Arithmetic spectral transition for the unitary almost Mathieu operator
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915701101428736 |
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| author | Yang, Fan |
| author_facet | Yang, Fan |
| contents | We study the unitary almost Mathieu operator (UAMO), a one-dimensional quasi-periodic unitary operator arising from a two-dimensional discrete-time quantum walk on $\mathbb Z^2$ in a homogeneous magnetic field. In the positive Lyapunov exponent regime $0\le λ_1<λ_2\le 1$, we establish an arithmetic localization statement governed by the frequency exponent $β(ω)$. More precisely, for every irrational $ω$ with $β(ω)<L$, where $L>0$ denotes the Lyapunov exponent, and every non-resonant phase $θ$, we prove Anderson localization, i.e. pure point spectrum with exponentially decaying eigenfunctions. This extends our previous arithmetic localization result for Diophantine frequencies (for which $β(ω)=0$) to a sharp threshold in frequency. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_24616 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Arithmetic spectral transition for the unitary almost Mathieu operator Yang, Fan Spectral Theory Mathematical Physics Quantum Physics We study the unitary almost Mathieu operator (UAMO), a one-dimensional quasi-periodic unitary operator arising from a two-dimensional discrete-time quantum walk on $\mathbb Z^2$ in a homogeneous magnetic field. In the positive Lyapunov exponent regime $0\le λ_1<λ_2\le 1$, we establish an arithmetic localization statement governed by the frequency exponent $β(ω)$. More precisely, for every irrational $ω$ with $β(ω)<L$, where $L>0$ denotes the Lyapunov exponent, and every non-resonant phase $θ$, we prove Anderson localization, i.e. pure point spectrum with exponentially decaying eigenfunctions. This extends our previous arithmetic localization result for Diophantine frequencies (for which $β(ω)=0$) to a sharp threshold in frequency. |
| title | Arithmetic spectral transition for the unitary almost Mathieu operator |
| topic | Spectral Theory Mathematical Physics Quantum Physics |
| url | https://arxiv.org/abs/2512.24616 |