Mathematical Physics

We look at connection Laplacians L,g defined by a field h:G to K, where G is a finite set of sets and K is a normed division ring which does not need to be commutative, nor associative but has a conjugation leading to the norm as the square root of h^* h. The target space K can be a normed real division algebra like the quaternions or an algebraic number field like a quadratic field. For parts of the results we can even assume K to be a Banach algebra like an operator algebra on a Hilbert space. The K-valued function h on G then defines connection matrices L,g in which the entries are in K. We show that the Dieudonne determinants of L and g are both equal to the abelianization of the product of all the field values on G. If G is a simplicial complex and h takes values in the units U of K, then g^* is the inverse of L and the sum of the energy values is equal to the sum of the Green function entries g(x,y). If K is the field C of complex numbers, we can study the spectrum of L(G,h) in dependence of the field h. The set of matrices with simple spectrum defines a |G|-dimensional non-compact Kaehler manifold that is disconnected in general and for which we can compute the fundamental group of each connected component.

We further develop the method of dressing the boundary for the focusing nonlinear Schrödinger equation (NLS) on the half-line to include the new boundary condition presented by Zambon. Additionally, the foundation to compare the solutions to the ones produced by the mirror-image technique is laid by explicitly computing the change of scattering data under the Darboux transformation. In particular, the developed method is applied to insert pure soliton solutions.

We present a new dynamical proof of the Thouless-Anderson-Palmer (TAP) equations for the classical Sherrington-Kirkpatrick spin glass at sufficiently high temperature. In our derivation, the TAP equations are a simple consequence of the decay of the two point correlation functions. The methods can also be used to establish the decay of higher order correlation functions. We illustrate this by proving a suitable decay bound on the three point functions from which we derive an analogue of the TAP equations for the two point functions.

The framework of dynamical C*-algebras for scalar fields in Minkowski space, based on local scattering operators, is extended to theories with locally perturbed kinetic terms. These terms encode information about the underlying spacetime metric, so the causality relations between the scattering operators have to be adjusted accordingly. It is shown that the extended algebra describes scalar quantum fields, propagating in locally deformed Minkowski spaces. Concrete representations of the abstract scattering operators, inducing this motion, are known to exist on Fock space. The proof that these representers also satisfy the generalized causality relations requires, however, novel arguments of a cohomological nature. They imply that Fock space representations of the extended dynamical C*-algebra exist, involving linear as well as kinetic and pointlike quadratic perturbations of the field.

The success of Gross--Pitevskii and Bogoliubov theories in the description of large systems of interacting bosons led to a substantial effort into rigorously deriving these effective theories. In this work we shall review the related existing literature in the context of dynamics of large bosonic systems.

The first work of Dyson relating to random matrix theory, "The dynamics of a disordered linear chain", is reviewed. Contained in this work is an exact solution of a so-called Type I chain in the case of the disorder variables being given by a gamma distribution. The exact solution exhibits a singularity in the density of states about the origin, which has since been shown to be universal for one-dimensional tight binding models with off diagonal disorder. We discuss this context and also point out some universal features of the weak disorder expansion of the exact solution near the band edge. Further, a link between the exact solution, and a tridiagonal formalism of anti symmetric Gaussian β -ensembles with β proportional to 1/N , is made.

Paralogarithms constitute a family of special functions, which are some generalizations of hyperlogarithms. They have been introduced by Jean Ecalle in the context of the classification of complex analytic dynamical systems with irregular singularities, to solve the so-called "synthesis problem" in an effective and very general way. We describe the formalism of resurgence monomials, introduce the paralogarithmic family and present the effective synthesis with paralogarithmic monomials, of analytic vector fields having a saddle-node singularity, following Ecalle.

This paper is concerned with the explicit computation of the limiting distribution function of the largest real eigenvalue in the real Ginibre ensemble when each real eigenvalue has been removed independently with constant likelihood. We show that the recently discovered integrable structures in \cite{BB} generalize from the real Ginibre ensemble to its thinned equivalent. Concretely, we express the aforementioned limiting distribution function as a convex combination of two simple Fredholm determinants and connect the same function to the inverse scattering theory of the Zakharov-Shabat system. As corollaries, we provide a Zakharov-Shabat evaluation of the ensemble's real eigenvalue generating function and obtain precise control over the limiting distribution function's tails. The latter part includes the explicit computation of the usually difficult constant factors.

We consider interaction energies E f [L] between a point O∈ R d , d≥2 , and a lattice L containing O , where the interaction potential f is assumed to be radially symmetric and decaying sufficiently fast at infinity. We investigate the conservation of optimality results for E f when integer sublattices kL are removed (periodic arrays of vacancies) or substituted (periodic arrays of substitutional defects). We consider separately the non-shifted ( O∈kL ) and shifted ( O∉kL ) cases and we derive several general conditions ensuring the (non-)optimality of a universal optimizer among lattices for the new energy including defects. Furthermore, in the case of inverse power laws and Lennard-Jones type potentials, we give necessary and sufficient conditions on non-shifted periodic vacancies or substitutional defects for the conservation of minimality results at fixed density. Different examples of applications are presented, including optimality results for the Kagome lattice and energy comparisons of certain ionic-like structures.

Built upon the hypoelliptic analysis of the effective Mori-Zwanzig (EMZ) equation for observables of stochastic dynamical systems, we show that the obtained semigroup estimates for the EMZ equation can be used to drive prior estimates of the observable statistics for system in the equilibrium and non-equilibrium state. In addition, we introduce both first-principle and data-driven methods to approximate the EMZ memory kernel, and prove the convergence of the data-driven parametrization schemes using the regularity estimate of the memory kernel. The analysis results are validated numerically via the Monte-Carlo simulation of the Langevin dynamics for a Fermi-Pasta-Ulam chain model. With the same example, we also show the effectiveness of the proposed memory kernel approximation methods.