Mathematical Physics, Analysis and Geometry最新文献

We prove two conjectures on the Korteweg-de Vries (KdV) and modified KdV (mKdV) hierarchies and Schur Q-functions presented by Yamada. The first one is that the functions defined by Sato and Mori in 1980 coincide with Schur Q-functions indexed by even or odd strict partitions. Mizukawa, Nakajima, and Yamada gave an expression for this function using symmetric functions and Littlewood-Richardson coefficients. We prove that this expression coincides with the Schur Q-function by using the formula of Lascoux, Leclerc, and Thibon. The second one is that Schur Q-functions indexed by strict partitions which have odd parts form a basis for the space of Hirota polynomials of the KdV hierarchy, and that Schur Q-functions indexed by strict partitions which have even parts form a basis for the space of Hirota polynomials of the mKdV hierarchy. This conjecture is verified by rewriting the generating series of the KdV and mKdV hierarchies using the techniques of symmetric functions.

We study the real hyperelliptic solutions of the focusing modified KdV (MKdV) equation of genus three. Since the complex hyperelliptic solutions of the focusing MKdV equation over ({{mathbb {C}}}) are associated with the real gauged MKdV equation, we present a novel construction related to the real hyperelliptic solutions of the gauged MKdV equation. When the gauge field is constant, it can be regarded as the real solution of the focusing MKdV equation, and thus we also discuss the behavior of the gauge field numerically.

We show that for every complex simple Lie algebra (mathfrak {g}), the equations of Schubert divisors on the flag variety (G/B^-) give a complete integrable system of the minimal nilpotent orbit (mathcal {O}_{min }). The approach is motivated by the integrable system on Coulomb branch as reported by Braverman (arXiv preprint arXiv:1604.03625, 2016).We give explicit computations of these Hamiltonian functions, using Chevalley basis and a so-called Heisenberg algebra basis. For classical Lie algebras we rediscover the lower order terms of the celebrated Gelfand-Zeitlin system. For exceptional types we computed the number of Hamiltonian functions associated to each vertex of Dynkin diagram. They should be regarded as analogs of Gelfand-Zeitlin functions on exceptional type Lie algebras.

We derive a two-term asymptotic expansion for the exchange energy of the free electron gas on strictly tessellating polytopes and fundamental domains of lattices in the thermodynamic limit. This expansion comprises a bulk (volume-dependent) term, the celebrated Dirac exchange, and a novel surface correction stemming from a boundary layer and finite-size effects. Furthermore, we derive analogous two-term asymptotic expansions for semi-local density functionals. By matching the coefficients of these asymptotic expansions, we obtain an integral constraint for semi-local approximations of the exchange energy used in density functional theory.

On a d-dimensional Riemannian, spin manifold (M, g) we consider non-linear, stochastic partial differential equations for spinor fields, driven by a Dirac operator and coupled to an additive Gaussian, vector-valued white noise. We extend to the case in hand a procedure, introduced in Dappiaggi et al (Commun Contemp Math 27(07):2150075, 2022), for the scalar counterpart, which allows to compute at a perturbative level the expectation value of the solutions as well as the associated correlation functions accounting intrinsically for the underlying renormalization freedoms. This framework relies strongly on tools proper of microlocal analysis and it is inspired by the algebraic approach to quantum field theory. As a concrete example we apply it to a stochastic version of the Thirring model proving in particular that it lies in the subcritical regime if (dle 2).

We study the limiting spectral distribution of quantum channels whose Kraus operators sampled as ( ntimes n) random Hermitian matrices satisfying certain assumptions. We show that when the Kraus rank goes to infinity with n, the limiting spectral distribution (suitably rescaled) of the corresponding quantum channel coincides with the semi-circle distribution. When the Kraus rank is fixed, the limiting spectral distribution is no longer the semi-circle distribution. It corresponds to an explicit law, which can also be described using tools from free probability.

In this paper, studying the inverse problem, we establish a curvature compatibility condition on a spherically symmetric Finsler metric. As an application, we characterize the spherically symmetric metrics of scalar curvature. We construct a Berwald frame for a spherically symmetric Finsler surface and calculate some associated geometric objects. Several examples are provided and discussed. Finally, we give a note on a certain general ((alpha ,beta ))-metric which appears in the literature.

In this paper we study the foundations of the algebraic treatment of classical and quantum field theories for Dirac fermions under external backgrounds following the initial contributions already present in various places in the literature. The treatment is restricted to contractible spacetimes of globally hyperbolic nature in dimensions (dge 4) and to external fields modelled with trivial principal bundles. In particular, we construct the classical Møller maps intertwining the configuration spaces of charged and uncharged fermions, and we show some of its properties in the case of a U(1) gauge charge. In the last part, as a first step towards a quantization of the theory, we explore the combination of the classical Møller maps with Hadamard bidistributions and prove that they are involutive isomorphisms (algebraically and topologically) between suitable (formal) algebras of functionals (observables) over the configuration spaces of charged and uncharged Dirac fields.

For a family of self-adjoint Dirac operators (-i c (alpha cdot nabla ) + frac{c^2}{2}) subject to generalized MIT bag boundary conditions on domains in (mathbb {R}^3), it is shown that the nonrelativistic limit in the norm resolvent sense is the Dirichlet Laplacian. This allows to transfer spectral geometry results for Dirichlet Laplacians to Dirac operators for large c.

We present a much shorter and streamlined proof of an improved version of the results previously given in [A. Posilicano: On the Self-Adjointness of (H+A^{*}+A). Math. Phys. Anal. Geom. 23 (2020)] concerning the self-adjoint realizations of formal QFT-like Hamiltonians of the kind (H+A^{*}+A), where H and A play the role of the free field Hamiltonian and of the annihilation operator respectively. We give explicit representations of the resolvent and of the self-adjointness domain; the consequent Kreĭn-type resolvent formula leads to a characterization of these self-adjoint realizations as limit (with respect to convergence in norm resolvent sense) of cutoff Hamiltonians of the kind (H+A^{*}_{n}+A_{n}-E_{n}), the bounded operator (E_{n}) playing the role of a renormalizing counter term. These abstract results apply to various concrete models in Quantum Field Theory.