Strong continuity of eigen-pairs of the Schrödinger operator with integrable potentials.

In this paper, we consider the eigenvalue problem of the Schrödinger operator with an integrable potential. Our first result states that each eigenvalue is continuously depending on the potential in the weak topology of some Lebesgue space. Furthermore, if potentials are convergent weakly in the Lebesgue space, then for each m ≥ 1 the m -th normalized eigenfunction is strongly convergent to the eigen-space of the m -th eigenvalue. Our second result obtains the convergence rate of eigenvalues, provided the potential is a cubic periodic integrable function. [ABSTRACT FROM AUTHOR]