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

Albeverio S, Cognola G, Spreafico M, Zerbini S (2010)

Journal of Mathematical Physics 51(6).

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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 effect related to a massless scalar field in the flat space-time with an infinite conducting plate and in the presence of a pointlike "impurity." We use the relative zeta determinant (as defined in the works of Muller ["Relative zeta functions, relative determinants and scattering theory," Commun. Math. Phys. 192, 309 (1998)] and Spreafico and Zerbini ["Finite temperature quantum field theory on noncompact domains and application to delta interactions," Rep. Math. Phys. 63, 163 (2009)]) in order to regularize the partition function of this model. We study the analytic extension of the associated relative zeta function, and we present explicit results for the partition function and for the Casimir force. (C) 2010 American Institute of Physics. [doi: 10.1063/1.3397551]

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Albeverio S, Cognola G, Spreafico M, Zerbini S. Singular perturbations with boundary conditions and the Casimir effect in the half space.

*Journal of Mathematical Physics*. 2010;51(6).Albeverio, S., Cognola, G., Spreafico, M., & Zerbini, S. (2010). Singular perturbations with boundary conditions and the Casimir effect in the half space.

*Journal of Mathematical Physics*,*51*(6).Albeverio, S., Cognola, G., Spreafico, M., and Zerbini, S. (2010). Singular perturbations with boundary conditions and the Casimir effect in the half space.

*Journal of Mathematical Physics*51.Albeverio, S., et al., 2010. Singular perturbations with boundary conditions and the Casimir effect in the half space.

*Journal of Mathematical Physics*, 51(6). S. Albeverio, et al., “Singular perturbations with boundary conditions and the Casimir effect in the half space”,

*Journal of Mathematical Physics*, vol. 51, 2010. Albeverio, S., Cognola, G., Spreafico, M., Zerbini, S.: Singular perturbations with boundary conditions and the Casimir effect in the half space. Journal of Mathematical Physics. 51, (2010).

Albeverio, Sergio, Cognola, G., Spreafico, M., and Zerbini, S. “Singular perturbations with boundary conditions and the Casimir effect in the half space”.

*Journal of Mathematical Physics*51.6 (2010).
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