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A functional model for a family of operators induced by Laguerre operator

Hatamleh Ra'ed (2003)

Archivum Mathematicum

The paper generalizes the instruction, suggested by B. Sz.-Nagy and C. Foias, for operatorfunction induced by the Cauchy problem T t : t h ' ' ( t ) + ( 1 - t ) h ' ( t ) + A h ( t ) = 0 h ( 0 ) = h 0 ( t h ' ) ( 0 ) = h 1 A unitary dilatation for T t is constructed in the present paper. then a translational model for the family T t is presented using a model construction scheme, suggested by Zolotarev, V., [3]. Finally, we derive a discrete functional model of family T t and operator A applying the Laguerre transform f ( x ) 0 f ( x ) P n ( x ) e - x d x where P n ( x ) are Laguerre polynomials [6, 7]. We show that the Laguerre transform...

A Simple Example of Localized Parametric Resonance for the Wave Equation

Colombini, Ferruccio, Rauch, Jeffrey (2008)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 35L05, 35P25, 47A40.The problem studied here was suggested to us by V. Petkov. Since the beginning of our careers, we have benefitted from his insights in partial differential equations and mathematical physics. In his writings and many discussions, the conjuction of deep analysis and specially interesting problems has been a source inspiration for us.The research of J. Rauch is partially supported by the U.S. National Science Foundation under grant NSF-DMS-0104096...

Asymptotic expansion in time of the Schrödinger group on conical manifolds

Xue Ping Wang (2006)

Annales de l’institut Fourier

For Schrödinger operator P on Riemannian manifolds with conical end, we study the contribution of zero energy resonant states to the singularity of the resolvent of P near zero. Long-time expansion of the Schrödinger group U ( t ) = e - i t P is obtained under a non-trapping condition at high energies.

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