{
  "id": "0912.3183",
  "version": "v5",
  "published": "2009-12-16T16:24:54.000Z",
  "updated": "2011-05-09T11:01:59.000Z",
  "title": "The Berry-Keating operator on $L^2(\\rz_>,\\ud x)$ and on compact quantum graphs with general self-adjoint realizations",
  "authors": [
    "Sebastian Endres",
    "Frank Steiner"
  ],
  "comment": "33pp",
  "journal": "J. Phys. A: Math. Theor. 43 (2010) 095204",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "The Berry-Keating operator $H_{\\mathrm{BK}}:= -\\ui\\hbar(x\\frac{\\ud\\phantom{x}}{\\ud x}+{1/2})$ [M. V. Berry and J. P. Keating, SIAM Rev. 41 (1999) 236] governing the Schr\\\"odinger dynamics is discussed in the Hilbert space $L^2(\\rz_>,\\ud x)$ and on compact quantum graphs. It is proved that the spectrum of $H_{\\mathrm{BK}}$ defined on $L^2(\\rz_>,\\ud x)$ is purely continuous and thus this quantization of $H_{\\mathrm{BK}}$ cannot yield the hypothetical Hilbert-Polya operator possessing as eigenvalues the nontrivial zeros of the Riemann zeta function. A complete classification of all self-adjoint extensions of $H_{\\mathrm{BK}}$ acting on compact quantum graphs is given together with the corresponding secular equation in form of a determinant whose zeros determine the discrete spectrum of $H_{\\mathrm{BK}}$. In addition, an exact trace formula and the Weyl asymptotics of the eigenvalue counting function are derived. Furthermore, we introduce the \"squared\" Berry-Keating operator $H_{\\mathrm{BK}}^2:= -x^2\\frac{\\ud^2\\phantom{x}}{\\ud x^2}-2x\\frac{\\ud\\phantom{x}}{\\ud x}-{1/4}$ which is a special case of the Black-Scholes operator used in financial theory of option pricing. Again, all self-adjoint extensions, the corresponding secular equation, the trace formula and the Weyl asymptotics are derived for $H_{\\mathrm{BK}}^2$ on compact quantum graphs. While the spectra of both $H_{\\mathrm{BK}}$ and $H_{\\mathrm{BK}}^2$ on any compact quantum graph are discrete, their Weyl asymptotics demonstrate that neither $H_{\\mathrm{BK}}$ nor $H_{\\mathrm{BK}}^2$ can yield as eigenvalues the nontrivial Riemann zeros. Some simple examples are worked out in detail.",
  "revisions": [
    {
      "version": "v5",
      "updated": "2011-05-09T11:01:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "81Q35"
    ],
    "keywords": [
      "compact quantum graph",
      "general self-adjoint realizations",
      "berry-keating operator",
      "corresponding secular equation",
      "self-adjoint extensions"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1088/1751-8113/43/9/095204",
      "journal": "Journal of Physics A Mathematical General",
      "year": 2010,
      "month": "Mar",
      "volume": 43,
      "number": 9,
      "pages": "095204"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010JPhA...43i5204E"
    }
  }
}