Sharp estimates for eigenvalues of localization operators with applications to area laws
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866908912300589056 |
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| author | Kulikov, Aleksei Larsen, Martin Dam |
| author_facet | Kulikov, Aleksei Larsen, Martin Dam |
| contents | We study the eigenvalues of the localization operator $S_{A, B} = P_A\mathcal{F}^{-1}P_B\mathcal{F} P_A$, where $\mathcal{F}$ is the Fourier transform and $A = cA_0, B = B_0$ for some fixed sets $A_0, B_0\subset \mathbb{R}^d$ and a large parameter $c > 0$. For the counting function of the eigenvalues $|\{n: \varepsilon < λ_n(A,B)\le 1-\varepsilon\}|$ we obtain a sharp uniform upper bound if one of the sets is a finite disjoint union of parallelepipeds and a bound which is only a single logarithm off the conjectural optimal bound in the general case. These bounds are applied to the estimation of traces ${\rm{Tr}}\, f(S_{A,B})$ for functions $f$ with a very low regularity, in particular establishing an enhanced area law in the former case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_23832 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sharp estimates for eigenvalues of localization operators with applications to area laws Kulikov, Aleksei Larsen, Martin Dam Spectral Theory Classical Analysis and ODEs Functional Analysis We study the eigenvalues of the localization operator $S_{A, B} = P_A\mathcal{F}^{-1}P_B\mathcal{F} P_A$, where $\mathcal{F}$ is the Fourier transform and $A = cA_0, B = B_0$ for some fixed sets $A_0, B_0\subset \mathbb{R}^d$ and a large parameter $c > 0$. For the counting function of the eigenvalues $|\{n: \varepsilon < λ_n(A,B)\le 1-\varepsilon\}|$ we obtain a sharp uniform upper bound if one of the sets is a finite disjoint union of parallelepipeds and a bound which is only a single logarithm off the conjectural optimal bound in the general case. These bounds are applied to the estimation of traces ${\rm{Tr}}\, f(S_{A,B})$ for functions $f$ with a very low regularity, in particular establishing an enhanced area law in the former case. |
| title | Sharp estimates for eigenvalues of localization operators with applications to area laws |
| topic | Spectral Theory Classical Analysis and ODEs Functional Analysis |
| url | https://arxiv.org/abs/2603.23832 |