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Nonlinearities in finite dimensions can be linearized by projecting them into infinite dimensions. Unfortunately, often the linear operator techniques that one would then use simply fail since the operators cannot be diagonalized. This…
We define a class of discrete operators acting on infinite, finite or periodic sequences mimicking the standard properties of pseudo-differential operators. In particular we can define the notion of order and regularity, and we recover the…
We construct linear and quadratic Darboux matrices compatible with the reduction group of the Lax operator for each of the seven known non-Abelian derivative nonlinear Schr\"odinger equations that admit Lax representations. The…
In this paper we derive structure theorems that characterize the spaces of linear and non-linear differential operators that preserve finite dimensional subspaces generated by polynomials in one or several variables. By means of the useful…
Irreducible second-order Darboux transformations are applied to the periodic Schrodinger's operators. It is shown that for the pairs of factorization energies inside of the same forbidden band they can create new non-singular potentials…
The Bol operators are unary differential operators between spaces of weighted densities on the 1-dimensional manifold invariant under projective transformations of the manifold. On the $1|n$-dimensional supermanifold (superstring)…
We prove a general black box result which produces algebras of pseudodifferential operators (ps.d.o.s) on noncompact manifolds, together with a precise principal symbol calculus. Our construction (which also applies in parameter-dependent…
In this paper we consider the fourth order operators A1u := (au")" and A2u := au"" in divergence form and non divergence form, respectively, where a, defined in [0, 1] with values in R+, degenerates in an interior point of the interval.…
Let $X$ be a separable Banach space and let $Q:X^*\rightarrow X$ be a linear, bounded, non-negative and symmetric operator and let $A:D(A)\subseteq X\rightarrow X$ be the infinitesimal generator of a strongly continuous semigroup of…
We give an algorithm to write down all conformally invariant differential operators acting between scalar functions on Minkowski space. All operators of order k are nonlinear and are functions on a finite family of functionally independent…
Differential operators commuting with integral operators were discovered in the work of C. Tracy and H. Widom [37, 38] and used to derive asymptotic expansions of the Fredholm determinants of integral operators arising in random matrix…
We use the $\zeta$-function regularization and an integral representation of the complex power of a pseudo differential operator, to give an unambiguous definition of the determinant of the Dirac operator. We bring this definition to a…
Denote by $SL_3(\mathbb R)$ the special linear group of degree 3 over the real numbers, $A$ the subgroup consisting of the diagonal matrices with positive entries. In this paper, we study the algebraic and analytic properties of the…
This article presents a survey of recent developments on pseudodifferential operators on noncommutative tori. We describe currently available constructions of those operators: by means of a $C^*$--dynamical system, by using an analogue of…
The technique of differential intertwining operators (or Darboux transformation operators) is systematically applied to the one-dimensional Dirac equation. The following aspects are investigated: factorization of a polynomial of Dirac…
First-order differential operators arising from the representation-theoretic decomposition of the covariant derivative play a central role in Riemannian geometry. In this paper, we study Stein-Weiss $O(n)$-gradients acting on covariant…
We study the ring of differential operators D(X) on the basic affine space X=G/U of a complex semisimple group G with maximal unipotent subgroup U. One of the main results shows that the cohomology group H^*(X,O_X) decomposes as a finite…
Following the definitions of the algebras of differential operators, $\beta$-differential operators, and the quantum differential operators on a noncommutative (graded) algebra given in \cite{LR}, we describe these operators on the free…
Let $R$ be a commutative ring, $\mathcal A$ an $R$-algebra (not necessarily commutative) and $V$ an $R$-subspace or $R$-submodule of $\mathcal A$. By the radical of $V$ we mean the set of all elements $a\in \mathcal A$ such that $a^m\in V$…
A complete classification of linear differential operators possessing finite-dimensional invariant subspace with a basis of monomials is presented.