arXiv subjects
Publications and source records attributed to Mauro Spreafico.
Motivated by the work of Cappell, Deturck, Gluch and Miller, we extend the notion of cohomology of harmonic forms (of a compact manifold with boundary) to the abstract setting of Hilbert complexes. Then, we present some geometric applications of our construction to incomplete Riemannian manifolds with particular interest to the case of smoothly stratified Thom-Mather spaces.
We study the global analytic properties of a space $X$ with a horn type singularity. In particular, we introduce some de Rham complex of square integrable forms and we describe its homology and the spectral properties of the associated Hodge Laplace operator. All this is applied to produce a suitable description of the analytic torsion of $X$ and to prove an extension of the Cheeger Müller theorem.
We prove that the anomaly term appearing in the ratio between Reidemeister torsion and analytic torsion for a space with a conical singularity vanishes in the smooth case. Moreover, we show that such anomaly term is non trivial for the cone over a torus.
We construct an explicit equivariant cellular decomposition of the $(4n-1)$-sphere with respect to binary polyhedral groups, and describe the associated cellular homology chain complex. As a corollary of the binary octahedral case, we deduce an $\mathfrak{S}_3$-equivariant decomposition of the flag manifold of $SL_3(\mathbb{R})$.
We prove an extension of the Cheeger-Müller theorem to spaces with isolated conical singularities: the $L^2$-analytic torsion coincides with the Ray-Singer intersection torsion on an even dimensional space, and they are trivial, while the ratio is non trivial on an odd dimensional space, and the anomaly depends only on the link of the singularities. For this aim, we develop on one side a combinatorial cellular theory whose homology coincides with the intersection homology of Gregory and Macpherson, and where the Ray-Singer intersection torsion is well defined. On the other side, we elaborate the spectral theory for the Hodge-Laplace operator on the square integrable forms on a space with conical singularities {\it á la} Cheeger, and we extend the classical results of the Hodge theory and the analytic torsion.
The orbit polytope for a finite group G acting linearly and freely on a sphere S is used to construct a cellularized fundamental domain for the action. A resolution of the integers over G results from the associated G-equivariant cellularization of S. This technique is applied to the generalized binary tetrahedral group family; the homology groups, the cohomology rings and the Reidemeister torsions of the related spherical space forms are determined.
The relative partition function and the relative zeta function of the perturbation of the Laplace operator by a Coulomb potential plus a point interaction centered in the origin is discussed. Applications to the study of the Casimir effect are indicated.
We prove the equality of the $L^2$-analytic torsion and the intersection R torsion of the even dimensional finite metric cone over an odd dimensional compact manifold.
We investigate the limit the R torsion of a conical frustum as one of the basis is shrunk to a point. We show that, if we take suitable regularization, such a limit gives the intersection torsion of the resulting cone.
We study the spectral functions, and in particular the zeta function, associated to a class of sequences of complex numbers, called of spectral type. We investigate the decomposability of the zeta function associated to a double sequence with respect to some simple sequence, and we provide a technique for obtaining the first terms in the Laurent expansion at zero of the zeta function associated to a double sequence. We particularize this technique to the case of sums of sequences of spectral type, and we give two applications: the first concerning some special functions appearing in number theory, and the second the functional determinant of the Laplace operator on a product space.
We use relative zeta functions technique of W. Muller \cite{Mul} to extend the classical decomposition of the zeta regularized partition function of a finite temperature quantum field theory on a ultrastatic space-time with compact spatial section to the case of non compact spatial section. As an application, we study the case of Schrödinger operators with delta like potential, as described by Albeverio & alt. in \cite{AGHH}.
We study the non homogeneous quadratic Bessel zeta function $ζ_{RB}(s,ν,a)$ defined as the sum of the square of the positive zeros of the Bessel function $J_ν(z)$ plus a positive constant. In particular, we give explicit formulas for the main associated zeta invariants, namely poles and residua, $ζ_{RB}(0,ν,a)$ and $ζ'_{RB}(0,ν,a)$.
A Hermite type formula is introduced and used to study the zeta function over the real and complex n-projective space. This approach allows to compute the residua at the poles and the value at the origin as well as the value of the derivative at the origin, that gives the regularized determinant of the associated Laplacian operator.