In this contribution to the study of one dimensional point potentials, we prove that if we take the limit q → 0 on a potential of the type v0δ(y) + 2v1δ (y) + w0δ(y − q) + 2w1δ (y − q), we obtain a new point potential of the type u0δ(y) + 2u1δ (y), when u0 and u1 are related to v0, v1, w0 and w1 by a law having the structure of a group. This is the Borel subgroup of SL2(R). We also obtain the non-abelian addition law from the scattering data. The spectra of the Hamiltonian in the decoupling cases emerging in the study are also described in full detail. It is shown that for the v1 = ±1, w1 = ±1 values of the δ couplings the singular Kurasov matrices become equivalent to Dirichlet at one side of the point interaction and Robin boundary conditions at the other side.

The effects induced by the quantum vacuum fluctuations of one massless real scalar field on a configuration of two partially transparent plates are investigated. The physical properties of the infinitely thin plates are simulated by means of Dirac-δ − δ point interactions. It is shown that the distortion caused on the fluctuations by this external background gives rise to a generalization of Robin boundary conditions. The T -operator for potentials concentrated on points with non defined parity is evaluated with total generality. The quantum vacuum interaction energy between the two plates is computed in several dimensions using the T GT G formula to find positive, negative, and zero Casimir energies. The parity properties of the δ − δ potential demands to distinguish between opposite and identical objects. It is shown that between identical sets of δ − δ plates repulsive, attractive, or null quantum vacuum forces arise. However there is always attraction between a pair of opposite δ − δ plates.

The cerebellum plays a key role in motor tasks, but its involvement in cognition is still being considered. Although there is an association of different psychiatric and cognitive disorders with cerebellar impairments, the lack of time-course studies has hindered the understanding of the involvement of cerebellum in cognitive and non-motor functions. Such association was here studied using the Purkinje Cell Degeneration mutant mouse, a model of selective and progressive cerebellar degeneration that lacks the cytosolic carboxypeptidase 1 (CCP1). The effects of the absence of this enzyme on the cerebellum of mutant mice were analyzed both in vitro and in vivo. These analyses were carried out longitudinally (throughout both the pre-neurodegenerative and neurodegenerative stages) and different motor and non-motor tests were performed. We demonstrate that the lack of CCP1 affects microtubule dynamics and flexibility, defects that contribute to the morphological alterations of the Purkinje cells (PCs), and to progressive cerebellar breakdown. Moreover, this degeneration led not only to motor defects but also to gradual cognitive impairments, directly related to the progression of cellular damage. Our findings confirm the cerebellar implication in non-motor tasks, where the formation of the healthy, typical PCs structure is necessary for normal cognitive and affective behavior.

Following the seminal works of Asorey-Ibort-Marmo and Muñoz-Castañeda-Asorey about selfadjoint extensions and quantum fields in bounded domains, we compute all the heat kernel coefficients for any strongly consistent selfadjoint extension of the Laplace operator over the finite line [0, L]. The derivative of the corresponding spectral zeta function at s = 0 (partition function of the corresponding quantum field theory) is obtained. To compute the correct expression for the a 1/2 heat kernel coefficient, it is necessary to know in detail which non-negative selfadjoint extensions have zero modes and how many of them they have. The answer to this question leads us to analyze zeta function properties for the Von Neumann-Krein extension, the only extension with two zero modes.Mathematics Subject Classification. 81S99, 81T20, 81T55, 81U99, 34B08, 35J25, 11M36, 47B25, 47A10.Keywords. Quantum Theory, Quantum field theory on curved space backgrounds, Casimir effect, Scattering theory, Parameter dependent boundary value problems, Boundary value problems for second-order elliptic equations, Zeta and L-functions: analytic theory, Symmetric and selfadjoint operators, General theory of linear operators.

We calculate the quantum vacuum interaction energy between two kinks of the sine-Gordon equation. Using the T GT G-formula, the problem is reduced to the known formulas for quantum fluctuations in the background of a single kink. This interaction induces an attractive force between the kinks in parallel to the Casimir force between conducting mirrors.