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On a property of weak resolvents and its application to a spectral problem

Yoichi Uetake (1997)

Annales Polonici Mathematici

We show that the poles of a resolvent coincide with the poles of its weak resolvent up to their orders, for operators on Hilbert space which have some cyclic properties. Using this, we show that a theorem similar to the Mlak theorem holds under milder conditions, if a given operator and its adjoint have cyclic vectors.

On the GBDT Version of the Bäcklund-Darboux Transformation and its Applications to Linear and Nonlinear Equations and Weyl Theory

A. Sakhnovich (2010)

Mathematical Modelling of Natural Phenomena

A general theorem on the GBDT version of the Bäcklund-Darboux transformation for systems depending rationally on the spectral parameter is treated and its applications to nonlinear equations are given. Explicit solutions of direct and inverse problems for Dirac-type systems, including systems with singularities, and for the system auxiliary to the N-wave equation are reviewed. New results on explicit construction of the wave functions for radial...

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