179 research outputs found
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
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
. As a comparison, similar statements for a 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
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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 and -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
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