ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

11. Mathematical Physics

1991 MS Classification: 47A10, 81Q10 Demuth, Michael, TU Clausthal, Germany Stable absolutely and singularly continuous spectra due to integral conditions Let A, B be selfadjoint operators in L^{2}(R^{n}) such that their associated semigroups e^{−tA}, e^{−tB} are integral operators. Denote by D = e^{−A} − e^{−B}, the difference of the semigroups at t = 1, with the kernel D(…) mapping R^{n} ×R^{n} to R. Then the absolutely continuous spectra of A and B coincide if ∫_{R^{n}} ∫_{R^{n}} |D(x, y)|dxdy < ∞. Moreover, if A is the Laplacian the singularly

continuous spectrum of B is also empty if sup_{|x|≥1} |x|^{α} ∫_{R^{n}} |D(x, y)|dy < ∞ with

α > 1. The results are obtained in collaboration with S. Eder (Clausthal) and K. B. Sinha (New Delhi).