Analytical Treatment of a Periodic $\delta $-Function Potential in the Path-Integral Formalism

Goovaerts, Marc; Broeckx, F.

doi:10.1137/0145028

Cited by 8 publications

(4 citation statements)

References 7 publications

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“…(2.23) can be used to describe periodic δ-functions in solid state physics, see e.g. Goovaerts and Broeckx [29] for such a discussion, including the determination of energy bands.…”

Section: Multiple δ-Function Perturbationsmentioning

confidence: 99%

δ‐function perturbations and boundary problems by path integration

Grosche

1993

Annalen der Physik

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Abstract. A wide class of boundary problems in quantum mechanics is discussed by using path integrals. This includes motion in half-spaces, radial boxes, rings, and moving boundaries. As a preparation the formalism for the incorporation of δ-function perturbations is outlined, which includes the discussion of multiple δ-function perturbations, δ-function perturbations along perpendicular lines and planes, and moving δ-function perturbations. The limiting process, where the strength of the δ-function perturbations gets infinite repulsive, has the effect of producing impenetrable walls at the locations of the δ-function perturbations, i.e. a consistent description for boundary problems with Dirichlet boundary-condition emerges. Several examples illustrate the formalism.

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“…(2.23) can be used to describe periodic δ-functions in solid state physics, see e.g. Goovaerts and Broeckx [29] for such a discussion, including the determination of energy bands.…”

Section: Multiple δ-Function Perturbationsmentioning

confidence: 99%

δ‐function perturbations and boundary problems by path integration

Grosche

1993

Annalen der Physik

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“…The problem at hand has been widely studied in the realm of quantum field theories (QFTs); the interested reader may consult [43][44][45][46][47][48][49][50][51][52][53][54]. Moreover, it has also been considered as a problem in a first quantization [55][56][57][58][59][60][61][62][63]. Recently, it has attracted much attention in connection with the related δ ′ problem, see [64][65][66] for δ − δ ′ potentials and [67] for some generalizations of semitransparent boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

Resummed heat-kernel and form factors for surface contributions: Dirichlet semitransparent boundary conditions

Franchino-Viñas

2023

J. Phys. A: Math. Theor.

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In this article we consider resummed expressions for the heat-kernel's trace of a Laplace operator, the latter including a potential and imposing Dirichlet semitransparent boundary conditions on a surface of codimension one in flat space. We obtain resummed expressions that correspond to the first and second order expansion of the heat-kernel in powers of the potential. We show how to apply these results to obtain the bulk and surface form factors of a scalar quantum field theory in $d=4$ with a Yukawa coupling to a background. Additionally, we discuss a connection between heat-kernels for Dirichlet semitransparent, Dirichlet and Robin boundary conditions.

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“…Since only the Green function G (V ) (E) of the unperturbed problem is relevant, the incorporation of multiple δ-function perturbations can be successively done with result [3,28,36] 16) with the matrix Γ…”

Section: Multiple δ-Function Perturbationsmentioning

confidence: 99%

Path integrals for two‐ and three‐dimensional δ‐function perturbations

Grosche

1994

Annalen der Physik

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Abstract. The incorporation of two-and three-dimensional δ-function perturbations into the path-integral formalism is discussed. In contrast to the onedimensional case, a regularization procedure is needed due to the divergence of the Green-function G (V ) (x, y; E), (x, y ∈ IR 2 , IR 3 ) for x = y, corresponding to a potential problem V (x). The known procedure to define proper self-adjoint extensions for Hamiltonians with deficiency indices can be used to regularize the path integral, giving a perturbative approach for δ-function perturbations in two and three dimensions in the context of path integrals. Several examples illustrate the formalism.

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Analytical Treatment of a Periodic $\delta $-Function Potential in the Path-Integral Formalism

Cited by 8 publications

References 7 publications

δ‐function perturbations and boundary problems by path integration

δ‐function perturbations and boundary problems by path integration

Resummed heat-kernel and form factors for surface contributions: Dirichlet semitransparent boundary conditions

Path integrals for two‐ and three‐dimensional δ‐function perturbations

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