Letters in Mathematical Physics最新文献

Impulsive gravitational waves are theoretical models of short but violent bursts of gravitational radiation. They are commonly described by two distinct spacetime metrics, one of local Lipschitz regularity and the other one even distributional. These two metrics are thought to be ‘physically equivalent’ since they can be formally related by a ‘discontinuous coordinate transformation’. In this paper we provide a mathematical analysis of this issue for the entire class of nonexpanding impulsive gravitational waves propagating in a background spacetime of constant curvature. We devise a natural geometric regularisation procedure to show that the notorious change of variables arises as the distributional limit of a family of smooth coordinate transformations. In other words, we establish that both spacetimes arise as distributional limits of a smooth sandwich wave taken in different coordinate systems which are diffeomorphically related.

In this letter, we obtain the precise range of the values of the parameter (alpha ) such that Petz–Rényi (alpha )-relative entropy (D_{alpha }(rho ||sigma )) of two faithful displaced thermal states is finite. More precisely, we prove that, given two displaced thermal states (rho ) and (sigma ) with inverse temperature parameters (r_1, r_2,ldots , r_n) and (s_1,s_2, ldots , s_n), respectively, (0<r_j,s_j<infty ), for all j, we have

where we adopt the convention that the minimum of an empty set is equal to infinity. This result is particularly useful in the light of operational interpretations of the Petz–Rényi (alpha )-relative entropy in the regime (alpha >1 ). Along the way, we also prove a special case of a conjecture of Seshadreesan et al. (J Math Phys 59(7):072204, 2018. https://doi.org/10.1063/1.5007167).

In this paper, we present the connection of two concepts as induced representation and partially reduced irreducible representations (PRIR) appear in the context of port-based teleportation protocols. Namely, for a given finite group G with arbitrary subgroup H, we consider a particular case of matrix irreducible representations, whose restriction to the subgroup H, as a matrix representation of H, is completely reduced to diagonal block form with an irreducible representation of H in the blocks. The basic properties of such representations are given. Then as an application of this concept, we show that the spectrum of the port-based teleportation operator acting on n systems is connected in a very simple way with the spectrum of the corresponding Jucys–Murphy operator for the symmetric group (S(n-1)subset S(n)). This shows on the technical level relation between teleporation and one of the basic objects from the point of view of the representation theory of the symmetric group. This shows a deep connection between the central object describing properties of deterministic PBT schemes and objects appearing naturally in the abstract representation theory of the symmetric group. In particular, we present a new expression for the eigenvalues of the Jucys–Murphy operators based on the irreducible characters of the symmetric group. As an additional but not trivial result, we give also purely matrix proof of the Frobenius reciprocity theorem for characters with explicit construction of the unitary matrix that realizes the reduction in the natural basis of induced representation to the reduced one.

The main purpose of this paper is to discuss the Cauchy problem of integrable nonlocal (reverse-space-time) Kundu–Eckhaus (KE) equation through the Riemann–Hilbert (RH) method. Firstly, based on the zero-curvature equation, we present an integrable nonlocal KE equation and its Lax pair. Then, we discuss the properties of eigenfunctions and scattering matrix, such as analyticity, asymptotic behavior, and symmetry. Finally, for the prescribed step-like initial value: (u(z,t)=o(1)), (zrightarrow -infty ) and (u(z,t)=R+o(1)), (zrightarrow +infty ), where (R>0) is an arbitrary constant, we consider the initial value problem of the nonlocal KE equation. The paramount techniques is the asymptotic analysis of the associated RH problem.

We provide a simple method to recognize a classical orthogonal polynomial sequence on a q-quadratic lattice defined only by the three-term recurrence relation. It is pointed out that this can be extended to all orthogonal polynomials in the q-Askey scheme.

Consider a locally compact quantum group (mathbb {G}) with a closed classical abelian subgroup (Gamma ) equipped with a 2-cocycle (Psi :hat{Gamma }times hat{Gamma }rightarrow mathbb {C}). We study in detail the associated Rieffel deformation (mathbb {G}^{Psi }) and establish a canonical correspondence between (Gamma )-invariant convolution semigroups of states on (mathbb {G}) and on (mathbb {G}^{Psi }).

A simple condition is given that is sufficient to determine whether a measure that is absolutely continuous with respect to a Gaußian measure on the space of distributions is reflection positive. It readily generalises conventional lattice results to an abstract setting, enabling the construction of many reflection positive measures that are not supported on lattices.

This paper discusses the mean-field limit for the quantum dynamics of N identical bosons in ({textbf{R}}^3) interacting via a binary potential with Coulomb-type singularity. Our approach is based on the theory of quantum Klimontovich solutions defined in Golse and Paul (Commun Math Phys 369:1021–1053, 2019) . Our first main result is a definition of the interaction nonlinearity in the equation governing the dynamics of quantum Klimontovich solutions for a class of interaction potentials slightly less general than those considered in Kato (Trans Am Math Soc 70:195–211, 1951). Our second main result is a new operator inequality satisfied by the quantum Klimontovich solution in the case of an interaction potential with Coulomb-type singularity. When evaluated on an initial bosonic pure state, this operator inequality reduces to a Gronwall inequality for a functional introduced in Pickl (Lett Math Phys 97:151-164, 2011), resulting in a convergence rate estimate for the quantum mean-field limit leading to the time-dependent Hartree equation.

We construct Lax matrices of superoscillator type that are solutions of the RTT-relation for the rational orthosymplectic R-matrix, generalizing orthogonal and symplectic oscillator type Lax matrices previously constructed by the authors in Frassek (Nuclear Phys B, 2020), Frassek and Tsymbaliuk (Commun Math Phys 392 (2):545–619, 2022), Frassek et al. (Commun Math Phys 400 (1):1–82, 2023). We further establish factorisation formulas among the presented solutions.

By principal representation of toroidal Lie algebra (mathrm{sl^{tor}_2}), we construct an integrable system: Bogoyavlensky–modified KdV (B–mKdV) hierarchy, which is ((2+1))-dimensional generalization of modified KdV hierarchy. Firstly, bilinear equations of B–mKdV hierarchy are obtained by fermionic representation of (mathrm{sl^{tor}_2}) and boson–fermion correspondence, which are rewritten into Hirota bilinear forms. Also Fay-like identities of B–mKdV hierarchy are derived. Then from B–mKdV bilinear equations, we investigate Lax structure, which is another equivalent formulation of B–mKdV hierarchy. Conversely, we also derive B–mKdV bilinear equations from Lax structure. Other equivalent formulations of wave functions and dressing operator are needed when discussing bilinear equations and Lax structure. After that, Miura links between Bogoyavlensky–KdV hierarchy and B–mKdV hierarchy are discussed. Finally, we construct soliton solutions of B–mKdV hierarchy.