Related papers: Ruijsenaars' commuting difference operators as com…
The two papers in this series analyze quantum invariant differential operators for quantum symmetric spaces in the maximally split case. In this paper, we complete the proof of a quantum version of Harish-Chandra's theorem: There is a…
In this monograph we develop magnetic pseudodifferential theory for operator-valued and equivariant operator-valued functions and distributions from first principles. These have found plentiful applications in mathematical physics,…
Pairs of $n\times n$ matrices whose commutator differ from the identity by a matrix of rank $r$ are used to construct bispectral differential operators with $r\times r$ matrix coefficients satisfying the Lax equations of the Matrix KP…
In this paper, we construct some examples of commuting differential operators $L_1$ and $L_2$ with rational coefficients of rank 3 corresponding to a curve of genus 2.
Let $(M,g)$ be a closed, smooth, Riemannian manifold of dimension $m \geq 1$. Let $\eta$ be a smooth $(0,1)$-tensor field on $M$. The divergence of $\eta$ is defined as $\text{div}_g(\eta):=g^{ij}(\nabla \eta)_{ij}$. Now let $\Delta_g$ be a…
The notion of $L^p$-distributions is introduced on Riemannian symmetric spaces of noncompact type and their main properties are established. We use a geometric description for the topology of the space of test functions in terms of the…
In this paper, we construct a new Lax operator for the elliptic $A_{n-1}$ Calogero-Moser model with general $n(2\leq n$) from the classical dynamical twisting,in which the corresponding r-matrix is purely numeric (nondynamical one). The…
We prove a Liouville-type theorem for bounded stable solutions $v \in C^2(\R^n)$ of elliptic equations of the type (-\Delta)^s v= f(v)\qquad {in $\R^n$,} where $s \in (0,1)$ {and $f$ is any nonnegative function}. The operator $(-\Delta)^s$…
The translation of an operator is defined by using conjugation with time-frequency shifts. Thus, one can define $\Lambda$-shift-invariant subspaces of Hilbert-Schmidt operators, finitely generated, with respect to a lattice $\Lambda$ in…
In this paper we propose a very effective method for constructing matrix commuting differential operators of rank 2 and vector rank (2,2). We find new matrix commuting differential operators L, M of orders 2 and 2g respectively.
We study second-order elliptic partial differential operators acting on sections of vector bundles over a compact manifold with boundary with a non-scalar positive definite leading symbol. Such operators, called non-Laplace type operators,…
The integrability of the one-dimensional (1D) fermion chain model is investigated in the framework of the Quantum Inverse Scattering Method (QISM). We introduce a new R-operator for the fermion chain model, which is expressed in terms of…
In this paper we introduce preHamiltonian pairs of difference operators and study their connections with Nijenhuis operators and the existence of weakly non-local inverse recursion operators for differential-difference equations. We begin…
Let $L=-\Delta+V$ be a Schr\"{o}dinger operator, where $\Delta $ is the Laplacian operator on $\rz$, while nonnegative potential $V$ belongs to the reverse H\"{o}lder class. In this paper, we establish the weighted norm inequalities for…
We develop a Fredholm alternative for a fractional elliptic operator~$\mathcal{L}$ of mixed order built on the notion of fractional gradient. This operator constitutes the nonlocal extension of the classical second order elliptic operators…
This work studies operators mapping vector and scalar fields defined over a manifold $\mathcal{M}$, and which commute with its group of diffeomorphisms $\text{Diff}(\mathcal{M})$. We prove that in the case of scalar fields…
We define the notion of C^{(2)}_{N+1} Ruijsenaars-Schneider models and construct their Lax formulation. They are obtained by a particular folding of the A_{2N+1} systems. Their commuting Hamiltonians are linear combinations of…
In this paper, we introduce the notion of a relative Rota-Baxter operator of weight $\lambda$ on a Lie triple system with respect to an action on another Lie triple system, which can be characterized by the graph of their semidirect…
We obtain the Baxter Q-operators in the $U_q(\hat{sl}_2)$ invariant integrable models as a special limits of the quantum transfer matrices corresponding to different spins in the auxiliary space both from the functional relations and from…
In this paper, let $L=L_{0}+V$ be a Schr\"{o}dinger type operator where $L_{0}$ is higher order elliptic operator with complex coefficients in divergence form and $V$ is signed measurable function, under the strongly subcritical assumption…