Almost Everything About the Unitary Almost Mathieu Operator
Fuente:
arXiv
Guardado en:
| Autores principales: | , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2021
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866912061060022272 |
|---|---|
| author | Cedzich, Christopher Fillman, Jake Ong, Darren C. |
| author_facet | Cedzich, Christopher Fillman, Jake Ong, Darren C. |
| contents | We introduce a unitary almost-Mathieu operator, which is obtained from a two-dimensional quantum walk in a uniform magnetic field. We exhibit a version of Aubry--André duality for this model, which partitions the parameter space into three regions: a supercritical region and a subcritical region that are dual to one another, and a critical regime that is self-dual. In each parameter region, we characterize the cocycle dynamics of the transfer matrix cocycle generated by the associated generalized eigenvalue equation. In particular, we show that supercritical, critical, and subcritical behavior all occur in this model. Using Avila's global theory of one-frequency cocycles, we exactly compute the Lyapunov exponent on the spectrum in terms of the given parameters. We also characterize the spectral type for each value of the coupling constant, almost every frequency, and almost every phase. Namely, we show that for almost every frequency and every phase the spectral type is purely absolutely continuous in the subcritical region, pure point in the supercritical region, and purely singular continuous in the critical region. In some parameter regions, we refine the almost-sure results. In the critical case for instance, we show that the spectrum is a Cantor set of zero Lebesgue measure for arbitrary irrational frequency and that the spectrum is purely singular continuous for all but countably many phases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_03216 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Almost Everything About the Unitary Almost Mathieu Operator Cedzich, Christopher Fillman, Jake Ong, Darren C. Spectral Theory Mathematical Physics Functional Analysis Quantum Physics We introduce a unitary almost-Mathieu operator, which is obtained from a two-dimensional quantum walk in a uniform magnetic field. We exhibit a version of Aubry--André duality for this model, which partitions the parameter space into three regions: a supercritical region and a subcritical region that are dual to one another, and a critical regime that is self-dual. In each parameter region, we characterize the cocycle dynamics of the transfer matrix cocycle generated by the associated generalized eigenvalue equation. In particular, we show that supercritical, critical, and subcritical behavior all occur in this model. Using Avila's global theory of one-frequency cocycles, we exactly compute the Lyapunov exponent on the spectrum in terms of the given parameters. We also characterize the spectral type for each value of the coupling constant, almost every frequency, and almost every phase. Namely, we show that for almost every frequency and every phase the spectral type is purely absolutely continuous in the subcritical region, pure point in the supercritical region, and purely singular continuous in the critical region. In some parameter regions, we refine the almost-sure results. In the critical case for instance, we show that the spectrum is a Cantor set of zero Lebesgue measure for arbitrary irrational frequency and that the spectrum is purely singular continuous for all but countably many phases. |
| title | Almost Everything About the Unitary Almost Mathieu Operator |
| topic | Spectral Theory Mathematical Physics Functional Analysis Quantum Physics |
| url | https://arxiv.org/abs/2112.03216 |