Seminários do Grupo de Física Matemática

9 de Março de 2005 (quarta-feira), 14:30, sala B1-01

On principal eigenvalue of Stokes operator in large random domain

Vadim Yurinsky
(Universidade da Beira Interior)

This communication deals with localization of the principal eigenvalue (PE) of the Stokes operator under the Dirichlet condition on a random fine-grained boundary of the flow domain F_t contained in a large cubic block [-t,t]^d, t→∞. In individual unit cubic cells, the complements to the flow domain are S_z = (z+[-1/2, 1/2]^d) F_t for each z from the integer lattice. The random microstructure is assumed identically distributed and essentially independent in distinct cells. In this setting, the PE exhibits deterministic behavior as the volume of the containing block goes to infinity. For the Laplace and Schrödinger operators, problems of this kind have been studied since mid-eighties (see [1,2]). A major part of exposition deals with adaptation to the Stokes operator of the approach used in [2] to characterize large volume asymptotic behaviour of the PE of the Schrödinger operator. One of the results is an extension of the upper bound on the Stokes PE obtained in [4] for planar flows to dimensions d>2. The method of [2] is used with later modifications [3].

1. Sznitman A.-S., Brownian Motion, Obstacles and Random Media, (Springer-Verlag, Berlin-Heidelberg-New York, 1998).
2. Merkl F., Wütrich N. V., "Infinite volume asymptotics of the ground state energy in scaled Poissonian potential", Ann.Inst. H. Poincaré - PR 38 (3) (2002), 253-284.
3. Yurinsky V. V., "On the principal eigenvalue of the Schrödinger operator with a scaled random potential", in Contemporary Mathematics (2003) vol. 339, 201-216.
4. Yurinsky V. V., "Localization of spectrum bottom for the Stokes operator in a random porous medium", Siber. Math. J. 42 (2) (2001), 386-413.

23 de Fevereiro de 2005 (quarta-feira) 14:30, sala B1-01

Smooth functional derivatives in Feynman path integrals by time slicing approximation

Naoto Kumano-go
(GFMUL / Dept. Mathematics, Kogakuin University, Japan)

We give a fairly general class of functionals on a path space so that Feynman path integral has a mathematically rigorous meaning. More precisely, for any functional belonging to our class, the time slicing approximation of Feynman path integral converges uniformly on compact subsets of the configuration space. Our class of functionals is closed under addition, multiplication, functional differentiation, translation and real linear transformation. The integration by parts and Taylor's expansion formula with respect to functional differentiation holds in Feynman path integral. Feynman path integral is invariant under translation and orthogonal transformation. The interchange of the order with Riemann-Stieltjes integrals, the interchange of the order with a limit, the semiclassical approximation and the fundamental theorem of calculus in Feynman path integral stay valid.

[1] N. Kumano-go, "Feynman path integrals as analysis on path space by time slicing approximation". Bull. Sci. Math 128, issue 3, pp.197-251 (2004).
[2] D. Fujiwara and N. Kumano-go, "Smooth functional derivatives in Feynman path integrals by time slicing approximation". Bull. Sci. Math. 129, issue 1, pp 57-79 (2005).

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Last modified: Tue Apr 19 08:38:01 WEST 2005