An estimate of the gap of the first two eigenvalues in the Schrödinger operator
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 4, Volume 12 (1985) no. 2, p. 319-333
@article{ASNSP_1985_4_12_2_319_0,
     author = {Singer, Isadore M. and Wong, Bun and Yau, Shing-Tung and Yau, Stephen S.-T.},
     title = {An estimate of the gap of the first two eigenvalues in the Schr\"odinger operator},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola normale superiore},
     volume = {Ser. 4, 12},
     number = {2},
     year = {1985},
     pages = {319-333},
     zbl = {0603.35070},
     mrnumber = {829055},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_1985_4_12_2_319_0}
}
Singer, I. M.; Wong, Bun; Yau, Shing-Tung; Yau, Stephen S.-T. An estimate of the gap of the first two eigenvalues in the Schrödinger operator. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 4, Volume 12 (1985) no. 2, pp. 319-333. http://www.numdam.org/item/ASNSP_1985_4_12_2_319_0/

[1] Brascamp - Lieb, On extensions of the Brunn-Minkowski and prékopa-Leindler theorems, including inequalities for Log concave functions, and with an application to Diffusion equation, Journal of Functional Analysis, 22 (1976), pp. 366-389. | Zbl 0334.26009

[2] S.Y. Cheng, Eigenvalue comparison theorems and its geometric applications, Math. Z., 143 (1975), pp. 289-297. | Zbl 0329.53035

[3] Courant-Hilbert, Method of Mathematical Physics, Vol. I. | Zbl 0051.28802

[4] P. Li - S.T. Yau, Estimate of eigenvalues of a compact Riemannian manifold, Proc. Symp. Pure Math., 36 (1980), pp. 205-240. | Zbl 0441.58014

[5] B. Malgrange, Ideals of differentiable functions, Oxford University Press, 1966. | Zbl 0177.17902

[6] Payne- Polya-Weinberger, On the ratio of consecutive eigenvalues, Journal of Math. and Physics, 35, No. 3 (Oct. 1956), pp. 289-298. | MR 84696 | Zbl 0073.08203

[7] B. Simon, The P(ϕ)2 Euclidean quantum field theory, Princeton Series in Physics. | Zbl 1175.81146