WIAS Preprint No. 1266, (2007)

Linear non-autonomous Cauchy problems and evolution semigroups



Authors

  • Neidhardt, Hagen
  • Zagrebnov, Valentin A.

2010 Mathematics Subject Classification

  • 35L90 34G10 47D06

Keywords

  • linear evolution equations, evolution semigroups, perturbation theory, time-dependent Schrödinger operators, moving potentials

Abstract

The paper is devoted to the problem of existence of propagators for an abstract linear non-autonomous evolution Cauchy problem of hyperbolic type in separable Banach spaces. The problem is solved using the so-called evolution semigroup approach which reduces the existence problem for propagators to a perturbation problem of semigroup generators. The results are specified to abstract linear non-autonomous evolution equations in Hilbert spaces where the assumption is made that the domains of the quadratic forms associated with the generators are independent of time. Finally, these results are applied to time-dependent Schrödinger operators with moving point interactions in 1D.

Appeared in

  • Adv. Differential Equations, 14 (2009) pp. 289--340.

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