Cwikel-Lieb-Rozenblum type inequalities for Hardy-Schrödinger operator

We prove a Cwikel-Lieb-Rozenblum type inequality for the number of negative eigenvalues of the Hardy-Schr\"odinger operator \(-\Delta - (d-2)^2/(4|x|^2) -W(x)\) on \(L^2(\mathbb{R}^d)\). The bound is given in terms of a weighted \(L^{d/2}-\)norm of \(W\) which is sharp in both large and small c...

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Bibliographic Details
Published inarXiv.org
Main Authors Duong, Giao Ky, Frank, Rupert L, Thi Minh Thao Le, Phan, Thành Nam, Phuoc-Tai Nguyen
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 19.06.2024
Subjects
Eigenvalues
Operators (mathematics)
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Summary:We prove a Cwikel-Lieb-Rozenblum type inequality for the number of negative eigenvalues of the Hardy-Schr\"odinger operator \(-\Delta - (d-2)^2/(4|x|^2) -W(x)\) on \(L^2(\mathbb{R}^d)\). The bound is given in terms of a weighted \(L^{d/2}-\)norm of \(W\) which is sharp in both large and small coupling regimes. We also obtain a similar bound for the fractional Laplacian.
ISSN:2331-8422