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**An estimate of the gap of the first two eigenvalues in the Schrödinger operator.**
*(English)*
Zbl 0603.35070

Let \(\Omega \subset\mathbb R^n\) be a bounded convex domain with a smooth boundary and \(V: {\bar \Omega}\to\mathbb R^a \)nonnegative convex potential. Denote by \(\lambda_ 1\) and \(\lambda_ 2\) the first and second eigenvalue of the Schrödinger operator \(-\Delta +V\) with Dirichlet boundary condition. In their remarkable paper the authors show that
\[
(\pi /2d)^ 2\leq \lambda_ 2-\lambda_ 1\leq n(2\pi /D)^ 2+(4/n)(\sup\, V-\inf\, V) \tag{*}
\]
where \(d:=\text{diam}\, \Omega\) and \(D\) is the diameter of the largest ball inscribed in \(\Omega\). The lower bound in (*) is of particular interest; the upper bound holds under more general circumstances and does not require \(\Omega\) to be convex. In proving the lower bound the authors seem to use more than the convexity of \(\Omega\), viz. that \(H\), the second fundamental form of \(\partial \Omega\) relative to an outward unit normal, is positive definite in the strict sense (see, in particular, Proposition 2.3). (\(\Omega\) is convex if and only if \(H\) is positive semi-definite.) Their proof rests on deriving an upper bound for \(| \nabla u|^ 2+(\lambda_ 2-\lambda_ 1)(\mu -u)^ 2\) where \(u:=f_ 2/f_ 1\) is the ratio of the first and second eigenfunction and \(\mu\geq \sup u\).

In order to follow the argument it is helpful also to consult the paper by Peter Li and the third author [Proc. Symp. Pure Math. 36, 205–239 (1980; Zbl 0441.58014)], in particular since formula (2.13) is misleading.

In an appendix the authors show the smoothness of \(u\) up to the boundary using the Malgrange preparation theorem, although this result could have been arrived at with less sophisticated means. The second part of the appendix is devoted to an elementary proof of a result of H. J. Brascamp and E. H. Lieb [J. Funct. Anal. 22, 366–389 (1976; Zbl 0334.26009)] that \(\log f_ 1\) is logarithmically concave, but the argument presented here does not appear to be conclusive. (There is a gap in the penultimate section of p. 332.)

{Reviewer’s remarks: Inspired by the paper under review, Qihuang Yu and Jia-Qing Zhong recently improved the lower bound in (*) by a factor of 4 [Trans. Am. Math. Soc. 294, 341–349 (1986; Zbl 0593.53030)]. These authors assume that \(\Omega\) is bounded and strictly convex, but it appears that the above-mentioned condition on \(H\) is again required (cf. p. 343).

A lower bound on \(\lambda_ n-\lambda_{n-1}\), the difference of two successive eigenvalues of the one-dimensional Schrödinger operator with Dirichlet boundary conditions, was recently given by W. Kirsch and B. Simon [Commun. Math. Phys. 97, 453–460 (1985; Zbl 0579.34014)] [see also W. Kirsch, On the difference between eigenvalues of Sturm-Liouville operators and the semiclassical limit, preprint Nr. 73/1986, University of Bochum]. In the higher-dimensional case \(\lambda_ 2-\lambda_ 1\) was estimated by W. Kirsch and B. Simon [Comparison theorems for the gap of Schrödinger operators (Caltech preprint 1986)].

The reviewer benefited greatly from discussions with Messrs. Hempel and Schindelbeck, Munich.}

In order to follow the argument it is helpful also to consult the paper by Peter Li and the third author [Proc. Symp. Pure Math. 36, 205–239 (1980; Zbl 0441.58014)], in particular since formula (2.13) is misleading.

In an appendix the authors show the smoothness of \(u\) up to the boundary using the Malgrange preparation theorem, although this result could have been arrived at with less sophisticated means. The second part of the appendix is devoted to an elementary proof of a result of H. J. Brascamp and E. H. Lieb [J. Funct. Anal. 22, 366–389 (1976; Zbl 0334.26009)] that \(\log f_ 1\) is logarithmically concave, but the argument presented here does not appear to be conclusive. (There is a gap in the penultimate section of p. 332.)

{Reviewer’s remarks: Inspired by the paper under review, Qihuang Yu and Jia-Qing Zhong recently improved the lower bound in (*) by a factor of 4 [Trans. Am. Math. Soc. 294, 341–349 (1986; Zbl 0593.53030)]. These authors assume that \(\Omega\) is bounded and strictly convex, but it appears that the above-mentioned condition on \(H\) is again required (cf. p. 343).

A lower bound on \(\lambda_ n-\lambda_{n-1}\), the difference of two successive eigenvalues of the one-dimensional Schrödinger operator with Dirichlet boundary conditions, was recently given by W. Kirsch and B. Simon [Commun. Math. Phys. 97, 453–460 (1985; Zbl 0579.34014)] [see also W. Kirsch, On the difference between eigenvalues of Sturm-Liouville operators and the semiclassical limit, preprint Nr. 73/1986, University of Bochum]. In the higher-dimensional case \(\lambda_ 2-\lambda_ 1\) was estimated by W. Kirsch and B. Simon [Comparison theorems for the gap of Schrödinger operators (Caltech preprint 1986)].

The reviewer benefited greatly from discussions with Messrs. Hempel and Schindelbeck, Munich.}

Reviewer: Hubert Kalf

### MSC:

35P15 | Estimates of eigenvalues in context of PDEs |

35J10 | Schrödinger operator, Schrödinger equation |

35A30 | Geometric theory, characteristics, transformations in context of PDEs |

### Keywords:

bounded convex domain; smooth boundary; nonnegative convex potential; first and second eigenvalue; Schrödinger operator; Dirichlet boundary condition; lower bound; upper bound; Malgrange preparation theorem
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\textit{I. M. Singer} et al., Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 12, 319--333 (1985; Zbl 0603.35070)

### References:

[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 · Zbl 0334.26009 · doi:10.1016/0022-1236(76)90004-5 |

[2] | S.Y. Cheng , Eigenvalue comparison theorems and its geometric applications , Math. Z. , 143 ( 1975 ), pp. 289 - 297 . Article | Zbl 0329.53035 · Zbl 0329.53035 · doi:10.1007/BF01214381 |

[3] | Courant - Hilbert , Method of Mathematical Physics , Vol. I . Zbl 0051.28802 · 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 · Zbl 0441.58014 |

[5] | B. Malgrange , Ideals of differentiable functions , Oxford University Press , 1966 . Zbl 0177.17902 · 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 · Zbl 0073.08203 |

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

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