Relative-Zeta and Casimir Energy for a Semitransparent Hyperplane Selecting Transverse Modes

  • Claudio CacciapuotiEmail author
  • Davide Fermi
  • Andrea Posilicano
Part of the Springer INdAM Series book series (SINDAMS, volume 18)


We study the relative zeta function for the couple of operators A0 and A α , where A0 is the free unconstrained Laplacian in L2(R d ) (d ≥ 2) and A α is the singular perturbation of A0 associated to the presence of a delta interaction supported by a hyperplane. In our setting the operatorial parameter α, which is related to the strength of the perturbation, is of the kind α = α(−Δ), where −Δ is the free Laplacian in L2(R d−1). Thus α may depend on the components of the wave vector parallel to the hyperplane; in this sense A α describes a semitransparent hyperplane selecting transverse modes.

As an application we give an expression for the associated thermal Casimir energy. Whenever α = χ I (−Δ), where χ I is the characteristic function of an interval I, the thermal Casimir energy can be explicitly computed.


Casimir effect Delta-interactions Finite temperature quantum fields Relative zeta function Zeta regularization 

MSC 2010

81Q10 81T55 81T10 



C.C. and D.F. acknowledge the support of the FIR 2013 project “Condensed Matter in Mathematical Physics”, Ministry of University and Research of Italian Republic (code RBFR13WAET).


