Series: Mathematics Colloquium

Date: Thursday, November 7, 2002

Time: 4:30 - 5:30 PM

Place: 102 McAllister Building

Host: Yuri Zarhin

Refreshments: 4:00 - 4:30 PM, in 212 McAllister

Speaker: Mikhail Shubin, Northeastern University

Title: Discreteness of Spectrum for Schr\"odinger Operators

Abstract:

Discreteness of spectrum of a self-adjoint operator in a Hilbert space
means that this operator has an orthonormal basis of eigenvectors, and
besides the eigenvalues form a discrete set and have finite
multiplicities.  In the talk I will discuss conditions for the
discreteness of spectrum of Schr\"odinger operators $H_V=-\Delta + V(x)$ in $L^2({\mathbb{R}}^n)$ with $V\ge 0$. A sufficient condition
$V(x)\to+\infty$ as $x\to\infty$ was first found by H. Weyl (1910) for
$n=1$ and K.Friedrichs (1934) for arbitrary $n$. A necessary and
sufficient condition expressed in terms of Wiener capacity was found
by A. Molchanov (1953).  Recently new criteria of discreteness of
spectrum were found for $H_V$ as well as for more general magnetic
Schr\"odinger operators. The equivalence of these conditions
constitutes a new property of the Wiener capacity. The only known
proof is very indirect: it follows from the fact that all these
conditions are equivalent to the discreteness of spectrum.