Related papers: Differential operators on Schur and Schubert polyn…

Haglund, Morse, and Zabrocki introduced a family of creation operators of Hall-Littlewood polynomials, $\{C_{a}\}$ for any $a\in \mathbb{Z}$, in their compositional refinement of the shuffle (ex-)conjecture. For any $\alpha\vDash n$, the…

We use Young's raising operators to give short and uniform proofs of several well known results about Schur polynomials and symmetric functions, starting from the Jacobi-Trudi identity.

This paper deals with a certain class of second-order conformally invariant operators acting on functions taking values in particular (finite-dimensional) irreducible representations of the orthogonal group. These operators can be seen as a…

We present certain results on the direct and inverse spectral theory of the Jacobi operator with complex periodic coefficients. For instance, we show that any $N$-th degree polynomial whose leading coefficient is $(-1)^N$ is the Hill…

This paper has been withdrawn by the authors. A class of pseudodifferential operators on the Heisenberg group is defined. As it should be, this class is an algebra containing the class of differential operators. Furthermore, those…

A class of pseudodifferential operators on the Heisenberg group is defined. As it should be, this class is an algebra containing the class of differential operators. Furthermore, those pseudodifferential operators act continuously on…

In this paper, we introduce an inverse problem of a Schr\"odinger type variable nonlocal elliptic operator $(-\nabla\cdot(A(x)\nabla))^{s}+q)$, for $0<s<1$. We determine the unknown bounded potential $q$ from the exterior partial…

We offer a detailed treatment of spectral and Weyl-Titchmarsh-Kodaira theory for all self-adjoint Jacobi operator realizations of the differential expression \begin{align*} \tau_{\alpha,\beta} = - (1-x)^{-\alpha} (1+x)^{-\beta}(d/dx)…

This paper presents an algebraic approach to characterizing higher-order differential operators. While the foundational Leibniz rule addresses first-order derivatives, its extension to higher orders typically involves identities relating…

In this article, the authors consider the Schr\"{o}dinger type operator $L:=-{\rm div}(A\nabla)+V$ on $\mathbb{R}^n$ with $n\geq 3$, where the matrix $A$ satisfies uniformly elliptic condition and the nonnegative potential $V$ belongs to…

We study the second-order differential operators \(\mathcal D_{\Xi}\) and \(\mathcal D_{\Lambda}\) associated with the rescaled polynomial families \((\widetilde{\Xi}_n)\) and \((\widetilde{\Lambda}_n)\), and more generally the polynomial…

We give several new formulas which are useful for Schubert Calculus associated with the orthogonal groups and related orthogonal degeneracy loci.

The double Schur functions form a distinguished basis of the ring \Lambda(x||a) which is a multiparameter generalization of the ring of symmetric functions \Lambda(x). The canonical comultiplication on \Lambda(x) is extended to…

In this study, we define discrete fractional Sturm-Liouville (DFSL) operators within Riemann-Liouville and Gr\"unwald-Letnikov fractional operators with both delta and nabla operators. We show selfadjointness of the DFSL operator for the…

We give a combinatorial rule for calculating the coefficients in the expansion of a product of two factorial Schur functions. It is a special case of a more general rule which also gives the coefficients in the expansion of a skew factorial…

We consider the operators \[ \nabla_X\cdot(A(X)\nabla_X),\ \nabla_X\cdot(A(X)\nabla_X)-\partial_t,\ \nabla_X\cdot(A(X)\nabla_X)+X\cdot\nabla_Y-\partial_t, \] where $X\in \Omega$, $(X,t)\in \Omega\times \mathbb R$ and $(X,Y,t)\in…

This note supplements the work of Gomez-Ullate, Kamran and Milson on the X_(1)-Jacobi polynomials which are orthogonal in a weighted Hilbert function space on the the interval (-1,+1) of the real line. These polynomials are generated by a…

For any Schur function $s_{\nu}$, the associated {\em delta operator} $\Delta'_{s_{\nu}}$ is a linear operator on the ring of symmetric functions which has the modified Macdonald polynomials as an eigenbasis. When $\nu = (1^{n-1})$ is a…

We continue the work begun by Mickler-Moll investigating the properties of the polynomial eigenfunctions of the Nazarov-Sklyanin quantum Lax operator. By considering products of these eigenfunctions, we produce a novel generalization of a…

Fix $d \geq 3$ and $1 < p < \infty$. Let $V : \mathbb{R}^{d} \rightarrow [0,\infty)$ belong to the reverse H\"{o}lder class $RH_{d/2}$ and consider the Schr\"{o}dinger operator $L_{V} := - \Delta + V$. In this article, we introduce classes…