Factorization problems and operator identities

Abstract

We show how the Dijkgraaf-Vafa matrix model proposal can be extended to describe five-dimensional gauge theories compactified on a circle to four dimensions. This involves solving a certain quantum mechanical matrix model. We do this for the lift of the N = 1 * theory to five dimensions. We show that the resulting expression for the superpotential in the confining vacuum is identical with the elliptic superpotential approach based on Nekrasov's five-dimensional generalization of Seiberg-Witten theory involving the relativistic elliptic Calogero-Moser, or Ruijsenaars-Schneider, integrable system.

Direct and inverse scattering for selfadjoint Hamiltonian systems on the line

“…(2.38) 39) and hence all(A) is invertible for all A e R. Thus, f+(x,A) is nonsingular and forms a fundamental matrix for (1.1). Similarly, we get 40) and hence with the help of (2.36) and (2.39), we conclude that f_ (x, 7~) is nonsingular and forms a fundamental matrix for (1.1).…”

“…The term "canonical differential equations" for the system (1.1) has been used by MelikAdamyan [32][33][34], L. A. Sakhnovieh [39,40], and A. L. Sakhnovich [38], who have studied the direct and inverse scattering problems for (1.1) on the half line. Under minor restrictions on the given so-called reflection function, a characterization of the scattering data corresponding to an Ll-potential on the half line was given by Melik-Adamyan [34], who also supplied a method to reduce the inverse spectral problem on the full line for a canonical equation of order 2n to an inverse spectral problem on the half line for a canonical equation of order 4n [32] (see also [41]).…”

Weyl functions, the inverse problem and special solutions for the system auxiliary to the nonlinear optics equation

“…The proof of (2.26) in [33] is based on the representation of the fundamental solution w: 27) where w A is the transfer matrix function in the Lev Sakhnovich form [38]- [40]: 28) and (x) = V Π (x). By the last relation, in view of (2.22) and (2.24), we have…”

Bäcklund-Darboux Transformation for Non-Isospectral Canonical System and Riemann-Hilbert Problem

“…Notice that the equation A x = A 2 is motivated by the similar equation λ x = λ 2 for the spectral parameter λ because A can be viewed as a generalized spectral parameter (see [14]). In the points of invertibility of S we can introduce a transfer matrix function in the Lev Sakhnovich form [17,18,19] …”