The trace formula for quantum graphs with general self adjoint boundary conditions

We consider compact metric graphs with an arbitrary self adjoint realisation of the differential Laplacian. After discussing spectral properties of Laplacians, we prove several versions of trace formulae, relating Laplace spectra to sums over periodic orbits on the graph. This includes trace formulae with, respectively, absolutely and conditionally convergent periodic orbit sums; the convergence depending on properties of the test functions used. We also prove a trace formula for the heat kernel and provide small-$t$ asymptotics for the trace of the heat kernel.Comment: 36 pages; improved, final version to appear in Ann. H. Poincar

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

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

Semiclassical Approach to Chaotic Quantum Transport

We describe a semiclassical method to calculate universal transport properties of chaotic cavities. While the energy-averaged conductance turns out governed by pairs of entrance-to-exit trajectories, the conductance variance, shot noise and other related quantities require trajectory quadruplets; simple diagrammatic rules allow to find the contributions of these pairs and quadruplets. Both pure symmetry classes and the crossover due to an external magnetic field are considered.Comment: 33 pages, 11 figures (appendices B-D not included in journal version

Zero Modes of Quantum Graph Laplacians and an Index Theorem

We study zero modes of Laplacians on compact and non-compact metric graphs with general self-adjoint vertex conditions. In the first part of the paper the number of zero modes is expressed in terms of the trace of a unitary matrix $\mathfrak{S}$ that encodes the vertex conditions imposed on functions in the domain of the Laplacian. In the second part a Dirac operator is defined whose square is related to the Laplacian. In order to accommodate Laplacians with negative eigenvalues it is necessary to define the Dirac operator on a suitable Kre\u{\i}n space. We demonstrate that an arbitrary, self-adjoint quantum graph Laplacian admits a factorisation into momentum-like operators in a Kre\u{\i}n-space setting. As a consequence, we establish an index theorem for the associated Dirac operator and prove that the zero-mode contribution in the trace formula for the Laplacian can be expressed in terms of the index of the Dirac operator

COVID-19 pandemic: Lessons for spatial development

Background and aims of this position paper: Since the COVID-19 pandemic began, it has become ever clearer that it poses an enormous challenge for society. The lockdown imposed on large parts of public life, which hit all social groups and institutions relatively abruptly with a wide range of impacts, as well as the measures adopted subsequently have resulted in radical changes in our living conditions. In some cases, the crisis has acted as an accelerator of trends affecting processes that were already ongoing: the digitalisation of communications and educational processes, the growth in working from home and mobile working arrangements, the expansion of online retail, changes in travel behaviour (in favour of cars and bicycles), and the establishment of regional service networks. At the same time, there has been a braking effect on sectors such as long-distance travel, global trade, trade fairs and cultural events, as well as on progress towards gender equality in the division of labour for household responsibilities and childcare. Socio-spatial, infrastructural, economic and ecological effects are becoming increasingly apparent. For those involved in spatial development and spatial planning, urgent questions arise not only about the weaknesses that have become apparent in our spatial uses in terms of infrastructure and public service provision, the economy and ecology, and in our ways of life in terms of housing and the supply of goods and services, but also about what opportunities have emerged for sustainable and self-determined lifestyles. What conclusions for anticipatory and preventive planning can be drawn from these (provisional) findings? Using a critical, multidisciplinary and integrative examination of the spatially-relevant effects of the COVID-19 pandemic, this paper establishes connections between the crisis management of today and crisis preparedness concepts for potential future pandemics. Building on that, it proposes corresponding recommended actions. These actions relate not only to insights for medium-term space-related crisis management but also to conclusions on long-term strategic challenges for spatial development in view of pandemics to be expected in the future. For this position paper, the 'Pandemic and Spatial Development' Ad hoc Working Group at the ARL - Academy for Territorial Development in the Leibniz Association has compiled interdisciplinary perspectives from spatial development and spatial planning, public health services, epidemiology, economics and social sciences, and has condensed them into transdisciplinary recommendations for action. These recommendations are directed at the various action levels for spatial development and spatial planning

GSE statistics without spin

Energy levels statistics following the Gaussian Symplectic Ensemble (GSE) of Random Matrix Theory have been predicted theoretically and observed numerically in numerous quantum chaotic systems. However in all these systems there has been one unifying feature: the combination of half-integer spin and time-reversal invariance. Here we provide an alternative mechanism for obtaining GSE statistics that is based on geometric symmetries of a quantum system which alleviates the need for spin. As an example, we construct a quantum graph with a particular discrete symmetry given by the quaternion group Q8. GSE statistics is then observed within one of its subspectra.Comment: 5 pages, 6 figure