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Modified Prüfer and EFGP transforms and the spectral analysis of one-dimensional Schrödinger operators. (English) Zbl 0912.34074
The authors discuss the spectral analysis of continuous and discrete half-time Schrödinger operators with slowly decaying potential. They investigate the spectral analysis of the operator \(H\), \[ (Hu)(x)= -{d^2\over dx^2}+ V(x) \] for the continuum case and \[ (Hu)(n)= u(n+ 1)+ u(n- 1)+ V(n) u(n) \] in the discrete case. The main theme is that there are perturbation techniques of remarkable power to control the growth of the transfer matrix. The authors show that, if \(V(x)= \sum^\infty_{n=1} a_n W(n- x_n)\), where \(W\) has a compact support and \((x_n/n_{n+ 1})\to 0\), then \(H\) has a purely a.c. (resp. purely s.c.) spectrum on \((0,\infty)\) if \(\sum a^2_n< \infty\) (resp. \(\sum a^2_n=\infty\)). For \(\lambda n^{1/2}a_n\) potentials, where \(a_n\) are independent, identically distributed random variables with \(E(a_n)= 0\), \(E(a^2_n)= 1\) and \(\lambda< 2\), they find a singular continuous spectrum with explicitly computable fractional Hausdorff dimension.

MSC:
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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