Abstract:
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].
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