{
  "id": "2009.05336",
  "version": "v1",
  "published": "2020-09-11T10:59:22.000Z",
  "updated": "2020-09-11T10:59:22.000Z",
  "title": "Mourre theory for unitary operators in two Hilbert spaces and quantum walks on trees",
  "authors": [
    "Rafael Tiedra de Aldecoa"
  ],
  "comment": "24 pages, 2 figures",
  "categories": [
    "math-ph",
    "math.FA",
    "math.MP",
    "math.SP"
  ],
  "abstract": "For unitary operators $U_0,U$ in Hilbert spaces ${\\mathcal H}_0,{\\mathcal H}$ and identification operator $J:{\\mathcal H}_0\\to{\\mathcal H}$, we present results on the derivation of a Mourre estimate for $U$ starting from a Mourre estimate for $U_0$ and on the existence and completeness of the wave operators for the triple $(U,U_0,J)$. As an application, we determine spectral and scattering properties of a class of anisotropic quantum walks on homogenous trees of degree $3$ with evolution operator $U$. In particular, we establish a Mourre estimate for $U$, obtain a class of locally $U$-smooth operators, and prove that the spectrum of $U$ covers the whole unit circle and is purely absolutely continuous, outside possibly a finite set where $U$ may have eigenvalues of finite multiplicity. We also show that (at least) three different choices of free evolution operators $U_0$ are possible for the proof of the existence and completeness of the wave operators.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-11T10:59:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A10",
      "47A40",
      "81Q10",
      "81Q12"
    ],
    "keywords": [
      "unitary operators",
      "hilbert spaces",
      "mourre theory",
      "mourre estimate",
      "wave operators"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}