Semiclassical Resolvent Estimates for the Magnetic Schr{Ö}dinger Operator
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866914090998300672 |
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| author | Vodev, Georgi |
| author_facet | Vodev, Georgi |
| contents | We obtain semiclassical resolvent estimates for the Schr{ö}dinger operator (ih$\nabla$ + b)^2 + V in R^d , d $\ge$ 3, where h is a semiclassical parameter, V and b are real-valued electric and magnetic potentials independent of h. Under quite general assumptions, we prove that the norm of the weighted resolvent is bounded by exp(Ch^{-2} log(h^{ -1} )) . We get better resolvent bounds for electric potentials which are H{ö}lder with respect to the radial variable and magnetic potentials which are H{ö}lder with respect to the space variable. For long-range electric potentials which are Lipschitz with respect to the radial variable and long-range magnetic potentials which are Lipschitz with respect to the space variable we obtain a resolvent bound of the form exp(Ch^{-1}) . |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_07271 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Semiclassical Resolvent Estimates for the Magnetic Schr{Ö}dinger Operator Vodev, Georgi Analysis of PDEs We obtain semiclassical resolvent estimates for the Schr{ö}dinger operator (ih$\nabla$ + b)^2 + V in R^d , d $\ge$ 3, where h is a semiclassical parameter, V and b are real-valued electric and magnetic potentials independent of h. Under quite general assumptions, we prove that the norm of the weighted resolvent is bounded by exp(Ch^{-2} log(h^{ -1} )) . We get better resolvent bounds for electric potentials which are H{ö}lder with respect to the radial variable and magnetic potentials which are H{ö}lder with respect to the space variable. For long-range electric potentials which are Lipschitz with respect to the radial variable and long-range magnetic potentials which are Lipschitz with respect to the space variable we obtain a resolvent bound of the form exp(Ch^{-1}) . |
| title | Semiclassical Resolvent Estimates for the Magnetic Schr{Ö}dinger Operator |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2501.07271 |