02-394 Christian Remling
Universal bounds on spectral measures of one-dimensional Schrödinger operators (38K, LaTeX) Sep 23, 02
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Abstract. Consider a Schr&ouml;dinger operator $H=-d^2/dx^2+V(x)$ on $L_2(0,\infty)$ and suppose that an initial piece of the potential $V(x)$, $0<x<N$ is known. We show that this information leads to upper and lower bounds on the spectral measure of intervals with a certain minimum length. This length scale is set by the eigenvalues of the problems on $[0,N]$. So in a sense (and perhaps somewhat surprisingly) the behavior of $V(x)$ becomes less important if $x$ grows. The results of this paper are developments of classical work of Chebyshev and Markov on orthogonal polynomials.

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