Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia
Academic Editor: Manfred H. Moller
Copyright © 2009 Peter Zhidkov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We consider the following eigenvalue problem: , , , , where is an arbitrary fixed parameter and is an odd smooth function. First, we prove that for each integer there exists a radially symmetric eigenfunction which possesses precisely zeros being regarded as a function of . For sufficiently small, such an eigenfunction is unique for each . Then, we prove that if is sufficiently small, then an arbitrary sequence of radial eigenfunctions , where for each the th eigenfunction possesses precisely zeros in , is a basis in ( is the subspace of that
consists of radial functions from . In addition, in the latter case, the sequence is a Bari basis in the same space.