We discuss correspondence between the predictions of quantum theories for rotation angle formulated in infinite and finite dimensional Hilbert spaces, taking as example, the calculation of matrix elements of phaseangular momentum commutator. A new derivation of the matrix elements is presented in infinite space, making use of a unitary transformation that maps from the state space of periodic f...

We give a reinterpretation of the matrix theory discussed by Moore, Nekrasov and Shatashivili (MNS) in terms of the second quantized operators which describes the homology class of the Hilbert scheme of points on surfaces. It naturally relate the contribution from each pole to the inner product of orthogonal basis of free boson Fock space. These basis can be related to the eigenfunctions of Cal...

Many important problems in the theory of integrable systems and approximation theory can be recast as Riemann-Hilbert problems for a matrix-valued unknown. Via the connection with approximation theory, and specifically the theory of orthogonal polynomials, one can also study problems from the theory of random matrix ensembles and combinatorics. Roughly speaking, solving a Riemann-Hilbert proble...

A matrix Riemann-Hilbert problem associated with the one-dimensional SchrBdinger equation is considered, and the existence and uniqueness of its solutions are studied. The solution of this Riemann-Hilbert problem yields the solution of the inverse scattering problem for a larger class of potentials than the usual Faddeev class. Some examples of explicit solutions of the Riemann-Hilbert problem ...

Th e interaction b etwee n t heoretical mathematics and practical computational expire-m en t is illustrated by a discussion of recent work, at v arious centers, concerning t he Hilber t matrix.

We explore a variety of pleasing connections between analysis, number theory and operator theory, while revisiting a number of beautiful inequalities originating with Hilbert, Hardy and others. We shall first establish the aforementioned Hilbert inequality [?, ?] and then apply it to various multiple zeta values. In consequence we obtain the norm of the classical Hilbert matrix, while illustrat...