Letters in Mathematical Physics最新文献

We revisit the perturbative theory of infinite dimensional integrable systems developed by P. Deift and X. Zhou [8], aiming to provide new and simpler proofs of some key (L^infty ) bounds and (L^p) a priori estimates. Our proofs emphasizes a further step towards understanding focussing problems and extends the applicability to other integrable models. As a concrete application, we examine the perturbation of the one-dimensional defocussing cubic nonlinear Schrödinger equation by a localized higher-order term. We introduce improved estimates to control the power of the perturbative term and demonstrate that the perturbed equation exhibits the same long-time behavior as the completely integrable nonlinear Schrödinger equation.

We consider real tensors of order D, that is D-dimensional arrays of real numbers (T_{a^1a^2 dots a^D}), where each index (a^c) can take N values. The tensor entries (T_{a^1a^2 dots a^D}) have no symmetry properties under permutations of the indices. The invariant polynomials built out of the tensor entries are called trace invariants. We prove that for a Gaussian random tensor with (Dge 3) indices (that is such that the entries (T_{a^1a^2 dots a^D}) are independent identically distributed Gaussian random variables) the cumulant, or connected expectation, of a product of trace invariants is not always suppressed in scaling in N with respect to the product of the expectations of the individual invariants. Said otherwise, not all the multi-trace expectations factor at large N in terms of the single-trace ones and the Gaussian scaling is not subadditive on the connected components. This is in stark contrast to the (D=2) case of random matrices in which the multi-trace expectations always factor at large N. The best one can do for (Dge 3) is to identify restricted families of invariants for which the large N factorization holds and we check that this indeed happens when restricting to the family of melonic observables, the dominant family in the large N limit.

This is a review of the relationship between Fay identities and Hirota equations in integrable systems, reformulated in a geometric language compatible with recent Topological Recursion formalism. We write Hirota equations as trans-series and Fay identities as spinor functional relations. We also recall several constructions of how some solutions to Fay/Hirota equations can be built from Riemann surface geometry.

We consider trapped Bose gases in three dimensions in the Gross–Pitaevskii regime whose low energy states are well known to exhibit Bose–Einstein condensation. That is, the majority of the particles occupies the same condensate state. We prove exponential control of the number of particles orthogonal to the condensate state, generalizing recent results from Nam and Rademacher (Trans Am Math Soc, 2023, arXiv:2307.10622) for translation invariant systems.

Time operators associated with an abstract semi-bounded self-adjoint operator H possessing a purely discrete spectrum are considered. The existence of a bounded self-adjoint time operator T for such an operator H is known as the Galapon time operator. In this paper, we construct a self-adjoint but unbounded time operator T for H with a dense CCR-domain, thereby extending the framework beyond the bounded setting.

We prove a correspondence, for Riemannian manifolds with self-dual Weyl tensor, between twistor functions and solutions to the Teukolsky equations for any conformal and spin weights. In particular, we give a contour integral formula for solutions to the Teukolsky equations, and we find a recursion operator that generates an infinite family of solutions and leads to the construction of Čech representatives and sheaf cohomology classes on twistor space. Apart from the general conformally self-dual case, examples include self-dual black holes, scalar-flat Kähler surfaces, and quaternionic-Kähler metrics, where we map the Teukolsky equation to the conformal wave equation, establish new relations to the linearised Przanowski equation, and find new classes of quaternionic deformations.

We consider an electronic bound state of the usual, non-relativistic, molecular Hamiltonian with Coulomb interactions, fixed nuclei, and N electrons ((N>1)). Near appropriate electronic collisions, we determine the regularity of the ((N-1))-particle electronic reduced density matrix.

We analyze oriented Riemannian 4-manifolds whose Weyl tensors W satisfy the conformally invariant condition (W(T,cdot ,cdot ,T) = 0) for some nonzero vector T. While this can be algebraically classified via W’s normal form, we find a further geometric classification by deforming the metric into a Lorentzian one via T. We show that such a W will have the analogue of Petrov Types from general relativity, that only Types I and D can occur, and that each is completely determined by the number of critical points of W’s associated Lorentzian quadratic form. A similar result holds for the Lorentzian version of this question, with T timelike.

In this paper, I study Wilson line operators in a certain type of “split” Chern–Simons theory for a Lie-algebra (mathfrak {g}={mathfrak {a}}oplus {mathfrak {a}}^*) on a manifold with boundaries. The resulting gauge theory is a 3d topological BF theory equivalent to a topologically twisted 3d ({mathcal {N}}=4) theory. I show that this theory realises solutions to the quantum Yang–Baxter equation all orders in perturbation theory as the expectation value of crossing Wilson lines.

We revisit the classical aspects of (mathcal {N}=(2,2)) supersymmetric sigma models with Hermitian symmetric target spaces, using the so-called Gross–Neveu (“first-order GLSM”) formalism. We reformulate these models for complex Grassmannians in terms of simple supersymmetric Lagrangians with polynomial interactions. For maximal isotropic Grassmannians we propose two types of equivalent Lagrangians, which make either supersymmetry or the geometry of target space manifest. These reformulations can be seen as current–current deformations of curved (beta gamma ) systems. The (textsf{CP}^{1}) supersymmetric sigma model is our prototypical example.