Heat-kernel and Resolvent Asymptotics for Schrödinger Operators on Metric Graphs

We consider Schroedinger operators on compact and non-compact (finite) metric graphs. For such operators we analyse their spectra, prove that their resolvents can be represented as integral operators and introduce trace-class regularisations of the resolvents. Our main result is a complete asymptotic expansion of the trace of the (regularised) heat-semigroup generated by the Schroedinger operator. We also determine the leading coefficients in the expansion explicitly.Comment: This article has been accepted for publication in Applied Mathematics Research Express Published by Oxford University Pres

The Berry-Keating operator on a lattice

We construct and study a version of the Berry-Keating operator with a built-in truncation of the phase space, which we choose to be a two-dimensional torus. The operator is a Weyl quantisation of the classical Hamiltonian for an inverted harmonic oscillator, producing a difference operator on a finite, periodic lattice. We investigate the continuum and the infinite-volume limit of our model in conjunction with the semiclassical limit. Using semiclassical methods, we show that a specific combination of the limits leads to a logarithmic mean spectral density as it was anticipated by Berry and Keating

Defining the spectral position of a Neumann domain

A Laplacian eigenfunction on a two-dimensional Riemannian manifold provides a natural partition into Neumann domains (a.k.a. Morse-Smale complexes). This partition is generated by gradient flow lines of the eigenfunction -- these bound the so-called Neumann domains. We prove that the Neumann Laplacian $\Delta$ defined on a single Neumann domain is self-adjoint and possesses a purely discrete spectrum. In addition, we prove that the restriction of the eigenfunction to any one of its Neumann domains is an eigenfunction of $\Delta$. As a comparison, similar statements for a $nodal$ domain of an eigenfunction (with the Dirichlet Laplacian) are basic and well-known. The difficulty here is that the boundary of a Neumann domain may have cusps and cracks, and hence is not necessarily continuous, so standard results about Sobolev spaces are not available

The Berry-Keating operator on a lattice

We construct and study a version of the Berry-Keating operator with a built-in truncation of the phase space, which we choose to be a two-dimensional torus. The operator is a Weyl quantisation of the classical Hamiltonian for an inverted harmonic oscillator, producing a difference operator on a finite, periodic lattice. We investigate the continuum and the infinite-volume limit of our model in conjunction with the semiclassical limit. Using semiclassical methods, we show that a specific combination of the limits leads to a logarithmic mean spectral density as it was anticipated by Berry and Keating

Adiabatic quantum simulations with driven superconducting qubits

We propose a quantum simulator based on driven superconducting qubits where the interactions are generated parametrically by a polychromatic magnetic flux modulation of a tunable bus element. Using a time-dependent Schrieffer-Wolff transformation, we analytically derive a multi-qubit Hamiltonian which features independently tunable $XX$ and $YY$-type interactions as well as local bias fields over a large parameter range. We demonstrate the adiabatic simulation of the ground state of a hydrogen molecule using two superconducting qubits and one tunable bus element. The time required to reach chemical accuracy lies in the few microsecond range and therefore could be implemented on currently available superconducting circuits. Further applications of this technique may also be found in the simulation of interacting spin systems.Comment: 11 pages, 6 figure