Dense point spectra of Schr\"odinger and Dirac operators

S. N. Naboko

Teoretičeskaâ i matematičeskaâ fizika, Tome 68 (1986) no. 1, pp. 18-28

Résumé

Examples are constructed of one-dimensional self-adjoint Schrödinger and Dirac operators with potential that decreases slightly slower than the Coulomb potential for which the point spectrum fills densely the half-axis $[0,\infty)$ and the complete axis $\mathbb R$, respectively. Examples are constructed of potentials $q$ for which the corresponding Schrödinger operator with decreasing potential $C\cdot q$ ($C=\operatorname{const}>0$ is the coupling constant) has a point spectrum that fills the interval $[0,C]$ densely while for $\lambda>C$ there are no eigenvalues at all. This example may be of interest for investigation of the metal – insulator phase transition in the Anderson model. References are given [1–7] to related discussions of the spectral rearrangement of the Schrödinger operator. The main results of the paper were presented briefly in an earlier note of the author [8].