Ichiro Shigekawa

Spectral properties of Schrödinger operators with magnetic fields for a spin 12 particle

Abstract We discuss the spectral properties of Schrodinger operators with magnetic fields, especially for a spin 1 2 particle. We are interested in the essential spectrum and the discrete spectrum. We classified the spectrum by the asymptotic behavior of the magnetic field at infinity. Our main tools are the supersymmetry and the complex analysis in several variables. We show that 0 is an eigenvalue with infinite multiplicity under suitable conditions.

Sobolev spaces of Banach-valued functions associated with a Markov process

SummaryWe discuss Sobolev spaces of Banach-valued functions. They are extensions of Sobolev spaces of scalar functions. We use a gamma transform of a semigroup associated with a Markov process. A typical example is the Ornstein-Uhlenbeck process on the Wiener space.

Itô-Wiener expansions of holomorphic functions on the complex Wiener space

Abstract We show Lp-convergence of Ito-Wiener expansions for holomorphic functions on a complex abstract Wiener space and the unicity theorem.

VANISHING THEOREM OF THE HODGE–KODAIRA OPERATOR FOR DIFFERENTIAL FORMS ON A CONVEX DOMAIN OF THE WIENER SPACE

We discuss the vanishing theorem on a convex domain of the Wiener space. We show that there is no harmonic form satisfying the absolute boundary condition. Our method relies on an expression of the bilinear form associated with the Hodge–Kodaira operator.

Semigroup Domination on a Riemannian Manifold with Boundary

We discuss the semigroup domination on a Riemannian manifold with boundary. Our main interest is the Hodge–Kodaira Laplacian for differential forms. We consider two kinds of boundary conditions; the absolutely boundary condition and the relative boundary condition. Our main tool is the square field operator. We also develop a general theory of semigroup commutation.

A Kähler metric on a based loop group and a covariant differentiation

Loop groups have been attracting many authors recently. In this paper, we are discussing a Kahler metric on a loop group. Let G be a d-dimensional compact Lie group and g be its Lie algebra (- the space of left invariant vector fields). Then, g admits an Ad(G)-invariant inner product (η, η)g and we fix it through the paper. We denote the G-valued path space on [0, 1] by

The Domain of a Generator and the Intertwining Property

PG:=\left\{ \gamma :\left[ 0,1 \right]\to G;continuous and \gamma \left( 0 \right)=e \right\}

New Trends in Stochastic Analysis: Proceedings of a Taniguchi International Workshop

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