Schrödinger operators with δ- and δ′-interactions on Lipschitz surfaces and chromatic numbers of associated partitions

@article{Behrndt2013SchrdingerOW,
  title={Schr{\"o}dinger operators with $\delta$- and $\delta$′-interactions on Lipschitz surfaces and chromatic numbers of associated partitions},
  author={Jussi Behrndt and Pavel Exner and Vladimir Lotoreichik},
  journal={Reviews in Mathematical Physics},
  year={2013},
  volume={26},
  pages={1450015}
}
We investigate Schrodinger operators with δ- and δ′-interactions supported on hypersurfaces, which separate the Euclidean space into finitely many bounded and unbounded Lipschitz domains. It turns out that the combinatorial properties of the partition and the spectral properties of the corresponding operators are related. As the main result, we prove an operator inequality for the Schrodinger operators with δ- and δ′-interactions which is based on an optimal coloring and involves the chromatic… 

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References

SHOWING 1-10 OF 57 REFERENCES

Schrödinger Operators with δ and δ′-Potentials Supported on Hypersurfaces

Self-adjoint Schrödinger operators with δ and δ′-potentials supported on a smooth compact hypersurface are defined explicitly via boundary conditions. The spectral properties of these operators are

Lower bound on the spectrum of the two-dimensional Schrödinger operator with a δ-perturbation on a curve

We consider the two-dimensional Schrödinger operator with a δ-potential supported by curve. For the cases of infinite and closed finite smooth curves, we obtain lower bounds on the spectrum of the

Spectral asymptotics of a strong δ′ interaction on a planar loop

We consider a generalized Schrödinger operator in L2(R2)?> with an attractive strongly singular interaction of δ′ type characterized by the coupling parameter β > 0 and supported by a C4 smooth

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

Bound states due to a strong δ interaction supported by a curved surface

We study the Schrodinger operator −Δ − αδ(x − Γ) in L2(3) with a δ interaction supported by an infinite non-planar surface Γ which is smooth and admits a global normal parametrization with a

Lower Bounds on the Lowest Spectral Gap of Singular Potential Hamiltonians

Abstract.We analyze Schrödinger operators whose potential is given by a singular interaction supported on a sub-manifold of the ambient space. Under the assumption that the operator has at least two

Weakly coupled bound state of 2-D Schrödinger operator with potential-measure

Leaky quantum graphs: approximations by point-interaction Hamiltonians

We prove an approximation result showing how operators of the type −Δ − γδ(x − Γ) in , where Γ is a graph, can be modelled in the strong resolvent sense by point-interaction Hamiltonians with an

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
...