Mathematical Physics

We study ground state properties of the Kondo lattice model with an electron-phonon interaction. The ground state is proved to be unique; in addition, the total spin of the ground state is determined according to the lattice structure. To prove the assertions, an extension of the method of spin reflection positivity is given in terms of order preserving operator inequalities.

We use classical Jacobi polynomials to identify the equilibrium configurations of charged particles confined to the unit circle. Our main result unifies two theorems from a 1986 paper of Forrester and Rogers.

We revisit an algorithm constructing elliptic tori, that was originally designed for applications to planetary hamiltonian systems. The scheme is adapted to properly work with models of chains of N+1 particles interacting via anharmonic potentials, thus covering also the case of FPU chains. After having preliminarily settled the Hamiltonian in a suitable way, we perform a sequence of canonical transformations removing the undesired perturbative terms by an iterative procedure. This is done by using the Lie series approach, that is explicitly implemented in a programming code with the help of a software package, which is especially designed for computer algebra manipulations. In the cases of FPU chains with N=4,8 , we successfully apply our new algorithm to the construction of elliptic tori for wide sets of the parameter ruling the size of the perturbation, i.e., the total energy of the system. Moreover, we explore the stability regions surrounding 1D elliptic tori. We compare our semi-analytical results with those provided by numerical explorations of the FPU-model dynamics, where the latter ones are obtained by using techniques based on the so called frequency analysis. We find that our procedure works up to values of the total energy that are of the same order of magnitude with respect to the maximal ones, for which elliptic tori are detected by numerical methods.

We study emergent behaviors of Cucker-Smale(CS) flocks on the hyperboloid H d in any dimensions. In a recent work \cite{H-H-K-K-M}, a first-order aggregation model on the hyperboloid was proposed and its emergent dynamics was analyzed in terms of initial configuration and system parameters. In this paper, we are interested in the second-order modeling of Cucker-Smale flocks on the hyperboloid. For this, we derive our second-order model from the abstract CS model on complete and smooth Riemannian manifolds by explicitly calculating the geodesic and parallel transport. Velocity alignment has been shown by combining general {velocity alignment estimates} for the abstract CS model on manifolds and verifications of a priori estimate of second derivative of energy functional. For the two-dimensional case H 2 , similar to the recent result in \cite{A-H-S}, asymptotic flocking admits only two types of asymptotic scenarios, either convergence to a rest state or a state lying on the same plane (coplanar state). We also provide several numerical simulations to illustrate an aforementioned dichotomy on the asymptotic dynamics of the hyperboloid CS model on H 2 .

We study emergent behaviors of the Lohe hermitian sphere(LHS) model with a time-delay for a homogeneous ensemble. The LHS model is a complex counterpart of the Lohe sphere(LS) aggregation model on the unit sphere in Euclidean space, and it describes the aggregation of particles on the unit hermitian sphere in C d with d?? , Recently it has been introduced by two authors of this work as a special case of the Lohe tensor model [23]. When the coupling gain pair satisfies a specific linear relation, namely the Stuart-Landau(SL) coupling gain pair, it can be embedded into the LS model on R 2d . In this work, we show that if the coupling gain pair is close to the SL coupling pair case, the dynamics of the LHS model exhibits an emergent aggregate phenomenon via the interplay between time-delayed interactions and nonlinear coupling between states. For this, we present several frameworks for complete aggregation and practical aggregation in terms of initial data and system parameters using the Lyapunov functional approach.

We study emergent dynamics of the Lohe hermitian sphere(LHS) model which can be derived from the Lohe tensor model \cite{H-P2} as a complex counterpart of the Lohe sphere(LS) model. The Lohe hermitian sphere model describes aggregate dynamics of point particles on the hermitian sphere $\bbh\bbs^d$ lying in C d+1 , and the coupling terms in the LHS model consist of two coupling terms. For identical ensemble with the same free flow dynamics, we provide a sufficient framework leading to the complete aggregation in which all point particles form a giant one-point cluster asymptotically. In contrast, for non-identical ensemble, we also provide a sufficient framework for the practical aggregation. Our sufficient framework is formulated in terms of coupling strengths and initial data. We also provide several numerical examples and compare them with our analytical results.

Projecting a quantum theory onto the Hilbert subspace of states with energies below a cutoff E ¯ ¯ ¯ ¯ may lead to an effective theory with modified observables, including a noncommutative space(time). Adding a confining potential well V with a very sharp minimum on a submanifold N of the original space(time) M may induce a dimensional reduction to a noncommutative quantum theory on N . Here in particular we briefly report on our application of this procedure to spheres S d ⊂ R D of radius r=1 ( D=d+1>1 ): making E ¯ ¯ ¯ ¯ and the depth of the well depend on (and diverge with) Λ∈N we obtain new fuzzy spheres S d Λ covariant under the {\it full} orthogonal groups O(D) ; the commutators of the coordinates depend only on the angular momentum, as in Snyder noncommutative spaces. Focusing on d=1,2 , we also discuss uncertainty relations, localization of states, diagonalization of the space coordinates and construction of coherent states. As Λ→∞ the Hilbert space dimension diverges, S d Λ → S d , and we recover ordinary quantum mechanics on S d . These models might be suggestive for effective models in quantum field theory, quantum gravity or condensed matter physics.

Free Fermions on vertices of distance-regular graphs are considered. Bipartition are defined by taking as one part all vertices at a given distance from a reference vertex. The ground state is constructed by filling all states below a certain energy. Borrowing concepts from time and band limiting problems, algebraic Heun operators and Terwilliger algebras, it is shown how to obtain, quite generally, a block tridiagonal matrix that commutes with the entanglement Hamiltonian. The case of the Hadamard graphs is studied in details within that framework and the existence of the commuting matrix is shown to allow for an analytic diagonalization of the restricted two-point correlation matrix and hence for an explicit determination of the entanglement entropy.

We study quantum computation relations on unital finite-dimensional CAR C ∗ -algebras. We prove an entropy power inequality (EPI) in a fermionic setting, which presumably will permit understanding the capacities in fermionic linear optics. Similar relations to the bosonic case are shown, and alternative proofs of known facts are given. Clifford algebras and the Grassmann representation can thus be used to obtain mathematical results regarding coherent fermion states.

We prove that the energy of any eigenvector of a sum of several independent large Wigner matrices is equally distributed among these matrices with very high precision. This shows a particularly strong microcanonical form of the equipartition principle for quantum systems whose components are modelled by Wigner matrices.