Universal Bounds on Spectral Measures of One-dimensional Schrödinger Operators

Let H = −d/dx + V (x) be a Schrödinger operator on L2(0,∞) with spectral measure ρ, and suppose that the potential V is known on an initial interval [0, N ]. We show that this information yields strong restrictions on ρ(I) for intervals I ⊂ R. More precisely, we prove upper and lower bounds on ρ(I). The upper bound is finite for any I that is bounded above and the lower bound is positive if the interior of I contains at least two eigenvalues of the operator on L2(0, N). These results are developments of classical work of Chebyshev and Markov on orthogonal polynomials.