# 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…

## 61 Citations

### Trace formulae for Schrödinger operators with singular interactions

- Mathematics
- 2017

Let Σ⊂ℝd be a C∞-smooth closed compact hypersurface, which splits the Euclidean space ℝd into two domains Ω±. In this note self-adjoint Schrodinger operators with δ and δ'-interactions supported on Σ…

### Approximation of Schrödinger operators with δ‐interactions supported on hypersurfaces

- Mathematics
- 2015

We show that a Schrödinger operator Aδ,α with a δ‐interaction of strength α supported on a bounded or unbounded C2‐hypersurface Σ⊂Rd,d≥2 , can be approximated in the norm resolvent sense by a family…

### Schr\"odinger operators with $\delta$-potentials supported on unbounded Lipschitz hypersurfaces

- Mathematics
- 2021

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…

### Schrödinger operators with δ-interactions supported on conical surfaces

- Mathematics
- 2014

We investigate the spectral properties of self-adjoint Schrödinger operators with attractive δ-interactions of constant strength &agr; > 0 ?> supported on conical surfaces in R 3 ?> . It is shown…

### Boundary triples for Schrödinger operators with singular interactions on hypersurfaces

- Mathematics
- 2016

The self-adjoint Schrodinger operator Aδ,α with a δ-interaction of constant strength α supported on a compact smooth hypersurface C is viewed as a self-adjoint extension of a natural underlying…

### On Schrödinger operators with δ′-potentials supported on star graphs

- MathematicsJournal of Physics A: Mathematical and Theoretical
- 2022

The spectral properties of two-dimensional Schrödinger operators with δ′-potentials supported on star graphs are discussed. We describe the essential spectrum and give a complete description of…

### On the bound states of Schrödinger operators with δ-interactions on conical surfaces

- Mathematics
- 2015

ABSTRACT In dimension greater than or equal to three, we investigate the spectrum of a Schrödinger operator with a δ-interaction supported on a cone whose cross section is the sphere of codimension…

### A spectral isoperimetric inequality for cones

- Mathematics
- 2015

In this note, we investigate three-dimensional Schrödinger operators with $$\delta $$δ-interactions supported on $$C^2$$C2-smooth cones, both finite and infinite. Our main results concern a…

### Asymptotics of the bound state induced by δ-interaction supported on a weakly deformed plane

- Mathematics
- 2018

In this paper, we consider the three-dimensional Schrodinger operator with a δ-interaction of strength α > 0 supported on an unbounded surface parametrized by the mapping R2∋x↦(x,βf(x)), where β∈0,∞…

### 2D Schrödinger operators with singular potentials concentrated near curves

- Mathematics, Computer ScienceApplicable Analysis
- 2020

The transmission conditions on γ for the eigenfunctions, which arise in the limit as reveal a nontrivial connection between spectral properties of and the geometry of γ, are analyzed.

## References

SHOWING 1-10 OF 57 REFERENCES

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

- Mathematics
- 2012

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

- Mathematics
- 2010

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

- Mathematics
- 2013

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

- Mathematics
- 2009

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

- Mathematics
- 2003

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

- Mathematics
- 2007

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

- MathematicsJournal of mathematical analysis and applications
- 2014

### Leaky quantum graphs: approximations by point-interaction Hamiltonians

- Mathematics
- 2003

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

- Mathematics
- 2008

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…