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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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