Mathematical Physics

While dealing with a Hamiltonian with continuous spectrum we use a tridiagonal method involving orthogonal polynomials to construct a set of coherent states obeying a Glauber-type condition. We perform a Bayesian decomposition of the weight function of the orthogonality measure to show that the obtained coherent states can be recast in the Gazeau-Klauder approach. The Hamiltonian of the ℓ -wave free particle is treated as an example to illustrate the method.

We present a second-order extension of the first-order Lohe hermitian sphere(LHS) model and study its emergent asymptotic dynamics. Our proposed model incorporates an inertial effect as a second-order extension. The inertia term can generate an oscillatory behavior of particle trajectory in a small time interval(initial layer) which causes a technical difficulty for the application of monotonicity-based arguments. For emergent estimates, we employ two-point correlation function which is defined as an inner product between positions of particles. For a homogeneous ensemble with the same frequency matrix, we provide two sufficient frameworks in terms of system parameters and initial data to show that two-point correlation functions tend to the unity which is exactly the same as the complete aggregation. In contrast, for a heterogeneous ensemble with distinct frequency matrices, we provide a sufficient framework in terms of system parameters and initial data, which makes two-point correlation functions close to unity by increasing the principal coupling strength.

The Darwin-Fowler method in combination with the steepest descent approach is a common tool in the asymptotic description of many models arising from statistical physics. In this work, we focus rather on the non-asymptotic behavior of the Darwin-Fowler procedure. By using a combinatorial approach based on Bell polynomials, we solve it exactly. Due to that approach, we also show relationships of typical models with combinatorial Lah and Stirling numbers.

We present a proof that the operator norm of the commutator of certain spectral projections associated with spin operators converges to 1 2 in the semiclassical limit. The ranges of the projections are spanned by all eigenvectors corresponding to positive eigenvalues. The proof involves the theory of Hankel operators on the Hardy space. A discussion of several analogous results is also included, with an emphasis on the case of finite Heisenberg groups.

We provide a complete classification and an explicit representation of rational solutions to the fourth Painlevé equation PIV and its higher order generalizations known as the A 2n -Painlevé or Noumi-Yamada systems. The construction of solutions makes use of the theory of cyclic dressing chains of Schrödinger operators. Studying the local expansions of the solutions around their singularities we find that some coefficients in their Laurent expansion must vanish, which express precisely the conditions of trivial monodromy of the associated potentials. The characterization of trivial monodromy potentials with quadratic growth implies that all rational solutions can be expressed as Wronskian determinants of suitably chosen sequences of Hermite polynomials. The main classification result states that every rational solution to the A 2n -Painlevé system corresponds to a cycle of Maya diagrams, which can be indexed by an oddly coloured integer sequence. Finally, we establish the link with the standard approach to building rational solutions, based on applying Bäcklund transformations on seed solutions, by providing a representation for the symmetry group action on coloured sequences and Maya cycles.

This work develops a new method, based on the use of Gustafson's integrals and on the evaluation of singular integrals, allowing one to establish the unitarity of the separation of variables transform for infinite-dimensional representations of rank one quantum integrable models. We examine in detail the case of the SL(2,R) spin chains.

Because of the isomorphism Cℓ 1,3 (C)≅ Cℓ 2,3 (R) , it is possible to complexify the spacetime Clifford algebra Cℓ 1,3 (R) by adding one additional timelike dimension to the Minkowski spacetime. In a recent work we showed how this treatment provide a particular interpretation of Dirac particles and antiparticles in terms of the new temporal dimension. In this article we thoroughly study the structure of the real Clifford algebra Cℓ 2,3 (R) paying special attention to the isomorphism Cℓ 1,3 (C)≅ Cℓ 2,3 (R) and the embedding Cℓ 1,3 (R)⊆ Cℓ 2,3 (R) . On the first half of this article we analyze the Pin and Spin groups and construct an injective mapping Pin(1,3)↪Spin(2,3) , obtaining in particular elements in Spin(2,3) that represent parity and time reversal. On the second half of this paper we study the spinor space of the algebra and prove that the usual structure of complex spinors in Cℓ 1,3 (C) is reproduced by the Clifford conjugation inner product for real spinors in Cℓ 2,3 (R) .

Mathieu functions of period π or 2π , also called elliptic cylinder functions, were introduced in 1868 by Émile Mathieu together with so-called modified Mathieu functions, in order to help understand the vibrations of an elastic membrane set in a fixed elliptical hoop. These functions still occur frequently in applications today: our interest, for instance, was stimulated by a problem of pulsatile blood flow in a blood vessel compressed into an elliptical cross-section. This paper surveys and recapitulates the historical development of the theory and methods of computation for Mathieu functions and modified Mathieu functions and identifies some gaps in current software capability, particularly to do with double eigenvalues of the Mathieu equation. We demonstrate how to compute Puiseux expansions of the Mathieu eigenvalues about such double eigenvalues, and give methods to compute the generalized eigenfunctions that arise there. In examining Mathieu's original contribution, we bring out that his use of anti-secularity predates that of Lindstedt. For interest, we also provide short biographies of some of the major mathematical researchers involved in the history of the Mathieu functions: Émile Mathieu, Sir Edmund Whittaker, Edward Ince, and Gertrude Blanch.

Birkhoff normal forms are commonly used in order to ensure the so called "effective stability" in the neighborhood of elliptic equilibrium points for Hamiltonian systems. From a theoretical point of view, this means that the eventual diffusion can be bounded for time intervals that are exponentially large with respect to the inverse of the distance of the initial conditions from such equilibrium points. Here, we focus on an approach that is suitable for practical applications: we extend a rather classical scheme of estimates for both the Birkhoff normal forms to any finite order and their remainders. This is made for providing explicit lower bounds of the stability time (that are valid for initial conditions in a fixed open ball), by using a fully rigorous computer-assisted procedure. We apply our approach in two simple contexts that are widely studied in Celestial Mechanics: the Hénon-Heiles model and the Circular Planar Restricted Three-Body Problem. In the latter case, we adapt our scheme of estimates for covering also the case of resonant Birkhoff normal forms and, in some concrete models about the motion of the Trojan asteroids, we show that it can be more advantageous with respect to the usual non-resonant ones.

We formulate the abelian six-dimensional N=(2,0) theory perturbatively, in a generalization of the Batalin-Vilkovisky formalism. Using this description, we compute the holomorphic and non-minimal twists at the perturbative level. This calculation hinges on the existence of an L ∞ action of the supersymmetry algebra on the abelian tensor multiplet, which we describe in detail. Our formulation appears naturally in the pure spinor superfield formalism, but understanding it requires developing a presymplectic generalization of the BV formalism, inspired by Dirac's theory of constraints. The holomorphic twist consists of symplectic-valued holomorphic bosons from the N=(1,0) hypermultiplet, together with a degenerate holomorphic theory of holomorphic coclosed one-forms from the N=(1,0) tensor multiplet, which can be interpreted as representing the intermediate Jacobian. We check that our formulation and our results match with known ones under various dimensional reductions, as well as comparing the holomorphic twist to Kodaira-Spencer theory. Matching our formalism to five-dimensional Yang-Mills theory after reduction leads to some issues related to electric-magnetic duality; we offer some speculation on a nonperturbative resolution.