On the homotopy classification of elliptic operators on stratified manifolds

We find the stable homotopy classification of elliptic operators on stratified manifolds. Namely, we establish an isomorphism of the set of elliptic operators modulo stable homotopy and the $K$-homology group of the singular manifold. As a corollary, we obtain an explicit formula for the obstruction of Atiyah--Bott type to making interior elliptic operators Fredholm.Comment: 28 pages; submitted to Izvestiya Ross. Akad. Nau

Guillemin Transform and Toeplitz Representations for Operators on Singular Manifolds

An approach to the construction of index formulas for elliptic operators on singular manifolds is suggested on the basis of K-theory of algebras and cyclic cohomology. The equivalence of Toeplitz and pseudodifferential quantizations, well known in the case of smooth closed manifolds, is extended to the case of manifolds with conical singularities. We describe a general construction that permits one, for a given Toeplitz quantization of a C^*-algebra, to obtain a new equivalent Toeplitz quantization provided that a resolution of the projection determining the original quantization is given.Comment: 26 page

Wigner phase space distribution as a wave function

We demonstrate that the Wigner function of a pure quantum state is a wave function in a specially tuned Dirac bra-ket formalism and argue that the Wigner function is in fact a probability amplitude for the quantum particle to be at a certain point of the classical phase space. Additionally, we establish that in the classical limit, the Wigner function transforms into a classical Koopman-von Neumann wave function rather than into a classical probability distribution. Since probability amplitude need not be positive, our findings provide an alternative outlook on the Wigner function's negativity.Comment: 6 pages and 2 figure