Borg-type uniqueness theorems for matrix-valued Jacobi operators H and supersymmetric Dirac difference operators D are proved. More precisely, assuming reflectionless matrix coefficients A,B in the self-adjoint Jacobi operator H = AS + AS + B (with S the right/left shift operators on the lattice Z) and the spectrum of H to be a compact interval [E−, E+], E− < E+, we prove that A and B are certa...

We consider semi-infinite Jacobi matrices with discrete spectrum. prove that a operator can be uniquely recovered from one spectrum and subsets of another norming constants c...

The second variation operator of minimal submanifolds of Riemannian manifolds (the Jacobi operator) carries information about stability properties of the submanifold when it is thought of as a critical point for the area functional. When the ambient Riemannian manifold is a sphere S, Simons [S] characterized the totally geodesic submanifolds as the minimal submanifolds of S either with the lowe...

Quantum canonical transformations have attracted interest since the beginning of quantum theory. Based on their classical analogues, one would expect them to provide a powerful quantum tool. However, the difficulty of solving a nonlinear operator partial differential equation such as the quantum Hamilton-Jacobi equation (QHJE) has hindered progress along this otherwise promising avenue. We over...

The operator Lμ : f 7→ ∫ f(x)−f(y) x−y dμ(y) is, for a compactly supported measure μ with an L density, a closed, densely defined operator on L(μ). We show that the operator Q = pLμ−qLμ has polynomial eigenfunctions if and only if μ is a free Meixner distribution. The only time Q has orthogonal polynomial eigenfunctions is if μ is a semicircular distribution. More generally, the only time the o...

with respect to the Euclidian norm, where F :Rn →Rn is a nonlinear operator. Section 2 is devoted to a brief description of asynchronous parallel algorithm. In Section 3, we prove the main result concerning the convergence of the general algorithm in the synchronous case to a fixed point of a nonlinear operator fromRn to Rn. A particular case of this algorithm (algorithm of Jacobi) is applied i...

We study the Cauchy singular integral operator SwI on (−1, 1), where |w| is a generalized Jacobi weight. This operator is considered in pairs of weighted spaces of continuous functions, where the weights u and v are generalized Jacobi weights with nonnegative exponents such that |w| = u/v. We introduce a certain polynomial approximation space which is well appropriated to serve as domain of def...