Letters in Mathematical Physics最新文献

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.

We study a Hermitian matrix model with a kinetic term given by ( Tr (H Phi ^2 )), where H is a positive definite Hermitian matrix, similar as in the Kontsevich Matrix model, but with its potential (Phi ^3) replaced by (Phi ^4). We show that its partition function solves an integrable Schrödinger-type equation for a non-interacting N-body Harmonic oscillator system.

We present a special model of random band matrices where, at zero energy, the famous Fyodorov and Mirlin (sqrt{N})-conjecture (Phys Rev Lett 67(18):2405, 1991) can be established very simply.

We develop an invariant theory of quasi-split (imath )quantum groups ({textbf {U}} _n^imath ) of type AIII on a tensor space associated to (imath )Howe dualities. The first and second fundamental theorems for ({textbf {U}} _n^imath )-invariants are derived.

We prove a formula, first obtained by Kleban, Simmons and Ziff using conformal field theory methods, for the (renormalized) density of a critical percolation cluster in the upper half-plane “anchored” to a point on the real line. The proof is inspired by the method of images. We also show that more general bulk-boundary connection probabilities have well-defined, scale-covariant scaling limits and prove a formula for the scaling limit of the (renormalized) density of the critical percolation gasket in any domain conformally equivalent to the unit disk.

In this note we use the Matsuo–Cherednik duality between the solutions to the Knizhnik–Zamolodchikov (KZ) equations and eigenfunctions of Calogero–Moser Hamiltonians to get the polynomial (p^s)-truncation of the Calogero–Moser eigenfunctions at a rational coupling constant. The truncation procedure uses the integral representation for the hypergeometric solutions to KZ equations. The (srightarrow infty ) limit to the pure p-adic case has been analyzed in the (n=2) case.