# Singular perturbations with boundary conditions and the Casimir effect in the half space

@article{Albeverio2010SingularPW, title={Singular perturbations with boundary conditions and the Casimir effect in the half space}, author={Sergio Albeverio and Guido Cognola and Mauro Spreafico and Sergio Zerbini}, journal={Journal of Mathematical Physics}, year={2010}, volume={51}, pages={063502} }

We study the self-adjoint extensions of a class of nonmaximal multiplication operators with boundary conditions. We show that these extensions correspond to singular rank 1 perturbations (in the sense of Albeverio and Kurasov [Singular Perturbations of Differential Operaters (Cambridge University Press, Cambridge, 2000)]) of the Laplace operator, namely, the formal Laplacian with a singular delta potential, on the half space. This construction is the appropriate setting to describe the Casimir…

## 15 Citations

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The local Casimir effect with a point delta-type potential is analyzed, computing the renormalized VEV of the stress-energy tensor at any point of $\mathbb{R}^3\setminus \mathbf{0}\}$; to this purpose the zeta regularization approach is followed, in the formulation already employed for different configurations in previous works of the authors'.

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## References

SHOWING 1-10 OF 34 REFERENCES

### Singular Perturbations of Differential Operators

- Physics
- 2000

Singular perturbations of Schrodinger type operators are of interest in mathematics, e.g. to study spectral phenomena, and in applications of mathematics in various sciences, e.g. in physics,…

### Quantum fields and extended objects in space-times with constant curvature spatial section

- Physics
- 1996

### Zeta function regularization of path integrals in curved spacetime

- Mathematics
- 1977

This paper describes a technique for regularizing quadratic path integrals on a curved background spacetime. One forms a generalized zeta function from the eigenvalues of the differential operator…

### Zeta-function approach to Casimir energy with singular potentials

- Physics
- 2006

In the framework of zeta-function approach the Casimir energy for three simple model system: single delta potential, step function potential and three delta potentials are analyzed. It is shown that…

### Zeta function regularization for a scalar field in a compact domain

- Mathematics
- 2004

We express the zeta function associated with the Laplacian operator on S1r × M in terms of the zeta function associated with the Laplacian on M, where M is a compact connected Riemannian manifold.…

### Casimir buoyancy

- Mathematics, Physics
- 2005

: We study the Casimir force on a single surface immersed in an inhomogeneous medium. Speciﬁcally we study the vacuum ﬂuctuations of a scalar ﬁeld with a spatially varying squared mass, m 2 + λ ∆( x…

### Zeta-Function Regularization, the Multiplicative Anomaly and the Wodzicki Residue

- Mathematics
- 1998

Abstract:The multiplicative anomaly associated with the zeta-function regularized determinant is computed for the Laplace-type operators V1=−Δ+V1 and V2=−Δ+V2, with V1, V2 constant, in a…

### The Casimir Effect and Its Applications

- Physics
- 1997

The Casimir effect is analyzed. This effect consists of a polarization of the vacuum of quantized fields which arises as a result of a change in the spectrum of vacuum oscillations when the…