The connection of the canonical systems with the classical systems 8.3. Livsic-Brodskii nodes and spectral theory 116 123 Table of Contents VII 804. Spectral problems on the axis 124 8.5. Weyl-Titchmarsh matrix functions 127 8.6. The inverse spectral problem on the axis 131 Chapter 9. INTEGRABLE NONLINEAR EQUATIONS (DISCRETE (DISCRETE CASE). .. . 9.1. Evolution law of spectral data 137 9 .2. Construction of hierarchy Chapter 10. ON SEMI-INFINITE TODA CHAIN 10.1. Inverse problem and evolution of spectral data 10.2. Semi-infinite chain with a free end 10.3. The evolution law for the Toda chain with a fixed end lOA. Analytical properties of the Weyl-Titchmarsh function 10.5. A half-finite chain with a fixed end, the solving procedure 10.6. Finite chain, solution procedure Chapter 11. FUNCTIONS WITH AN OPERATOR ARGUMENT 11.1. Nevanlinna class functions with an operator argument 17.'5 11.2. Positive functions with an operator' argument 11.3. On Sarason scheme. 1104. Factorization formula 185 Commentaries and Remarks 188 Bibliography 191 Index. ' 111 9 =-ig, ~2g = g, ' 11 2 =-i N*(x)g (0.11) Using the notation (0.9) and the fact that B = A*, we can rewrite (0.9) in the form of the operator identity (0.3). If M(x) = N*(x), then we get identity (0.5). Let us note that for WI = W 2 = ... = Wn an operator S of form (0.6), (0.7) is an operator with a difference kernel sex, t) = .s(xt). If WI = w 2 = ... = wp = a,
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.
The spectrum of the Sturm-Liouville problem is given. The potential on the half-interval is known. It is necessary to reconstruct the potential on the whole interval. The uniqueness of the solution of the formulated problem is proved (
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