Mathematical Physics

We propose the systematic construction of classical and quantum two dimensional space-time lattices primarily based on algebraic considerations, i.e. on the existence of associated r-matrices and underlying spatial and temporal classical and quantum algebras. This is a novel construction that leads to the derivation of fully discrete integrable systems governed by sets of consistent integrable non-linear space-time difference equations. To illustrate the proposed methodology, we derive two versions of the fully discrete non-linear Schrodinger type system. The first one is based on the existence of a rational r-matrix, whereas the second one is the fully discrete Ablowitz-Ladik model and is associated to a trigonometric r-matrix. The Darboux-dressing method is also applied for the first discretization scheme, mostly as a consistency check, and solitonic as well as general solutions, in terms of solutions of the fully discrete heat equation, are also derived. The quantization of the fully discrete systems is then quite natural in this context and the two dimensional quantum lattice is thus also examined.

We initiate the study of an algebra of symmetries for the 3D Dirac-Dunkl operator associated with the Weyl group of the exceptional root system G 2 . For this symmetry algebra, we give both an abstract definition and an explicit realisation. We then construct ladder operators, using an intermediate result we prove for the Dirac-Dunkl symmetry algebra associated with arbitrary finite reflection group acting on a three-dimensional space.

In two phase materials, each phase having a non-local response in time, it has been found that for some driving fields the response somehow untangles at specific times, and allows one to directly infer useful information about the geometry of the material, such as the volume fractions of the phases. Motivated by this, and to obtain an algorithm for designing appropriate driving fields, we find approximate, measure independent, linear relations between the values that Markov functions take at a given set of possibly complex points, not belonging to the interval [-1,1] where the measure is supported. The problem is reduced to simply one of polynomial approximation of a given function on the interval [-1,1] and to simplify the analysis Chebyshev approximation is used. This allows one to obtain explicit estimates of the error of the approximation, in terms of the number of points and the minimum distance of the points to the interval [-1,1]. Assuming this minimum distance is bounded below by a number greater than 1/2, the error converges exponentially to zero as the number of points is increased. Approximate linear relations are also obtained that incorporate a set of moments of the measure. In the context of the motivating problem, the analysis also yields bounds on the response at any particular time for any driving field, and allows one to estimate the response at a given frequency using an appropriately designed driving field that effectively is turned on only for a fixed interval of time. The approximation extends directly to Markov-type functions with a positive semidefinite operator valued measure, and this has applications to determining the shape of an inclusion in a body from boundary flux measurements at a specific time, when the time-dependent boundary potentials are suitably tailored.

We define an index for 2d G -invariant invertible states of bosonic lattice systems in the thermodynamic limit for a finite symmetry group G . We show that this index is an invariant of SPT phase.

This work is devoted to review the modern geometric description of the Lagrangian and Hamiltonian formalisms of the Hamilton--Jacobi theory. The relation with the "classical" Hamiltonian approach using canonical transformations is also analyzed. Furthermore, a more general framework for the theory is also briefly explained. It is also shown how, from this generic framework, the Lagrangian and Hamiltonian cases of the theory for dynamical systems are recovered, and how the model can be extended to other types of physical systems, such as higher-order dynamical systems and (first-order) classical field theories in their multisymplectic formulation.

We present a Helmholtz Decomposition for many n-dimensional, continuously differentiable vector fields on unbounded domains that do not decay at infinity. Existing methods are restricted to fields not growing faster than polynomially and require solving n-dimensional volume integrals over unbounded domains. With our method only one-dimensional integrals have to be solved to derive gradient and rotation potentials. Analytical solutions are obtained for smooth vector fields f(x) whose components are separable into a product of two functions: f k (x)= u k ( x k )??v k ( x ?�k ) , where u k ( x k ) depends only on x k and v k ( x ?�k ) depends not on x k . Additionally, an integer λ k must exist such that the 2 λ k -th integral of one of the functions times the λ k -th power of the Laplacian applied to the other function yields a product that is a multiple of the original product. A similar condition is well-known from repeated partial integration, where the calculation can be finalized if the shifting of derivatives yields a multiple of the original integrand. Also linear combinations of such vector fields can be decomposed. These conditions include periodic and exponential functions, combinations of polynomials with arbitrary integrable functions, their products and linear combinations, and examples such as Lorenz or Rössler attractor.

Wave fields obeying the 2D Helmholtz equation on branched surfaces (Sommerfeld surfaces) are studied. Such surfaces appear naturally as a result of applying the reflection method to diffraction problems with straight scatterers bearing ideal boundary conditions. This is for example the case for the classical canonical problems of diffraction by a half-line or a segment. In the present work, it is shown that such wave fields admit an analytical continuation into the domain of two complex coordinates. The branch sets of such continuation are given and studied in detail. For a generic scattering problem, it is shown that the set of all branches of the multi-valued analytical continuation of the field has a finite basis. Each basis function is expressed explicitly as a Green's integral along so-called double-eight contours. The finite basis property is important in the context of coordinate equations, introduced and utilised by the authors previously, as illustrated in this article for the particular case of diffraction by a segment.

We give a new criterion for a classical gas with a repulsive pair potential to exhibit uniqueness of the infinite volume Gibbs measure and analyticity of the pressure. Our improvement on the bound for analyticity is by a factor e 2 over the classical cluster expansion approach and a factor e over the known limit of cluster expansion convergence. The criterion is based on a contractive property of a recursive computation of the density of a point process. The key ingredients in our proofs include an integral identity for the density of a Gibbs point process and an adaptation of the algorithmic correlation decay method from theoretical computer science. We also deduce from our results an improved bound for analyticity of the pressure as a function of the density.

We present a new type of approximation of a second-order elliptic operator in a planar domain with a point interaction. It is of a geometric nature, the approximating family consists of operators with the same symbol and regular coefficients on the domain with a small hole. At the boundary of it Robin condition is imposed with the coefficient which depends on the linear size of a hole. We show that as the hole shrinks to a point and the parameter in the boundary condition is scaled in a suitable way, nonlinear and singular, the indicated family converges in the norm-resolvent sense to the operator with the point interaction. This resolvent convergence is established with respect to several operator norms and order-sharp estimates of the convergence rates are provided.

In this paper, we focus on a class of quantum channels which are covariant for symmetries from free orthogonal quantum groups O + N . These quantum channels are called O + N -Temperley-Lieb channels, and their information-theoretic properties such as Holevo information and coherent information were analyzed in [BCLY20], but their additivity questions remained open. The main result of this paper is to approximate O + N -Temperley-Lieb quantum channels by much simpler ones in terms Bures distance. As applications, we study strong additivity questions for O + N -Temperley-Lieb quantum channels, and their classical capacity, private classical capacity and quantum capacity in the asymptotic regime N?��? .