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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Analysis with weak trace ideals and the number of bound states of Schrödinger operators
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by Barry Simon PDF
Trans. Amer. Math. Soc. 224 (1976), 367-380 Request permission

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 $\epsilon > 0:N(V) \leqslant C_\epsilon (\left \| V \right \|_{n/2 + \epsilon } + \left \| V \right \|_{n/2 - \epsilon })^{n/2}$ where ${\left \| \cdot \right \|_p}$ is an ${L^p}$ norm and that ${\lim \nolimits _{\lambda \to \infty }}N(\lambda V)/{\lambda ^{n/2}} = {(2\pi )^{ - n}}{\tau _n}\smallint |{V_ - }(x){|^{n/2}}{d^n}x$ 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 ${{\mathbf {R}}^n}$.
References
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Additional Information
  • © Copyright 1976 American Mathematical Society
  • 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