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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Analysis with weak trace ideals and the number of bound states of Schrödinger operators


Author: Barry Simon
Journal: Trans. Amer. Math. Soc. 224 (1976), 367-380
MSC: Primary 47F05; Secondary 81.35
DOI: https://doi.org/10.1090/S0002-9947-1976-0423128-X
MathSciNet review: 0423128
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Abstract: We discuss interpolation theory for the operator ideals $ I_p^w$ defined on a separable Hilbert space as those operators A whose singular values $ {\mu _n}(A)$ obey $ {\mu _n} \leqslant c{n^{ - 1/p}}$ for some c. As an application we consider the functional $ N(V) = \dim $ (spectral projection on $ ( - \infty ,0)$ for $ - \Delta + V$) on functions V on $ {{\mathbf{R}}^n},n \geqslant 3$. We prove that for any $ \epsilon > 0:N(V) \leqslant C_\epsilon (\left\Vert V \right\Vert _{n/2 + \epsilon } + \left\Vert V \right\Vert _{n/2 - \epsilon})^{n/2}$ where $ {\left\Vert \cdot \right\Vert _p}$ is an $ {L^p}$ norm and that $ {\lim\nolimits _{\lambda \to \infty }}N(\lambda V)/{\lambda ^{n/2}} = {(2\pi )^{ - n}}{\tau _n}\smallint \vert{V_ - }(x){\vert^{n/2}}{d^n}x$ for any $ V \in {L^{n/2 - }} \cap {L^{n/2 + }}$. Here $ {V_ - }$ is the negative part of V and $ {\tau _n}$ is the volume of the unit ball in $ {{\mathbf{R}}^n}$.


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DOI: https://doi.org/10.1090/S0002-9947-1976-0423128-X
Keywords: Weak trace ideals, Schrödinger operators
Article copyright: © Copyright 1976 American Mathematical Society