2017/01/01 by Claudio Cacciapuoti, Davide Fermi, Andrea Posilicano
Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Casimir effect #Cosmology and Gravitation Theories #Geometry #Hyperplane #Laplace operator #Mathematical analysis #Mathematics #Multiplicity (mathematics) #Perturbation (astronomy) #Physics #Quantum Electrodynamics and Casimir Effect #Quantum mechanics #Riemann zeta function #Thermal #Thermodynamics #math-ph #math.MP #msc:81Q10 #msc:81T10 #msc:81T55
paper · pdf · doi:10.1007/978-3-319-58904-6_5
published as pp. 71-97 in G. F. Dell'Antonio, A. Michelangeli (Eds.), ''Advances in Quantum Mechanics: contemporary trends and open problems'', Springer (2017) · 21 pages
openalex publication_date 2017/01/01 · arxiv created 2017/02/17 · arxiv updated 2020/08/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We study the relative zeta function for the couple of operators A0 and Aα, where A0 is the free unconstrained Laplacian in L2(Rd) (d ≥ 2) and Aα 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(Rd-1). Thus α may depend on the components of the wave vector parallel to hyperplane; in this sense Aα describes a semitransparent hyperplane selecting transverse modes. As an application we give an expression for the associated thermal Casimir energy. Whenever α=χI(-Δ∥), where χI is the characteristic function of an interval I, the thermal Casimir energy can be explicitly computed.