Spectral analysis of one-dimensional Dirac operators with slowly decreasing potentials. (English) Zbl 1053.34075
Here, Dirac systems \(H(\alpha)\),
\[
H(\alpha)u=\left(\begin{matrix} 0 & -1 \\ -1 & 0 \end{matrix} \right) u' + \left(\begin{matrix} p_1 & q \\ q & p_2 \end{matrix}\right)u
\]
on the half axis \((0,\infty)\) for a vector function \((u_1,u_2)^{ t}\) with the boundary condition \(u_1(0)\cos \alpha + u_2(0) \sin\alpha=0\) with square integrable potential, are studied. The author shows that the absolutely continuous spectrum of \(H(\alpha)\) covers the whole real line. For Coulomb-like potentials, it is shown that \(H(\alpha)\) has at most \(4C^2\) distinct eigenvalues, where \(C=\limsup_{x\to\infty} x | V(x)| \).
Reviewer: Christiane Tretter (Bremen)
MSC:
34L05 | General spectral theory of ordinary differential operators |
34L40 | Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) |
34B15 | Nonlinear boundary value problems for ordinary differential equations |
34B20 | Weyl theory and its generalizations for ordinary differential equations |
34L15 | Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators |
47E05 | General theory of ordinary differential operators |