Letters in Mathematical Physics最新文献

In this paper, we investigate the Cauchy problem of the following complex cubic Camassa–Holm equation

where (b>0) is an arbitrary positive real constant. Long-time asymptotics of the equation is obtained through the (bar{partial })-steepest descent method. Firstly, based on the spectral analysis of the Lax pair and scattering matrix, the solution of the equation is able to be constructed via solving the corresponding Riemann–Hilbert problem. Then, we present different long-time asymptotic expansions of the solution u(y, t) in different space-time solitonic regions of (xi =y/t). The half-plane ({(y,t):-infty<y< infty , t > 0}) is divided into four asymptotic regions: (xi in (-infty ,-1)), (xi in (-1,0)), (xi in (0,frac{1}{8})) and (xi in (frac{1}{8},+infty )). When (xi ) falls in ((-infty ,-1)cup (frac{1}{8},+infty )), no stationary phase point of the phase function (theta (z)) exists on the jump profile in the space-time region. In this case, corresponding asymptotic approximations can be characterized with an (N(Lambda ))-solitons with diverse residual error order (O(t^{-1+2varepsilon })). There are four stationary phase points and eight stationary phase points on the jump curve as (xi in (-1,0)) and (xi in (0,frac{1}{8})), respectively. The corresponding asymptotic form is accompanied by a residual error order (O(t^{-frac{3}{4}})).

Quasi-periodic solutions of the KP hierarchy acted by vertex operators are studied. We show, with the aid of the Sato Grassmannian, that solutions thus constructed correspond to torsion free rank one sheaves on some singular algebraic curves whose normalizations are the non-singular curves corresponding to the seed quasi-periodic solutions. It means that the action of the vertex operator has an effect of creating singular points on an algebraic curve. We further check, by examples, that solutions obtained here can be considered as solitons on quasi-periodic backgrounds, where the soliton matrices are determined by parameters in the vertex operators.

We recast the physics discussions in the paper of Van den Bleeken (J High Energy Phys 2012(2):67, 2012) within the context of wall-crossing structure à la Kontsevich and Soibelman (Homol Mirror Symmetry Trop Geom, 2014. https://doi.org/10.1007/978-3-319-06514-4_6). In particular, we compare the Hesse flow given in Van den Bleeken (J High Energy Phys 2012(2):67, 2012) and the attractor flow on the base of the complex integrable system, and show that both can be used in the formalism of wall-crossing structure. We also propose the notions of dual Hesse flow and dual attractor flow, and show that under the rotation of the (mathbb {Z})-affine structure, the Hesse flow can be transformed into the dual attractor flow, while the attractor flow into the dual Hesse flow. This suggests its possible use in Mirror Symmetry.

Quantum field theories can be applied to compute various statistical properties of random tensors. In particular, signed distributions of tensor eigenvalues/vectors are the easiest, which can be computed as partition functions of four-fermi theories. Though signed distributions are different from genuine ones because of extra signs of weights, they are expected to coincide in vicinities of ends of distributions. In this paper, we perform a case study of the signed eigenvalue/vector distribution of the real symmetric order-three random tensor. The correct critical point and the correct end in the large N limit are obtained from the four-fermi theory, for which a method using the Schwinger-Dyson equation is very efficient. Since locations of ends are particularly important in applications, such as the largest eigenvalues and the best rank-one tensor approximations, signed distributions are the easiest and highly useful through the Schwinger-Dyson method.

In this paper, new self-adjoint realizations of the Dirac operator in dimension two and three are introduced. It is shown that they may be associated with the formal expression (mathcal {D}_0+|Fdelta _Sigma rangle langle Gdelta _Sigma |), where (mathcal {D}_0) is the free Dirac operator, F and G are matrix valued coefficients, and (delta _Sigma ) stands for the single layer distribution supported on a hypersurface (Sigma ), and that they can be understood as limits of the Dirac operators with scaled non-local potentials. Furthermore, their spectral properties are analysed.

