On the Approximation of the Solution of the Schrödinger Equation by Superpositions of Stationary Solutions

Abstract

Let S be a symmetric operator with gap J. Suppose in addition that the deficiency indices of S are infinite, the Hamiltonian H is a selfadjoint extension of S and the support of the spectral measure μ _f0 , H of the initial state f ₀ is a compact subset of J. Then there exist other self-adjoint extensions H _n of S and finite sums f _n of eigenvectors of H _n such that

$$ e^{ - itHn} f_n \to e^{ - itH} f_0 , as n \to \infty , $$

locally uniformly in time. Upper estimates for the rate of convergence will be given.