Spectral optimization for strongly singular Schrödinger operators with a star-shaped interaction

@article{Exner2019SpectralOF,
  title={Spectral optimization for strongly singular Schr{\"o}dinger operators with a star-shaped interaction},
  author={Pavel Exner and Sylwia Kondej},
  journal={Letters in Mathematical Physics},
  year={2019},
  volume={110},
  pages={735-751}
}
  • P. Exner, S. Kondej
  • Published 2 June 2019
  • Mathematics, Physics
  • Letters in Mathematical Physics
We discuss the spectral properties of singular Schrödinger operators in three dimensions with the interaction supported by an equilateral star, finite or infinite. In the finite case, the discrete spectrum is nonempty if the star arms are long enough. Our main result concerns spectral optimization: we show that the principal eigenvalue is uniquely maximized when the arms are arranged in one of the known five sharp configurations. 
An optimization problem for finite point interaction families
  • P. Exner
  • Mathematics
    Journal of Physics A: Mathematical and Theoretical
  • 2019
We consider the spectral problem for a family of N point interactions of the same strength confined to a manifold with a rotational symmetry, a circle or a sphere, and ask for configurations that
Schr\"odinger operators with $\delta$-potentials supported on unbounded Lipschitz hypersurfaces
In this note we consider the self-adjoint Schrödinger operator Aα in L(R), d ≥ 2, with a δ-potential supported on a Lipschitz hypersurface Σ ⊆ R of strength α ∈ L(Σ) + L∞(Σ). We show the uniqueness

References

SHOWING 1-10 OF 35 REFERENCES
Hiatus perturbation for a singular Schrödinger operator with an interaction supported by a curve in R3
We consider Schrodinger operators in L2(R3) with a singular interaction supported by a finite curve Γ. We present a proper definition of the operators and study their properties, in particular, we
On geometric perturbations of critical Schr\"odinger operators with a surface interaction
We study singular Schrodinger operators with an attractive interaction supported by a closed smooth surface A in R^3 and analyze their behavior in the vicinity of the critical situation where such an
Strong Coupling Asymptotics for a Singular Schrödinger Operator with an Interaction Supported by an Open Arc
We consider a singular Schrödinger operator in L 2(ℝ2) written formally as − Δ − βδ(x − γ) where γ is a C 4 smooth open arc in ℝ2 of length L with regular ends. It is shown that the jth negative
Optimization of the lowest eigenvalue for leaky star graphs
We consider the problem of geometric optimization for the lowest eigenvalue of the two-dimensional Schr\"odinger operator with an attractive $\delta$-interaction of a fixed strength, the support of
Curvature-Induced Bound States for a δ Interaction Supported by a Curve in R 3
We study the Laplacian in L2( 3) perturbed on an infinite curve Γ by a δ interaction defined through boundary conditions which relate the corresponding generalized boundary values. We show that if Γ
Curvature-Induced Bound States for a $ \delta $ Interaction Supported by a Curve in $ \mathbb{R}^3 $
Abstract. We study the Laplacian in $ L^{2}(\mathbb{R}^3) $ perturbed on an infinite curve $ \Gamma $ by a $ \delta $ interaction defined through boundary conditions which relate the corresponding
The Five-Electron Case of Thomson’s Problem
TLDR
A fairly efficient energy estimate is given that works for any number of points and any power-law potential that globally minimizes the Coulomb (1/r) potential.
Optimisation of the Lowest Robin Eigenvalue in the Exterior of a Compact Set, II: Non-Convex Domains and Higher Dimensions
We consider the problem of geometric optimisation of the lowest eigenvalue of the Laplacian in the exterior of a compact set in any dimension, subject to attractive Robin boundary conditions. As an
Inequalities for Means of Chords, with Application to Isoperimetric Problems
AbstractWe consider a pair of isoperimetric problems arising in physics. The first concerns a Schrödinger operator in $$L^2(\mathbb{R}^2)$$ with an attractive interaction supported on a closed curve
...
...