We provide a general expression of the Haar measure—that is, the essentially unique translation-invariant measure—on a p-adic Lie group. We then argue that this measure can be regarded as the measure naturally induced by the invariant volume form on the group, as it happens for a standard Lie group over the reals. As an important application, we next consider the problem of determining the Haar measure on the p-adic special orthogonal groups in dimension two, three and four (for every prime number p). In particular, the Haar measure on (text {SO}(2,mathbb {Q}_p)) is obtained by a direct application of our general formula. As for (text {SO}(3,mathbb {Q}_p)) and (text {SO}(4,mathbb {Q}_p)), instead, we show that Haar integrals on these two groups can conveniently be lifted to Haar integrals on certain p-adic Lie groups from which the special orthogonal groups are obtained as quotients. This construction involves a suitable quaternion algebra over the field (mathbb {Q}_p) and is reminiscent of the quaternionic realization of the real rotation groups. Our results should pave the way to the development of harmonic analysis on the p-adic special orthogonal groups, with potential applications in p-adic quantum mechanics and in the recently proposed p-adic quantum information theory.

The concept of antidistinguishability of quantum states has been studied to investigate foundational questions in quantum mechanics. It is also called quantum state elimination, because the goal of such a protocol is to guess which state, among finitely many chosen at random, the system is not prepared in (that is, it can be thought of as the first step in a process of elimination). Antidistinguishability has been used to investigate the reality of quantum states, ruling out (psi )-epistemic ontological models of quantum mechanics (Pusey et al. in Nat Phys 8(6):475–478, 2012). Thus, due to the established importance of antidistinguishability in quantum mechanics, exploring it further is warranted. In this paper, we provide a comprehensive study of the optimal error exponent—the rate at which the optimal error probability vanishes to zero asymptotically—for classical and quantum antidistinguishability. We derive an exact expression for the optimal error exponent in the classical case and show that it is given by the multivariate classical Chernoff divergence. Our work thus provides this divergence with a meaningful operational interpretation as the optimal error exponent for antidistinguishing a set of probability measures. For the quantum case, we provide several bounds on the optimal error exponent: a lower bound given by the best pairwise Chernoff divergence of the states, a single-letter semi-definite programming upper bound, and lower and upper bounds in terms of minimal and maximal multivariate quantum Chernoff divergences. It remains an open problem to obtain an explicit expression for the optimal error exponent for quantum antidistinguishability.

Using Zhelobenko–Stern formulas for the action of the generators of orthogonal Lie algebra in corresponding Gelfand–Tsetlin basis, we derive Mellin–Barnes presentations for the wave functions of (B_n) Toda lattice. They are in accordance with Iorgov–Shadura formulas.

In this paper, we investigate the spectrum of a class of weighted Laplacians on Cayley graphs and determine under what conditions the corresponding eigenspaces are generically irreducible. Specifically, we analyze the spectrum on left-invariant Cayley graphs endowed with an invariant metric, and we give some criteria for generically irreducible eigenspaces. Additionally, we introduce an operator that is comparable to the Laplacian and show that the same criterion holds.

This paper is dedicated to extending the focusing/defocusing Calogero–Moser–Sutherland cubic derivative Schrödinger equations (CMSdNLS)

which were initially introduced in Matsuno (Phys Lett A 278(1–2):53–58, 2000; Inverse Probl 18:1101–1125, 2002; J Phys Soc Jpn 71(6):1415–1418, 2002; Inverse Prob 20(2):437–445, 2004), Abanov et al. (J Phys A 42(13): 135201, 2009), Gérard and Lenzmann (The Calogero–Moser derivative nonlinear Schrödinger equation, Communications on Pure and Applied Mathematics. arXiv:2208.04105) and Badreddine (On the global well-posedness of the Calogero–Sutherland derivative nonlinear Schrödinger equation, Pure and Applied Analysis. arXiv:2303.01087; Traveling waves and finite gap potentials for the Calogero–Sutherland derivative nonlinear Schrödinger equation, Annales de l’Institut Henri Poincaré, Analyse Non Linéaire. arXiv:2307.01592), to a system of two matrix-valued variables, leading to the following intertwined system,

This system enjoys a Lax pair structure, enabling the establishment of an explicit formula for general solutions on both the 1-dimensional torus and the real line. Consequently, this system can be regarded as an integrable perturbation and extension of both the linear Schrödinger equation and the CMSdNLS equations.