Access

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

login

Log in to your personal account or through your institution.

If You Use a Screen Reader

This content is available through Read Online (Free) program, which relies on page scans. Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.

Weak Type Estimates for Singular Values and the Number of Bound States of Schrodinger Operators

Michael Cwikel
Annals of Mathematics
Second Series, Vol. 106, No. 1 (Jul., 1977), pp. 93-100
Published by: Annals of Mathematics
DOI: 10.2307/1971160
Stable URL: http://www.jstor.org/stable/1971160
Page Count: 8
  • Read Online (Free)
  • Subscribe ($19.50)
  • Cite this Item
Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Weak Type Estimates for Singular Values and the Number of Bound States of Schrodinger Operators
Preview not available

Abstract

If A is an operator defined by the kernel f(x - y)g(y) with g ε Lp (Rn) and f ε Weak Lp' (Rn) where $1/p + 1/p \' = 1 and 2

}1 K^{1/p} \mu_k \underline {\underline<} C_p \Vert f \Vert L^p \', \infty \Vert g \Vert L^p.$ One application of this is the following result: Let $n \underline{\underline>}$ and let N(V) - dime (spectral projection on (- ∞, 0] for - Δ + V) where V is any function in Ln/2(Rn), then $N(V) \underline{\underline <}$ const. $\int \vert V (x) \vert^{n/2}d^nx$ where V is the negative part of V. Furthermore $lim _{\lambda \rightarrow \infty} N(\lambda V)/ \lambda ^{n/2} = (2\pi) ^n {\tau _n} \int \vert V (x) \vert ^{n/2}d^nx,$ where τn is the volume of the unit ball of Rn.

Page Thumbnails

  • Thumbnail: Page 
[93]
    [93]
  • Thumbnail: Page 
94
    94
  • Thumbnail: Page 
95
    95
  • Thumbnail: Page 
96
    96
  • Thumbnail: Page 
97
    97
  • Thumbnail: Page 
98
    98
  • Thumbnail: Page 
99
    99
  • Thumbnail: Page 
100
    100