Renormalization of multi-delta-function point scatterers in two and three dimensions, the coincidence-limit problem, and its resolution

Loran, Farhang; Mostafazadeh, Alí

doi:10.1016/j.aop.2022.168966

Cited by 4 publications

(3 citation statements)

References 17 publications

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“…Being f λ reg ∈ H 2 we know that (−∆ + λ)f λ reg ∈ L 2 (R 3 ). From (18) we have which is an alternative way to compute the values of the regular part of f using the explicit form (19) of matrix Γ θ (−λ).…”

Section: Discussionmentioning

confidence: 99%

“…For fixed t θ , the resolvent (18) converges in norm when r → 0 to the resolvent of a one-center point interaction with integral kernel…”

Section: Characterization Of the Hamiltonianmentioning

confidence: 99%

“…It was recognized (for recent contribution see, e.g. [2,18]) that the assumption resulted in a non-additive property of point interactions. In particular, a local multi-center point interaction Hamiltonian with fixed strength parameters acts as a sum of single point interactions only when points are well spaced, but a ground state of lower and lower energy shows up when two or more points get closer and closer.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On a family of finitely many point interaction Hamiltonians free of ultraviolet pathologies

Figari,

Saberbaghi,

Teta

2024

J. Phys. A: Math. Theor.

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We re-investigate the entire family of many center point interaction Hamiltonians. Under the assumption of exchange symmetry with respect to the point positions, we show that a large sub- family of point interaction Hamiltonian operators does not become either singular or trivial when the positions of two or more scattering centers tend to coincide. In this sense, they appear to be renormalized by default as opposed to the point interaction Hamiltonians usually considered in the literature. Functions in their domains satisfy regularized boundary conditions which turn out to be very similar to the ones proposed recently in many-body quantum mechanics to define three-particle system Hamiltonians with contact interactions bounded from below.In the two-center case, we study the behavior of the negative eigenvalues as a function of the center distance. The result is used to analyze a formal Born-Oppenheimer approximation of a three- particle system with two heavy bosons and one light particle. We demonstrate that this simplified model describes a stable system (no “fall to the center” problem is present). Furthermore, in the unitary limit, the energy spectrum is characterized by an infinite sequence of negative energy eigenvalues accumulating at zero according to the geometrical Efimov law.

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Section: Discussionmentioning

confidence: 99%

“…For fixed t θ , the resolvent (18) converges in norm when r → 0 to the resolvent of a one-center point interaction with integral kernel…”

Section: Characterization Of the Hamiltonianmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On a family of finitely many point interaction Hamiltonians free of ultraviolet pathologies

Figari,

Saberbaghi,

Teta

2024

J. Phys. A: Math. Theor.

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Bound states of a one-dimensional Dirac equation with multiple delta-potentials

Gusynin

Sobol

Zolotaryuk

et al. 2022

Low Temperature Physics

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Two approaches are developed for the study of the bound states of a one-dimensional Dirac equation with the potential consisting of N δ-function centers. One of these uses Green’s function method. This method is applicable to a finite number N of δ-point centers, reducing the bound state problem to finding the energy eigenvalues from the determinant of a 2 N × 2 N matrix. The second approach starts with the matrix for a single delta-center that connects the two-sided boundary conditions for this center. This connection matrix is obtained from the squeezing limit of a piecewise constant approximation of the delta-function. Having then the connection matrices for each center, the transmission matrix for the whole system is obtained by multiplying the one-center connection matrices and the free transfer matrices between neighbor centers. An equation for bound state energies is derived in terms of the elements of the total transfer matrix. Within both approaches, the transcendental equations for bound state energies are derived, the solutions to which depend on the strength of delta-centers and the distance between them, and this dependence is illustrated by numerical calculations. The bound state energies for the potentials composed of one, two, and three delta-centers ( N = 1, 2, 3) are computed explicitly. The principle of strength additivity is analyzed in the limits as the delta-centers merge at a single point or diverge to infinity.

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Fundamental transfer matrix for electromagnetic waves, scattering by a planar collection of point scatterers, and anti- PT -symmetry

Loran

Mostafazadeh

2023

Phys. Rev. A

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Renormalization of multi-delta-function point scatterers in two and three dimensions, the coincidence-limit problem, and its resolution

Cited by 4 publications

References 17 publications

On a family of finitely many point interaction Hamiltonians free of ultraviolet pathologies

On a family of finitely many point interaction Hamiltonians free of ultraviolet pathologies

Bound states of a one-dimensional Dirac equation with multiple delta-potentials

Fundamental transfer matrix for electromagnetic waves, scattering by a planar collection of point scatterers, and anti- PT -symmetry

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