Let $X$ be a compact Kähler manifold with integral Kähler class and $L\to X$ a holomorphic Hermitian line bundle whose curvature is the symplectic form of $X$. Let $H\in C^{\infty }(X,ℝ)$ be a Hamiltonian, and let $T_k$ be the Toeplitz operator with multiplier $H$ acting on the space ${\mathscr{H}}_k=H^0(X,L^{\otimes k})$. We obtain estimates on the eigenvalues and eigensections of $T_k$ as $k\to \infty$, in terms of the classical Hamilton flow of $H$. We study in some detail the case when $X$ is an integral coadjoint orbit of a Lie group.

We consider systems of weakly coupled Schrödinger equations with nonconstant potentials and investigate the existence of nontrivial nonnegative solutions which concentrate around local minima of the potentials. We obtain sufficient and necessary conditions for a sequence of least energy solutions to concentrate.

Using some perturbation results in critical point theory, we prove that a class of nonlinear Schrödinger equations possesses semiclassical states that concentrate near the critical points of the potential $V$.

We prove a formula for the current in an electron gas in a semiclassical limit corresponding to strong, constant, magnetic fields. Little regularity is assumed for the scalar potential $V$. In particular, the result can be applied to the mean field from magnetic Thomas-Fermi theory $V_{\mathrm{MTF}}$. The proof is based on an estimate on the density of states in the second Landau band.

We describe representations of certain superconformal algebras in the semi-infinite Weil complex related to the loop algebra of a complex finite-dimensional Lie algebra and in the semi-infinite cohomology. We show that in the case where the Lie algebra is endowed with a non-degenerate invariant symmetric bilinear form, the relative semi-infinite cohomology of the loop algebra has a structure, which is analogous to the classical structure of the de Rham cohomology in Kähler...

We study a family of commuting selfadjoint operators $={\left(A_k\right)}_{k=1}^n$, which satisfy, together with the operators of the family $={\left(B_j\right)}_{j=1}^n$, semilinear relations ${⅀}_i{f}_{ij}\left(\right)B_j{g}_{ij}\left(\right)=h\left(\right)$, ($f_{ij}$, $g_{ij}$, $h_j:{ℝ}^n\to ℂ$ are fixed Borel functions). The developed technique is used to investigate representations of deformations of the universal enveloping algebra U(so(3)), in particular, of some real forms of the Fairlie algebra $U_q^{\text{'}}\left(so\left(3\right)\right)$.

Γ):Γ ∈ 𝒜, ℋ1(Γ) = l}, where ℋ1D1,...,Dk }  ⊂ Rd . The cost functional ℰ(Γ) is the Dirichlet energy of Γ defined through the Sobolev functions on Γ vanishing on the points Di. We analyze the existence of a solution in both the families of connected sets and of metric graphs. At the end, several explicit examples are discussed.

In this paper we prove a two-term asymptotic formula for the spectral counting function for a $2$D magnetic Schrödinger operator on a domain (with Dirichlet boundary conditions) in a semiclassical limit and with strong magnetic field. By scaling, this is equivalent to a thermodynamic limit of a $2$D Fermi gas submitted to a constant external magnetic field.The original motivation comes from a paper by H. Kunz in which he studied, among other things, the boundary correction for the grand-canonical pressure...

The goal of our work is to study the spaces of primitive elements of some combinatorial Hopf algebras, whose underlying vector spaces admit linear basis labelled by subsets of the set of maps between finite sets. In order to deal with these objects we introduce the notion of shuffle algebras, which are coloured algebras where composition is not always defined. We define bialgebras in this framework and compute the subpaces of primitive elements associated to them. These spaces of primitive elements...

We discuss additional supersymmetries for $𝒩=(2,2)$ supersymmetric non-linear sigma models described by left and right semichiral superfields.