The Berry-Keating operator on $L^2(\rz_>,\ud x)$ and on compact quantum graphs with general self-adjoint realizations

TL;DR

This paper analyzes the Berry-Keating operator and its squared form on $L^2( z_>)$ and quantum graphs, showing their spectra are continuous or discrete but do not produce Riemann zeros, and provides a full classification of self-adjoint extensions.

Contribution

It offers a complete classification of all self-adjoint extensions of the Berry-Keating operator on quantum graphs and derives explicit spectral formulas.

Findings

Spectrum of $H_{BK}$ on $L^2( z_>)$ is purely continuous.

Neither $H_{BK}$ nor $H_{BK}^2$ can produce Riemann zeros as eigenvalues.

Explicit secular equations, trace formulas, and Weyl asymptotics are derived.

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.