학술논문
Weak type estimates for singular values and the number of bound states of Schrödinger operators.
Document Type
Journal
Author
Cwikel, Michael AMS Author Profile
Source
Subject
81 Quantum theory
81.35Partial differential equations
81.35
Language
English
Abstract
In $L^2({\bf R}^n)$ consider the operator $B_{u,g}$ defined by $(B_{u,g}f)(\xi)=\int\exp(2\pi i\xi\cdot x)u(\xi)g(x)f(x)\,d^nx$. The following theorem is proved: Let $20}\alpha[\mu\{k\colon\lambda_k(B_{u,g})>\alpha\}]^{1/p}<\infty$. Here $\lambda_k$ are the singular values of $B_{u,g}$ (arranged in non-increasing order) and $\mu(E)$ is the cardinality of the set $E$. (b) The singular values of $B_{u,g}$ satisfy $\sup_kk^{1/p}[k^{-1}\sum_{m=1}^k\lambda_m(B_{u,g})^2]^{1/2}\leq K_p\|u\|_{p,\text{weak}}\|g\|_p$, where $K_p$ is a constant depending on $p$. As a corollary, let $n\geq 3$ and let $N(V)$ denote the number of nonpositive eigenvalues (multiplicities counted) of the Schrödinger operator $-\Delta+V$ in $L^2({\bf R}^n)$, where $V(x)$ is a negative potential in $L^{n/2}({\bf R}^n)$. Then $N(V)\leq A_n\int|V(x)|^{n/2}\,d^nx$ for some constant $A_n$ depending on $n$.