• Corpus ID: 220936540

Approximation of point interactions by geometric perturbations in two-dimensional domains

@article{Borisov2020ApproximationOP,
  title={Approximation of point interactions by geometric perturbations in two-dimensional domains},
  author={Denis I. Borisov and Pavel Exner},
  journal={arXiv: Mathematical Physics},
  year={2020}
}
We present a new type of approximation of a second-order elliptic operator in a planar domain with a point interaction. It is of a geometric nature, the approximating family consists of operators with the same symbol and regular coefficients on the domain with a small hole. At the boundary of it Robin condition is imposed with the coefficient which depends on the linear size of a hole. We show that as the hole shrinks to a point and the parameter in the boundary condition is scaled in a… 
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References

SHOWING 1-10 OF 23 REFERENCES
The Norm Resolvent Convergence for Elliptic Operators in Multi-Dimensional Domains with Small Holes
We consider a second order elliptic operator with variable coefficients in a multidimensional domain with a small hole and some classical boundary condition on the hole boundary. We show that the…
Homogenization and norm-resolvent convergence for elliptic operators in a strip perforated along a curve
We consider an infinite planar straight strip perforated by small holes along a curve. In such a domain, we consider a general second-order elliptic operator subject to classical boundary conditions…
Uniform resolvent convergence for strip with fast oscillating boundary
On a PT-symmetric waveguide with a pair of small holes
A planar PT-symmetric waveguide with a pair of small holes is considered. The waveguide is modeled by a planar infinite strip in which a pair of symmetric small holes is cut out. The operator is the…
Waveguide with non-periodically alternating Dirichlet and Robin conditions: homogenization and asymptotics
We consider a magnetic Schrödinger operator in a planar infinite strip with frequently and non-periodically alternating Dirichlet and Robin boundary conditions. Assuming that the homogenized boundary…
Potential Approximations to δ': An Inverse Klauder Phenomenon with Norm-Resolvent Convergence
Abstract: We show that there is a family Schrödinger operators with scaled potentials which approximates the δ'-interaction Hamiltonian in the norm-resolvent sense. This approximation, based on a…
Approximation of general zero-range potentials
A norm resolvent convergence result is proved for approximations of general Schrodinger operators with zero-range potentials. An approximation of the δ’-interaction by nonlocal non-Hermitian…
Elliptic Partial Differential Equa-tions of Second Order
Chapter 1. Introduction Part I: Linear Equations Chapter 2. Laplace's Equation 2.1 The Mean Value Inequalities 2.2 Maximum and Minimum Principle 2.3 The Harnack Inequality 2.4 Green's Representation…
Mukhametrakhimova: On norm resolvent convergence for elliptic operators in multi-dimensional domains with small holes
  • J. Math. Sci
  • 2018
Durante: Homogenization and uniform resolvent convergence for elliptic operators in a strip perforated along a curve
  • Proc. Roy. Soc. Edinburgh. Sec. A Math
  • 2016
...
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