Algebraic Problems in Quantum Mechanics on Graphs

Abstract

Quantum mechanics on graphs utilizes to a great extent the knowledge and methods of linear algebra and matrix theory. In this work we illustrate the connections between the above fields and demonstrate, among others, that those purely mathematical dis ciplines can draw inspiration from the theory of quantum graphs. The thesis consists of three main chapters. In the first of them, we introduce a series of original results concerning scattering properties of vertices in quantum graphs. We find a nontrivial parametric family of vertex couplings whose reflection and transmission probabilities are indistinguishable from those of the free coupling, and we introduce couplings with permutation-symmetric scattering probabilities. Then we construct two exotic types of couplings in vertices of degrees 3 and 4 such that the transmission probabilities depend in a remarkable way on an external poten tial, which thus allows to control the passage of electrons through the vertex. Finally, we introduce a parametric family of couplings that continuously interpolate between the δ coupling and a certain recently discovered rotationally symmetric coupling with anomalous spectral properties. These results are followed by a study of two special matrix families. At first we ex amine the existence and construction of Hermitian unitary matrices whose diagonal entries have absolute value r ≥ 0 and off-diagonal entries have absolute value t > 0. Then we focus on the family of circulant matrices with the value d ≥ 0 on the diago nal, 1 or −1 off the diagonal and with mutually orthogonal rows. If d ≥ 0 is even, we prove that such matrices exist only of order 2d + 2. This is a generalization of a classi cal theorem saying that circulant conference matrices (case d = 0) exist only of order 2 [Stanton and Mullin 1976]. For any d ≥ 0 we disprove the existence of a symmet ric matrix of the given type of order n > 2d + 2, which generalizes another classical theorem [Johnsen 1964] about the nonexistence of a symmetric circulant Hadamard matrix of order n > 4. In the last part of the thesis we address spectral properties of periodic quantum graphs. We examine a square lattice with a general coupling in the vertices, a dilated honeycomb network with the δ coupling, and finally we focus on a rectangular lat tice with the δ coupling in the vertices, for which we examine in detail the number of spectral gaps. After recalling the Bethe–Sommerfeld conjecture, we present one of the main achievements in this thesis, namely, the proof of existence of a periodic quantum graph featuring a finite nonzero number of gaps in the spectrum.