The behaviour of the three-dimensional Hamiltonian -Δ+λ[δ(x+x0)+δ(x-x0)] as the distance between the two centres vanishes

Albeverio, S.; Fassari, Silvestro; Rinaldi, Fabio

doi:10.17586/2220-8054-2017-8-2-153-159

Cited by 5 publications

(19 citation statements)

References 10 publications

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“…in the norm resolvent convergence sense defined in [66]. As in the Dirichlet case, described in the previous Section, the limit is analogous to those discussed in the aforementioned papers [12,17,20,28,65,67,68,69]. The Hamiltonian in the right hand side in (3.8) is exactly the x-space counterpart of the Aronszajn-Donoghue Hamiltonian of type II considered in 5.4.1 in [53].…”

Section: Resonancesmentioning

confidence: 88%

“…D + as x 0 → 0 + , the spectrum of which is purely absolutely continuous. Here, we have performed an x 0 -limit as in [12,17,20,28,65,67,68,69].…”

Section: Now Takementioning

confidence: 99%

“…Before doing that, we wish to remind the reader that −∆ ∞,0 and −∆ 0,0 are the very extensions used by Albeverio and collaborators in [72] to investigate a class of exactly solvable problems in Quantum Mechanics and their relation to the Efimov effect. It is worth reminding the reader that the self-adjoint extension −∆ 0,0 has the unique property of having a zero energy resonance, see [1,68].…”

Section: Resonancesmentioning

confidence: 99%

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The Schrödinger particle on the half-line with an attractive $δ$-interaction: bound states and resonances

Fassari

Gadella²,

Nieto

et al. 2021

Preprint

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In this paper we provide a detailed description of the eigenvalue E D (x 0 ) ≤ 0 (respectively E N (x 0 ) ≤ 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 − x 0 ) 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 x 0 . 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 r 0 .

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Section: Resonancesmentioning

confidence: 88%

“…D + as x 0 → 0 + , the spectrum of which is purely absolutely continuous. Here, we have performed an x 0 -limit as in [12,17,20,28,65,67,68,69].…”

Section: Now Takementioning

confidence: 99%

Section: Resonancesmentioning

confidence: 99%

See 1 more Smart Citation

The Schrödinger particle on the half-line with an attractive $δ$-interaction: bound states and resonances

Fassari

Gadella²,

Nieto

et al. 2021

Preprint

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confidence: 99%

“…In particular, as a consequence of the treatment in [1] suitable for a Minkowski spacetime, the discrete spectrum of the ground state and the first exited state in the above mentioned cosmic framework can be regained. Thus, the coupling constant λ must be choosen as a function of the cosmic comooving time t as λ/a 2 (t), with λ be the one of the static Hamiltonian studied in [1]. In this way we can introduce a time…”

mentioning

confidence: 99%

The 3D Perturbed Schrödinger Hamiltonian in a Friedmann Flat Spacetime Testing the Primordial Universe in a Non Commutative Spacetime

Fassari

Rinaldi

Viaggiu

2019

Math Phys Anal Geom

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In this paper we adapt the mathematical machinery presented in [1] to get, by means of the Laplace-Beltrami operator, the discrete spectrum of the Hamiltonian of the Schrödinger operator perturbed by an actractive 3D delta interaction in a Friedmann flat universe. In particular, as a consequence of the treatment in [1] suitable for a Minkowski spacetime, the discrete spectrum of the ground state and the first exited state in the above mentioned cosmic framework can be regained. Thus, the coupling constant λ must be choosen as a function of the cosmic comooving time t as λ/a 2 (t), with λ be the one of the static Hamiltonian studied in [1]. In this way we can introduce a time * sifassari@gmail.com † f.rinaldi@unimarconi.it ‡ s.viaggiu@unimarconi.it and viaggiu@axp.mat.uniroma2.it dependent delta interaction which is relevant in a primordial universe, where a(t) → 0 and becomes negligible at late times, with a(t) >> 1. We investigate, with the so obtained model, quantum effects provided by point interactions in a strong gravitational regime near the big bang. In particular, as a physically interesting application, we present a method to depict, in a semi-classical approximation, a test particle in a (non commutative) quantum spacetime where, thanks to Planckian effects, the initial classical singularity disappears and as a consequnce a ground state with negative energy emerges. Conversely, in a scenario where the scale factor a(t) follows the classical trajectory, this ground state is instable and thus physically cannot be carried out.

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On the behaviour of the two-dimensional Hamiltonian $-\,{\rm{\Delta }}+\lambda [\delta (\vec{x}+{\vec{x}}_{0})+\delta (\vec{x}-{\vec{x}}_{0})]$ as the distance between the two centres vanishes

2020

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In this note we continue our analysis of the behaviour of self-adjoint Hamiltonians with a pair of identical point interactions symmetrically situated around the origin perturbing various types of ‘free Hamiltonians’ as the distance between the two centres shrinks to zero. In particular, by making the coupling constant to be renormalised dependent also on the separation distance between the centres of the two point interactions, we prove that also in two dimensions it is possible to define the unique self-adjoint Hamiltonian that, differently from the one studied in detail in Albeverio’s monograph on point interactions, behaves smoothly as the separation distance vanishes. In fact, we rigorously prove that such a two-dimensional Hamiltonian converges in the norm resolvent sense to the one of the negative two-dimensional Laplacian perturbed by a single attractive point interaction situated at the origin having double strength, thus making this two-dimensional model similar to its one-dimensional analogue (not requiring the renormalisation procedure).

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The behaviour of the three-dimensional Hamiltonian -Δ+λ[δ(x+x0)+δ(x-x0)] as the distance between the two centres vanishes

Cited by 5 publications

References 10 publications

The Schrödinger particle on the half-line with an attractive $δ$-interaction: bound states and resonances

The Schrödinger particle on the half-line with an attractive $δ$-interaction: bound states and resonances

The 3D Perturbed Schrödinger Hamiltonian in a Friedmann Flat Spacetime Testing the Primordial Universe in a Non Commutative Spacetime

On the behaviour of the two-dimensional Hamiltonian $-\,{\rm{\Delta }}+\lambda [\delta (\vec{x}+{\vec{x}}_{0})+\delta (\vec{x}-{\vec{x}}_{0})]$ as the distance between the two centres vanishes

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