Exact solution to the Lippmann-Schwinger equation for an elliptical billiard

Maioli, Alan C.; Schmidt, Alexandre G. M.

doi:10.1016/j.physe.2019.02.025

Cited by 14 publications

(12 citation statements)

References 16 publications

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“…Taking into account the first relation in equation (17) for the successive integrals in equation (19), we get…”

Section: Applications For the Domain V As The Rectangular Semi-infini...mentioning

confidence: 99%

“…Lastly, rewritten the δ function in equation ( 3) in terms of the second relation in equation (17), we find for…”

Section: Applications For the Domain V As The Rectangular Semi-infini...mentioning

confidence: 99%

“…A very comprehensive mathematical and numerical (for this latter see also [12]) review of the BWM can be found in [15]. For instance, reference [15] discusses that formally the framework corresponds to a reformulation of standard single-layer boundary integral methods [4], leading to a first-kind Fredholm equation (a feature of the BWM recently used in concrete calculations [16,17]). Further conceptual aspects of the BWM have been examined in [18].…”

Section: Introductionmentioning

confidence: 99%

“…We can cite straight lines [12] and circles [12,18], with this latter shape being recently revisited in terms of particular integral equation solution techniques applied to the BWM [16]. Moreover, a 2D elliptical billiard [17] (for numerics see [27]) and a 3D spheroidal barrier [19] have being analyzed through ingenious transformations and appropriate coordinate system choices for the BWM associated expressions. In all these cases, V is taken as the whole 2D or 3D space, thus with the polar or spherical symmetries of G 0 greatly complying with those of the investigated C's.…”

Section: Introductionmentioning

confidence: 99%

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The flexibility in choosing distinct Green’s functions for the boundary wall method: waveguides and billiards

Teston¹,

Azevedo²,

Sales³

et al. 2022

J. Phys. A: Math. Theor.

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The boundary wall method (BWM) is a general purpose procedure to treat boundary value problems for wave equations, specially Helmholtz’s (the case addressed here). Similarly to most protocols for boundary value problems, the BWM may be computationally demanding for large borders C, at which the wave function must satisfies determined boundary conditions. Also, although an exact approach, it is rarely amenable to closed solutions. The BWM employs the Green’s function G 0 of the embedding domain V of C. However, in many instances — like for C modeling a billiard — the specific V is not really fundamental and thus one has a certain freedom to choose distinct domains and so G 0 ’s. Here we consider this characteristic of the BWM and show how to obtain some analytical results and solve numerically semi- infinite waveguides by exploring proper Green’s functions. Explicit calculations for Dirichlet and leaking boundaries for rectangular, triangular and trapezoidal structures are presented as well as scattering states within semi-infinite rectangular waveguides.

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“…Taking into account the first relation in equation (17) for the successive integrals in equation (19), we get…”

Section: Applications For the Domain V As The Rectangular Semi-infini...mentioning

confidence: 99%

“…Lastly, rewritten the δ function in equation ( 3) in terms of the second relation in equation (17), we find for…”

Section: Applications For the Domain V As The Rectangular Semi-infini...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The flexibility in choosing distinct Green’s functions for the boundary wall method: waveguides and billiards

Teston¹,

Azevedo²,

Sales³

et al. 2022

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

show abstract

“…The integral formulation for scattering problems is based on the celebrated Lippmann-Schwinger equation [22] which on its turn encompasses the boundary conditions at once. However, few analytic solutions are known [23][24][25][26][27][28][29]. One of our objectives is to present an analytical solution for the Lippmann-Schwinger equation with a linear potential.…”

Section: Introductionmentioning

confidence: 99%