Point interactions: boundary conditions or potentials with the Dirac delta function

Vincenzo, Salvatore De; Sánchez, C

doi:10.1139/p10-060

Cited by 8 publications

(8 citation statements)

References 18 publications

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“…This suggestion is also motivated by the studies [53,54,55] which demonstrate that the plausible averaging with η = 1/2 at the point of singularity in general does not work. The δ ′ -potential with intensity γ is defined by the boundary conditions ψ s (0) = γψ r (0) and ψ ′ s (0) = −γψ ′ r (0) [31].…”

Section: Introductionmentioning

confidence: 95%

Families of one-point interactions resulting from the squeezing limit of the sum of two- and three-delta-like potentials

Zolotaryuk¹

2017

J. Phys. A: Math. Theor.

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Abstract. Several families of one-point interactions are derived from the system consisting of two and three δ-potentials which are regularized by piecewise constant functions. In physical terms such an approximating system represents two or three extremely thin layers separated by some distance. The two-scale squeezing of this heterostructure to one point as both the width of δ-approximating functions and the distance between these functions simultaneously tend to zero is studied using the power parameterization through a squeezing parameter ε → 0, so that the intensity of each δ-potential is c j = a j ε 1−µ , a j ∈ R, j = 1, 2, 3, the width of each layer l = ε and the distance between the layers r = cε τ , c > 0. It is shown that at some values of intensities a 1 , a 2 and a 3 , the transmission across the limit point interactions is non-zero, whereas outside these (resonance) values the one-point interactions are opaque splitting the system at the point of singularity into two independent subsystems. Within the interval 1 < µ < 2, the resonance sets consist of two curves on the (a 1 , a 2 )-plane and three disconnected surfaces in the (a 1 , a 2 , a 3 )-space. While approaching the parameter µ to the critical value µ = 2, three types of splitting these sets into countable families of resonance curves and surfaces are observed.

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Section: Introductionmentioning

confidence: 95%

Families of one-point interactions resulting from the squeezing limit of the sum of two- and three-delta-like potentials

Zolotaryuk¹

2017

J. Phys. A: Math. Theor.

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“…The group-theoretical aspect of the model brings the following parallel with a problem of self-adjoint extension [33][34][35][36]. Consider a single particle problem, where the kinetic energy operator is defined as K = d 2 /dx 2 .…”

Section: Sl(2) Group Structure Of the Modelmentioning

confidence: 99%

“…The most general potential v(x) respecting these internal jump boundary conditions is a combination of δ(x), δ (x) and δ (x) [33][34][35][36],…”

Section: Sl(2) Group Structure Of the Modelmentioning

confidence: 99%

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Something interacting and solvable in 1D

Stouten¹,

Claeys²,

Zvonarev³

et al. 2018

J. Phys. A: Math. Theor.

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We present a two-parameter family of exactly solvable quantum many-body systems in one spatial dimension containing the Lieb-Liniger model of interacting bosons as a particular case. The principal building block of this construction is the previously-introduced [27] family of two-particle scattering matrices. We discuss an SL(2) transformation connecting the models within this family and make a correspondence with generalized point interactions. The Bethe equations for the ground state are discussed with a special emphasis on "non-interacting modes" connected by the modular subgroup of SL(2). The bound state solutions are discussed and are conjectured to follow some correlated version of the string hypothesis. The excitation spectrum of the new models in this family is derived in analogy to the Lieb-Liniger model and we show that for certain choices of parameters a spectrum inversion occurs such that the Umklapp solutions become the new ground state. arXiv:1804.10935v2 [cond-mat.other]

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“…There are a lot of articles which can help one to become familiar with the topic of zero-range potentials [35,36,37,38,39,40,41,24,42,43,34,44,45,46,47,48,49,50,51,52,53].…”

Section: Introductionmentioning

confidence: 99%

Application of a quantum wave impedance method for zero-range singular potentials

Hryhorchak¹

2020

Preprint

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An application of a quantum wave impedance method for a study of quantummechanical systems which contain singular zero-range potentials is considered. It was shown how to reformulate the problem of an investigation of mentioned systems in terms of a quantum wave impedance. As a result both the scattering and bound states problems are solved for systems of single δ, double δ and single δ − δ ′ potentials. The formalization of solving systems with an arbitrary combination of a piesewise constant potential and a δ-potentials with the help of a quantum wave impedance approach is described.

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Point interactions: boundary conditions or potentials with the Dirac delta function

Cited by 8 publications

References 18 publications

Families of one-point interactions resulting from the squeezing limit of the sum of two- and three-delta-like potentials

Families of one-point interactions resulting from the squeezing limit of the sum of two- and three-delta-like potentials

Something interacting and solvable in 1D

Application of a quantum wave impedance method for zero-range singular potentials

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