Schrodinger Type Operators with Continuous Spectra

✍ Scribed by M.S.P. Eastham, H. Kalf

• Publisher: Pitman Publishing
• Year: 1982
• Tongue: English
• Leaves: 298
• Series: Chapman & Hall/CRC Research Notes in Mathematics Series
• Category: Library

✦ Synopsis

In 1929 J. v. Neumann and Wigner, in a short paper entitled "Uber merkwUrdige
diskrete Eigenwerte", drew attention to the fact that the Schr6dinger equat-
ion

• u + qu = AU

( 1 )

may give rise to an eigenvalue which is embedded in a "continuous ll spectrum.
Physically this would correspond to the strange situation that a particle
remained in a stationary state although having enough energy to escape the
attraction of the surrounding system. Thus one faces the problem of finding
natural conditions on the potential q such that (1) has no non-trivial solu-
tion of integrable square when A belongs to a certain interval.
As it was with other specific problems concerning the nature of the spec-
trum of Schr6dinger operators, this problem was not attacked until the late
19405, first in the one-dimensional and then in the general case. The pro-
blem turns out to be a very intricate one and it has attracted many mathe-
maticians over the years as our list of references documents. By and large
it can now be considered as solved and we have tried in this book to give a
connected account of most of the results which have been obtained. In sev-
eral instances we have rounded off results or simplified their proof in some
detail. A few results are presented here for the first time. Clearly there
are also questions which have not found an answer yet and these are mentioned
as the occasion arises. Here we note that the most conspicuous of these. It
is unknown, even in one dimension, whether every potential q which is