Quantum Mixing for Schrödinger eigenfunctions in Benjamini-Schramm limit
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| Format: | Preprint |
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2026
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| _version_ | 1866910160425844736 |
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| author | Hippi, Kai Lequen, Félix Mikkelsen, Søren Sahlsten, Tuomas Ueberschär, Henrik |
| author_facet | Hippi, Kai Lequen, Félix Mikkelsen, Søren Sahlsten, Tuomas Ueberschär, Henrik |
| contents | Let $-Δ_{\mathbb{H}}+V$ be the Schrödinger operator on $\mathbb{H}$ where $V \in L^p(\mathbb{H}) \cap L^\infty(\mathbb{H})$ for some $p > 0$. If $(X_n)$ is a uniformly discrete sequence of compact hyperbolic surfaces with a uniform spectral gap that Benjamini-Schramm converges to $\mathbb{H}$, we prove quantum mixing for the eigenfunctions of $-Δ_{X_n}+V_n$ in any sufficiently large spectral window $I$, where $V_n$ is the potential on $X_n$ induced by $V$. These apply to large degree lifts of a potential on a base surface such as congruence covers of arithmetic surfaces, with high probability to random hyperbolic surfaces in the Weil-Petersson model of large genus, and to Hartree one-particle operators arising in thermodynamic limit of many-body Bose gas on hyperbolic surfaces. The proof uses the Duhamel formula for the hyperbolic wave equation together with exponential mixing of the geodesic flow on $T^1 X_n$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_21582 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Quantum Mixing for Schrödinger eigenfunctions in Benjamini-Schramm limit Hippi, Kai Lequen, Félix Mikkelsen, Søren Sahlsten, Tuomas Ueberschär, Henrik Spectral Theory Mathematical Physics Differential Geometry Dynamical Systems 81Q50, 37D40, 11F72 Let $-Δ_{\mathbb{H}}+V$ be the Schrödinger operator on $\mathbb{H}$ where $V \in L^p(\mathbb{H}) \cap L^\infty(\mathbb{H})$ for some $p > 0$. If $(X_n)$ is a uniformly discrete sequence of compact hyperbolic surfaces with a uniform spectral gap that Benjamini-Schramm converges to $\mathbb{H}$, we prove quantum mixing for the eigenfunctions of $-Δ_{X_n}+V_n$ in any sufficiently large spectral window $I$, where $V_n$ is the potential on $X_n$ induced by $V$. These apply to large degree lifts of a potential on a base surface such as congruence covers of arithmetic surfaces, with high probability to random hyperbolic surfaces in the Weil-Petersson model of large genus, and to Hartree one-particle operators arising in thermodynamic limit of many-body Bose gas on hyperbolic surfaces. The proof uses the Duhamel formula for the hyperbolic wave equation together with exponential mixing of the geodesic flow on $T^1 X_n$. |
| title | Quantum Mixing for Schrödinger eigenfunctions in Benjamini-Schramm limit |
| topic | Spectral Theory Mathematical Physics Differential Geometry Dynamical Systems 81Q50, 37D40, 11F72 |
| url | https://arxiv.org/abs/2604.21582 |