  1. 1.
    S. Albeverio, G. Cognola, M. Spreafico, S. Zerbini, Singular perturbations with boundary conditions and the Casimir effect in the half space. J. Math. Phys. 51, 063502 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    S. Albeverio, C. Cacciapuoti, M. Spreafico, Relative partition function of Coulomb plus delta interaction, arXiv:1510.04976 [math-ph], 24pp. To appear in the book J. Dittrich, H. Kovařík, A. Laptev (Eds.): Functional Analysis and Operator Theory for Quantum Physics. A Festschrift in Honor of Pavel Exner. (European Mathematical Society Publishing House, Zurich, 2016).Google Scholar
  3. 3.
    J.-P. Antoine, F. Gesztesy, J. Shabani, Exactly Solvable Models of Sphere Interactions in Quantum Mechanics. J. Phys. A Math. Gen. 20, 3687–3712 (1987)MathSciNetCrossRefGoogle Scholar
  4. 4.
    M. Bordag, The Casimir Effect 50 Years Later: Proceedings of the Fourth Workshop on Quantum Field Theory Under the Influence of External Conditions, Leipzig, Germany, 14–18 September 1998 (World Scientific, Singapore, 1999)CrossRefzbMATHGoogle Scholar
  5. 5.
    M. Bordag, D. Hennig, D. Robaschik, Vacuum energy in quantum field theory with external potentials concentrated on planes. J. Phys. A Math. Gen. 25, 4483–4498 (1992)MathSciNetCrossRefGoogle Scholar
  6. 6.
    M. Bordag, G.L. Klimchitskaya, U. Mohideen, V.M. Mostepanenko, Advances in the Casimir Effect, vol. 145 (Oxford University Press, Oxford, 2009)CrossRefzbMATHGoogle Scholar
  7. 7.
    H.B. Casimir, On the attraction between two perfectly conducting plates. Proc. Kongl. Nedel. Akad. Wetensch 51, 793–795 (1948)zbMATHGoogle Scholar
  8. 8.
    I. Cavero-Pelaez, K.A. Milton, K. Kirsten, Local and global Casimir energies for a semitransparent cylindrical shell. J. Phys. A 40, 3607–3632 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    L. Dabrowski, J. Shabani, Finitely many sphere interactions in quantum mechanics: nonseparated boundary conditions. J. Math. Phys. 29, 2241–2244 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    J.S. Dowker, R. Critchley, Effective Lagrangian and energy-momentum tensor in de Sitter space. Phys. Rev. D 13(12), 3224 (1976)Google Scholar
  11. 11.
    J. Dowker, G. Kennedy, Finite temperature and boundary effects in static space-times. J. Phys. A Math. Gen. 11(5), 895–920 (1978)MathSciNetCrossRefGoogle Scholar
  12. 12.
    E. Elizalde, S.D. Odintsov, A. Romeo, A.A. Bytsenko, S. Zerbini, Zeta Regularization Techniques with Applications (World Scientific, Singapore, 1994)CrossRefzbMATHGoogle Scholar
  13. 13.
    D. Fermi, L. Pizzocchero, Local zeta regularization and the Casimir effect. Prog. Theor. Phys. 126(3), 419–434 (2011)CrossRefzbMATHGoogle Scholar
  14. 14.
    D. Fermi, L. Pizzocchero, Local zeta regularization and the scalar Casimir effect I. A general approach based on integral kernels, 91pp. (2015), arXiv:1505.00711 [math-ph]Google Scholar
  15. 15.
    D. Fermi, L. Pizzocchero, Local zeta regularization and the scalar Casimir effect II. Some explicitly solvable cases, 54pp. (2015), arXiv:1505.01044 [math-ph]Google Scholar
  16. 16.
    D. Fermi, L. Pizzocchero, Local zeta regularization and the scalar Casimir effect III. the case with a background harmonic potential. Int. J. Mod. Phys. A 30(35), 1550213 (2015)Google Scholar
  17. 17.
    D. Fermi, L. Pizzocchero, Local zeta regularization and the scalar Casimir effect IV: the case of a rectangular box. Int. J. Mod. Phys. A 31, 1650003 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    M. Fierz, On the attraction of conducting planes in vacuum. Helv. Phys. Acta 33, 855 (1960)MathSciNetGoogle Scholar
  19. 19.
    N. Graham, R.L. Jaffe, V. Khemani, M. Quandt, M. Scandurra, H. Weigel, Calculating vacuum energies in renormalizable quantum field theories: a new approach to the Casimir problem. Nucl. Phys. B 645, 49–84 (2002)CrossRefzbMATHGoogle Scholar
  20. 20.
    S.W. Hawking, Zeta function regularization of path integrals in curved spacetime. Commun. Math. Phys. 55(2), 133–148 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    R.P. Kanwal, Generalized Functions: Theory and Applications (Springer Science + Business Media, Boston, 2011)Google Scholar
  22. 22.
    N.R. Khusnutdinov, Zeta-function approach to Casimir energy with singular potentials. Phys. Rev. D 73, 025003 (2006)MathSciNetCrossRefGoogle Scholar
  23. 23.
    S.G. Mamaev, N.N. Trunov, Vacuum expectation values of the energy-momentum tensor of quantized fields on manifolds of different topology and geometry. IV. Sov. Phys. J. 24(2), 171–174 (1981)CrossRefGoogle Scholar
  24. 24.
    J. Mehra, Temperature correction to the Casimir effect, Physica 37(1), 145–152 (1967)CrossRefGoogle Scholar
  25. 25.
    K.A. Milton, The Casimir effect: physical manifestations of zero-point energy (World Scientific, Singapore, 2001)CrossRefzbMATHGoogle Scholar
  26. 26.
    K.A. Milton, Casimir energies and pressures for δ-function potentials. J. Phys. A 37, 6391–6406 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    S. Minakshisundaram, A generalization of Epstein zeta functions. Can. J. Math. 1, 320–327 (1949)CrossRefzbMATHGoogle Scholar
  28. 28.
    S. Minakshisundaram, A. Pleijel, Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds. Canad. J. Math. 1, 242–256 (1949)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    W. Müller, Relative zeta functions, relative determinants and scattering theory. Commun. Math. Phys. 192, 309–347 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    J. Munoz-Castaneda, J.M. Guilarte, A.M. Mosquera, Quantum vacuum energies and Casimir forces between partially transparent δ-function plates. Phys. Rev. D 87, 105020 (2013)CrossRefGoogle Scholar
  31. 31.
    G. Ortenzi, M. Spreafico, Zeta function regularization for a scalar field in a compact domain. J. Phys. A 37, 11499–11518 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    A. Posilicano, A Krein-like formula for singular perturbations of self-adjoint operators and applications. J. Funct. Anal. 183(1), 109–147 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    D.B. Ray, I.M. Singer, R-torsion and the Laplacian on Riemannian manifolds. Adv. Math. 7(2), 145–210 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    R.T. Seeley, Complex powers of an elliptic operator, in Proceedings of Symposia in Pure Mathematics, vol. 10 (American Mathematical Society, Providence, 1967), pp. 288–307Google Scholar
  35. 35.
    J. Shabani, Finitely many δ interactions with supports on concentric spheres. J. Math. Phys. 29, 660–664 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    M. Spreafico, S. Zerbini, Finite temperature quantum field theory on noncompact domains and application to delta interactions. Rep. Math. Phys. 63(1), 163–177 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    R.M. Wald, On the Euclidean approach to quantum field theory in curved spacetime. Commun. Math. Phys. 70(3), 221–242 (1979)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Claudio Cacciapuoti
    • 1
    Email author
  • Davide Fermi
    • 2
  • Andrea Posilicano
    • 1
  1. 1.DiSAT, Sezione di MatematicaUniversità dell’InsubriaComoItaly
  2. 2.Dipartimento di MatematicaUniversità di MilanoMilanoItaly

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