Using invariance by fixed-endpoints homotopies and a generalized notion of symplectic Cayley transform, we prove a product formula for the Conley-Zehnder index of continuous paths with arbitrary endpoints in the symplectic group. We discuss two applications of the formula, to the metaplectic group and to periodic solutions of Hamiltonian systems.

Quantum Mechanical weak values are an interference effect measured by the cross-Wigner transform W(φ, ψ) of the post-and preselected states, leading to a complex quasi-distribution ρ φ,ψ (x, p) on phase space. We show that the knowledge of ρ φ,ψ (z) and of one of the two functions φ, ψ unambiguously determines the other, thus generalizing a recent reconstruction result of Lundeen and his collaborators.

We give an integral representation of the zeta-regularized determinant of Laplacians on three dimensional Heisenberg manifolds, and study a behavior of the values when we deform the uniform discrete subgroups. Heisenberg manifolds are the total space of a fiber bundle with a torus as the base space and a circle as a typical fiber, then the deformation of the uniform discrete subgroups means that the "radius" of the fiber goes to zero. We explain the lines of the calculations precisely for three dimensional cases and state the corresponding results for five dimensional Heisenberg manifolds. We see that the values themselves are of the product form with a factor which is that of the flat torus. So in the last half of this paper we derive general formulas of the zeta-regularized determinant for product type manifolds of two Riemannian manifolds, discuss the formulas for flat tori and explain a relation of the formula for the two dimensional flat torus and the Kroneker's second limit formula.

We use the properties of the Leray index to give precise formulas in arbitrary dimensions for the Maslov index of the monodromy matrix arising in periodic Hamiltonian systems. We compare our index with other indices appearing in the literature.

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