Spectral Theory of Infinite Quantum Graphs

We investigate quantum graphs with infinitely many vertices and edges without the common restriction on the geometry of the underlying metric graph that there is a positive lower bound on the lengths of its edges. Our central result is a close connection between spectral properties of a quantum graph and the corresponding properties of a certain weighted discrete Laplacian on the underlying discrete graph. Using this connection together with spectral theory of (unbounded) discrete Laplacians on infinite graphs, we prove a number of new results on spectral properties of quantum graphs. Namely, we prove several self-adjointness results including a Gaffney type theorem. We investigate the problem of lower semiboundedness, prove several spectral estimates (bounds for the bottom of spectra and essential spectra of quantum graphs, CLR-type estimates) and study spectral types.Comment: Dedicated to the memory of M. Z. Solomyak (16.05.1931 - 31.07.2016

Quantum ground state isoperimetric inequalities for the energy spectrum of local Hamiltonians

We investigate the relationship between the energy spectrum of a local Hamiltonian and the geometric properties of its ground state. By generalizing a standard framework from the analysis of Markov chains to arbitrary (non-stoquastic) Hamiltonians we are naturally led to see that the spectral gap can always be upper bounded by an isoperimetric ratio that depends only on the ground state probability distribution and the range of the terms in the Hamiltonian, but not on any other details of the interaction couplings. This means that for a given probability distribution the inequality constrains the spectral gap of any local Hamiltonian with this distribution as its ground state probability distribution in some basis (Eldar and Harrow derived a similar result in order to characterize the output of low-depth quantum circuits). Going further, we relate the Hilbert space localization properties of the ground state to higher energy eigenvalues by showing that the presence of k strongly localized ground state modes (i.e. clusters of probability, or subsets with small expansion) in Hilbert space implies the presence of k energy eigenvalues that are close to the ground state energy. Our results suggest that quantum adiabatic optimization using local Hamiltonians will inevitably encounter small spectral gaps when attempting to prepare ground states corresponding to multi-modal probability distributions with strongly localized modes, and this problem cannot necessarily be alleviated with the inclusion of non-stoquastic couplings

The Analytic Torsion on Manifolds with Boundary and Conical Singularities

The analytic torsion was introduced by D.B. Ray and I.M. Singer as an analytic counterpart to the combinatorial Reidemeister torsion. In this thesis we are concerned with analytic torsion of manifolds with boundary and conical singularities. Our work is comprised basically of three projects. In the first project we discuss a specific class of regular singular Sturm Liouville operators with matrix coefficients. Their zeta determinants were studied by K. Kirsten, P. Loya and J. Park on the basis of the Contour integral method, with general boundary conditions at the singularity and Dirichlet boundary conditions at the regular boundary. Our main result in the first project is the explicit verification that the Contour integral method indeed applies in the regular singular setup, and the generalization of the zeta determinant computations by Kirsten, Loya and Park to generalized Neumann boundary conditions at the regular boundary. Moreover we apply our results to Laplacians on a bounded generalized cone with relative boundary conditions. In the second project we derive a new formula for analytic torsion of a bounded generalized cone, generalizing the computational methods of M. Spreafico and using the symmetry in the de Rham complex, as established by M. Lesch. We evaluate our result in lower dimensions and further provide a separate computation of analytic torsion of a bounded generalized cone over S1, since the standard cone over the sphere is simply a flat disc. Finally, in the third project we discuss the refined analytic torsion, introduced by M. Braverman and T. Kappeler as a canonical refinement of analytic torsion on closed manifolds. Unfortunately there seems to be no canonical way to extend their construction to compact manifolds with boundary. We propose a different refinement of analytic torsion, similar to Braverman and Kappeler, which does apply to compact manifolds with and without boundary. We establish a gluing formula for our construction, which in fact can also be viewed as a gluing law for the original definition of refined analytic torsion by Braverman and Kappeler.</p

Discrete Bakry-\'Emery curvature tensors and matrices of connection graphs

Connection graphs are natural extensions of Harary's signed graphs. The Bakry-\'Emery curvature of connection graphs has been introduced by Liu, M\"unch and Peyerimhoff in order to establish Buser type eigenvalue estimates for connection Laplacians. In this paper, we reformulate the Bakry-\'Emery curvature of a vertex in a connection graph in terms of the smallest eigenvalue of a family of unitarily equivalent curvature matrices. We further interpret this family of curvature matrices as the matrix representations of a new defined curvature tensor with respect to different orthonormal basis of the tangent space at a vertex. This is a strong extension of previous works of Cushing-Kamtue-Liu-Peyerimhoff and Siconolfi on curvature matrices of graphs. Moreover, we study the Bakry-\'Emery curvature of Cartesian products of connection graphs, strengthening the previous result of Liu, M\"unch and Peyerimhoff. While results of a vertex with locally balanced structure cover previous works, various interesting phenomena of locally unbalanced connection structure have been clarified.Comment: 57 pages,14 figures. All comments are welcome