Quantum Graphs via Exercises

Studying the spectral theory of Schroedinger operator on metric graphs (also known as quantum graphs) is advantageous on its own as well as to demonstrate key concepts of general spectral theory. There are some excellent references for this study such as a mathematically oriented book by Berkolaiko and Kuchment, a review with applications to theoretical physicsby Gnutzmann and Smilansky, and elementary lecture notes by Berkolaiko. Here, we provide a set of questions and exercises which can accompany the reading of these references or an elementary course on quantum graphs. The exercises are taken from courses on quantum graphs which were taught by the authors

Conductance and absolutely continuous spectrum of 1D samples

We characterize the absolutely continuous spectrum of the one-dimensional Schr\"odinger operators $h=-\Delta+v$ acting on $\ell^2(\mathbb{Z}_+)$ in terms of the limiting behavior of the Landauer-B\"uttiker and Thouless conductances of the associated finite samples. The finite sample is defined by restricting $h$ to a finite interval $[1,L]\cap\mathbb{Z}_+$ and the conductance refers to the charge current across the sample in the open quantum system obtained by attaching independent electronic reservoirs to the sample ends. Our main result is that the conductances associated to an energy interval $I$ are non-vanishing in the limit $L\to\infty$ iff ${\rm sp}_{\rm ac}(h)\cap I=\emptyset$. We also discuss the relationship between this result and the Schr\"odinger Conjecture

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