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One dimensional Schrödinger operators with high potential barriers. (English) Zbl 0883.34078

Demuth, Michael (ed.) et al., Operator calculus and spectral theory. Proceedings of a symposium, held in Lambrecht, Germany, December 9-14, 1991. Basel: Birkhäuser Verlag. Oper. Theory, Adv. Appl. 57, 163-170 (1992).
A well-known theorem of Molchanov implies that a one-dimensional Hamiltonian \[ H_\alpha=-d^2/dx^2+V \] in half-space, acting on \(L^2[0,\infty)\)-functions \(\psi(x)\) with boundary conditions \[ \psi(0)\cos\alpha+\psi'(0)\sin\alpha=0,\quad 0\leq\alpha\leq 2\pi, \] has a purely discrete spectrum if and only if \(\int^{x+1}_x|V(x)|dx\to\infty\) as \(x\to\infty\).
The authors assume (for the sake of simplicity) that \(V(\in L^2_{\text{loc}})\geq 0\), but it is clear that their arguments can be extended to more general cases. Following B. Simon and T. Spencer [Commun. Math. Phys. 125, No. 1, 113-125 (1989; Zbl 0684.47010)] they introduce a sequence \(x_n\nearrow\infty\), and a corresponding sequence \(h_n\) such that \(V\geq V_n\), on \([x_n- h_n,x_n+h_n]\) as \(V_n\to\infty\) and \(h_n\sqrt V_n\to\infty\). Let \(\ell_n\) denote \((x_{n+1} -x_n)>\beta>0\). The authors prove that if for some \(\gamma <1\) it is true that \(\Sigma_n(\ell_n^{3/2}+\ell_{n+1}^{3/2})\exp\{\gamma h_n\sqrt V_n)<\infty\), then \(H_\alpha\) has pure point spectrum. The authors use Kotani’s technique of proof. Introducing Green’s function \(R(x,y)\) they apply De-Valee-Poussin’s theorem on spectral measure, showing that the singular continuous spectrum of \(H_\alpha\) is concentrated on a set of Lebesgue measure zero, which is independent of \(\alpha\). An interesting resolvent estimate completes this rather intricate proof. Examples of application include one of a slowly oscillating potential considered in a subsequently published article by G. Stolz: \(V(x)=\cos(2\pi x^{1-\varepsilon})\) with \(\varepsilon<1\).
For the entire collection see [Zbl 0863.00037].
Reviewer: V.Komkov (Roswell)

MSC:

34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
47E05 General theory of ordinary differential operators
47N50 Applications of operator theory in the physical sciences

Citations:

Zbl 0684.47010
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