We discuss Schrodinger operators perturbed by measures and the limit relations between them. For a large family of real-valued Radon measures m, including the Kato class, the operators $-\Delta + C^2 \Delta^2 + m$ tend to Schrodinger operator $-\Delta +m$ in the norm resolvent sense as C tends to zero. On the other hand, weak convergence of finite measures m implies the norm resolvent convergence of the corresponding perturbed fourth-order operators, provided the dimension is smaller than four.
Thus the combination of both convergence results yields an approximation of Schrodinger operators and the presence of the fourth-order perturbation makes it possible to choose point potentials as the approximating measures. Since in that case the spectral problem is solvable, we obtain an alternative method for the numerical computation of the eigenvalues. We illustrate the approximation by numerical calculations of eigenvalues for one simple example of measure m. Joint work with Johannes Brasche.
- http://xxx.lanl.gov/abs/math-ph/0511029 - preprint archives