An estimate of the gap of the first two eigenvalues in the Schrdinger operator

IM Singer Bun Wong Shing-Tung Yau Stephen S-T Yau

Mathematical Physics mathscidoc:1912.43485

Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 12, (2), 319-333, 1985
We shall consider the following Dirichlet eigenvalue problem on a smooth bounded domain S~ eRn, I where V is a nonnegative function defined on,~. As is well-known, the eigenvalues of problem (1.1) can be interpreted as the energy levels of a particle travelling under an external force field of a potential q in Rn, where and the corresponding eigenfunctions are wave functions of the Schrodinger equation-J~+ qu= lu. Furthermore, the set of eigenvalues {A,} of (1.1) are nonnegative and can be arranged in a nondecreasing order as follows, It is a significant problem to find a lower bound for~, 1 in terms of the geometry of Q. This subject has been studied extensively by many authors. A rather precise bound in the case V== 0 was worked out not only for a bounded domain in but actually valid for a general Riemannian manifold with certain curvature conditions; we refer to [4] for these recent develop-ments. Nevertheless, very little is known about the obvious interesting question of how big the gap is between 2 and l. There are both physical
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@inproceedings{im1985an,
  title={An estimate of the gap of the first two eigenvalues in the Schrdinger operator},
  author={IM Singer, Bun Wong, Shing-Tung Yau, and Stephen S-T Yau},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224203539461906049},
  booktitle={Annali della Scuola Normale Superiore di Pisa-Classe di Scienze},
  volume={12},
  number={2},
  pages={319-333},
  year={1985},
}
IM Singer, Bun Wong, Shing-Tung Yau, and Stephen S-T Yau. An estimate of the gap of the first two eigenvalues in the Schrdinger operator. 1985. Vol. 12. In Annali della Scuola Normale Superiore di Pisa-Classe di Scienze. pp.319-333. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224203539461906049.
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