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

The Berry-Keating operator H_{\mathrm{BK}}:= -\ui\hbar(x\frac{ \phantom{x}}{ 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_>, x) and on compact quantum graphs. It is proved that the spectrum of $H_{\mathrm{BK}}$ defined on L^2(\rz_>, 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 $H_{\mathrm{BK}}^2:= -x^2\frac{ ^2\phantom{x}}{ x^2}-2x\frac{ \phantom{x}}{ x}-{1/4}$ 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.Comment: 33p

Spectral asymptotics for the third order operator with periodic coefficients

We consider the self-adjoint third order operator with 1-periodic coefficients on the real line. The spectrum of the operator is absolutely continuous and covers the real line. We determine the high energy asymptotics of the periodic, anti-periodic eigenvalues and of the branch points of the Lyapunov function. Furthermore, in the case of small coefficients we show that either whole spectrum has multiplicity one or the spectrum has multiplicity one except for a small spectral nonempty interval with multiplicity three. In the last case the asymptotics of this small interval is determined.Comment: 30 pages, 3 figure

On the resonances and eigenvalues for a 1D half-crystal with localised impurity

We consider the Schr\"odinger operator $H$ on the half-line with a periodic potential $p$ plus a compactly supported potential $q$. For generic $p$, its essential spectrum has an infinite sequence of open gaps. We determine the asymptotics of the resonance counting function and show that, for sufficiently high energy, each non-degenerate gap contains exactly one eigenvalue or antibound state, giving asymptotics for their positions. Conversely, for any potential $q$ and for any sequences (\s_n)_{1}^\iy, \s_n\in \{0,1\}, and (\vk_n)_1^\iy\in \ell^2, \vk_n\ge 0, there exists a potential $p$ such that \vk_n is the length of the $n$-th gap, $n\in\N$, and $H$ has exactly \s_n eigenvalues and 1-\s_n antibound state in each high-energy gap. Moreover, we show that between any two eigenvalues in a gap, there is an odd number of antibound states, and hence deduce an asymptotic lower bound on the number of antibound states in an adiabatic limit.Comment: 25 page

On the Physics of the Riemann Zeros

We discuss a formal derivation of an integral expression for the Li coefficients associated with the Riemann xi-function which, in particular, indicates that their positivity criterion is obeyed, whereby entailing the criticality of the non-trivial zeros. We conjecture the validity of this and related expressions without the need for the Riemann Hypothesis and discuss a physical interpretation of this result within the Hilbert-Polya approach. In this context we also outline a relation between string theory and the Riemann Hypothesis.Comment: 8 pages, LaTeX, Quantum Theory and Symmetries 6 conference proceeding

Universality of the Distribution Functions of Random Matrix Theory. II

This paper is a brief review of recent developments in random matrix theory. Two aspects are emphasized: the underlying role of integrable systems and the occurrence of the distribution functions of random matrix theory in diverse areas of mathematics and physics.Comment: 17 pages, 3 figure