JournalspmVol. 68, No. 3pp. 317–343

Product approximations for solutions to a class of evolution equations in Hilbert space

  • Pierre-A. Vuillermot

    Université Henri Poincaré, Vandoeuvre-lès-Nancy, France
  • Walter F. Wreszinski

    Universidade de São Paulo, São Paulo, Brazil
Product approximations for solutions to a class of evolution equations in Hilbert space cover

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Abstract

In this article we prove approximation formulae for a class of unitary evolution operators U(t,s)s,t[0,T]U(t,s)_{s,t\in [0,T] } associated with linear non-autonomous evolution equations of Schr\"{o}dinger type defined in a Hilbert space H\mathcal{H}. An important feature of the equations we consider is that both the corresponding self-adjoint generators and their domains may depend explicitly on time, whereas the associated quadratic form domains may not. Furthermore the evolution operators we are interested in satisfy the equations in a weak sense. Under such conditions the approximation formulae we prove for U(t,s)U(t,s) involve weak operator limits of products of suitable approximating functions taking values in L(H)\mathcal{L(H)}, the algebra of all linear bounded operators on H\mathcal{H}. Our results may be relevant to the numerical analysis of U(t,s)U(t,s) and we illustrate them by considering two typical examples, including one related to the theory of time-dependent singular perturbations of self-adjoint operators.

Cite this article

Pierre-A. Vuillermot, Walter F. Wreszinski, Product approximations for solutions to a class of evolution equations in Hilbert space. Port. Math. 68 (2011), no. 3, pp. 317–343

DOI 10.4171/PM/1894