Infinitesimal form boundedness and Trudinger's subordination for the Schrodinger operator

• 成果类型: Article
• 署名作者: Maz'ya, VG; Verbitsky, IE
• 署名单位: University System of Ohio; Ohio State University; University of Liverpool; University of Missouri System; University of Missouri Columbia
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-005-0439-y
• 发表日期: 2005
• 页码: 81-136
• 关键词: INEQUALITIES; compactness

摘要

We give explicit analytic criteria for two problems associated with the Schrodinger operator H = -Delta + Q on L-2(R-n) where Q is an element of D'(R-n) is an arbitrary real- or complex-valued potential. First, we obtain necessary and sufficient conditions on Q so that the quadratic form < Q(.), (.)> has zero relative bound with respect to the Laplacian. For Q is an element of L-loc(1)(R-n), this property can be expressed in the form of the integral inequality: vertical bar integral(Rn) vertical bar u(x)vertical bar(2) Q(x)dx vertical bar <= epsilon parallel to del u parallel to(2)(L2(Rn)) + C(epsilon) parallel to u parallel to(2)(L2(Rn)), for all u is an element of C-0(infinity)(R-n), for an arbitrarily small epsilon > 0 and some C(epsilon) > 0. One of the major steps here is the reduction to a similar inequality with nonnegative function vertical bar del(1-Delta)(-1) Q vertical bar(2) + vertical bar(1 - Delta)(-1) Q vertical bar in place of Q. This provides a complete solution to the infinitesimal form boundedness problem for the Schrodinger operator, and leads to new broad classes of admissible distributional potentials Q, which extend the usual L-p and Kato classes, as well as those based on the well-known conditions of Fefferman-Phong and Chang-Wilson-Wolff. Secondly, we characterize Trudinger's subordination property where C(epsilon) in the above inequality is subject to the condition C(epsilon) <= c epsilon(-beta) (beta > 0) as epsilon -> +0. Such quadratic form inequalities can be understood entirely in the framework of Morrey-Campanato spaces, using mean oscillations of del(1 - Delta)(-1) Q and (1 - Delta)(-1) Q on balls or cubes. A version of this condition where epsilon is an element of (0,+infinity) is equivalent to the multiplicative inequality: vertical bar integral(Rn) vertical bar u(x)vertical bar(2) Q(x) dx vertical bar <= = C parallel to del u parallel to(2p)(L2(Rn)) parallel to u parallel to(2(1- p))(L2(Rn)) , for all u is an element of C-0(infinity) (R-n), with p = beta/1+beta is an element of (0, 1). We show that this inequality holds if and only if del Delta (-1) Q is an element of BMO(R-n) if p = 1/2. For 0 < p < 1/2, it is valid whenever del Delta (-1) Q is Holder-continuous of order 1 - 2p, or respectively lies in the Morrey space L-2,L-lambda with lambda = n + 2 - 4p if 1/2 < p < 1. As a consequence, we characterize completely the class of those Q which satisfy an analogous multiplicative inequality of Nash's type, with parallel to u parallel to (L1) ((Rn)) in place of parallel to u parallel to(L2) ((Rn)). These results are intimately connectedwith spectral theory and dynamics of the Schrodinger operator, and elliptic PDE theory.