Mathematics Colloquia and Seminars

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Description

Until recently Schroedinger operators with mixed absolutely continuous (a.c.) were regarded as pathological. The situation has changed with the invention of physical models (e.g. Anderson's) with such spectra, e.g. Schroedingers with non-smooth long range potentials. Until about 1996 it was even unclear if the essential spectrum of every operator -d_x^2+q( x) on L2(R) with q( x) =O( |x|^{-alpha}) ,alpha <1, has a non-empty a.c. component. This problem has drawn a considerable interest in the Schroedinger operator community. A major contribution to the study of 1D Schroedingers with slowly decaying potentials belongs to Christ-Kiselev who have succeeded in a detailed WKB-analysis of -d_x^2+q(x) under the condition q in Lp with p<2 that yields a fairly complete spectral scattering theory for such potentials. However their methods, based on harmonic analysis, brake down on L2 potentials. On the other hand, in 1999, Deift-Killip proved that if q in L2(R) then the a.c. spectrum is (0,infinity). The Deift-Killip arguments are different from Christ-Kiselev's and rely upon the so-called second Faddeev-Zhakharov trace formula. Deift-Killip's approach is elegant but does not readily yield scattering theory for such potentials. In dimension 2 and higher very little is known so far. The present talk deals with spectral/scattering theory in the L2-case. Our approach combines some of the Deift-Killip ideas and certain complex analytical arguments which make it possible to define the stationary scattering matrix by-passing the Christ-Kiselev WKB-analysis. We actually establish that the natural class here is not L2 but the Birman-Solomyak class $l2(L1) of locally integrable and globally square summable potentials. Our main contribution is a new sum rule (a stronger version of the second Faddeev-Zhakharov trace formula) which immediately yields a full description of the spectrum generalizing the relevant Killip-Simon results for Jacobi matrices to the continuum case. As by-products, we also improve on the 3/2- Lieb-Thirring inequality and put forward a complete description of reflectioness L2 potentials.