than the constant energy of the previous even level. Finally, we consider an interaction of the type −aδ(x) + bδ (x), and analyse in detail the discrete spectrum of the resulting self-adjoint Hamiltonian.
2021
In this paper we provide a detailed description of the eigenvalue ED(x0) ≤ 0 (respectively EN (x0) ≤ 0) of the self-adjoint Hamiltonian operator representing the negative Laplacian on the positive half-line with a Dirichlet (resp. Neuman) boundary condition at the origin perturbed by an attractive Dirac distribution −λδ(x − x0) for any fixed value of the magnitude of the coupling constant. We also investigate the λ-dependence of both eigenvalues for any fixed value of x0. Furthermore, we show that both systems exhibit resonances as poles of the analytic continuation of the resolvent. These results will be connected with the study of the ground state energy of two remarkable three-dimensional self-adjoint operators, studied in depth in Albeverio’s monograph, perturbed by an attractive δ-distribution supported on the spherical shell of radius r0.
arXiv (Cornell University), 2023
The objective of the present paper is the study of a one-dimensional Hamiltonian inside which the potential is given by the sum of two nonlocal attractive δ ′ interactions of equal strength and symmetrically located with respect to the origin. We use the procedure known as renormalisation of the coupling constant in order to rigorously achieve a self-adjoint determination for this Hamiltonian. This model depends on two parameters, the interaction strength and the distance between the centre of each interaction and the origin. Once we have the self-adjoint determination, we obtain its discrete spectrum showing that it consists of two negative eigenvalues representing the energy levels. We analyse the dependence of these energy levels on the above-mentioned parameters. We investigate the possible resonances of the model. Furthermore, we analyse in detail the limit of our model as the distance between the supports of the two δ ′ interactions vanishes.