An Estimate of the Gap of the First Two Eigenvalues in the Schr
By Shing-Tung Yau. 2009
176 citations recorded by the source corpus.
We give a lower estimate of the gap of the first two eigenvalues of the Schrodinger operator in the case when the potential is strongly convex. In particular, if the Hessian of the potential is bounded from below by a positive constant, the gap has a lower bound independent of the dimension. We also estimate the gap when the potential is not necessarily convex.
View Paper: https://www.semanticscholar.org/paper/d666693d12573181668028496621b7df47571c04
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
- Serena Dipierro
- Enrico Valdinoci
- Stefano Biagi
- Luis Caffarelli
- P. Garain
- Xinan Ma
- Pengfei Guan
- E. Vecchi
Citations in the corpus
- A Faber-Krahn inequality for mixed local and nonlocal operators (2021, background): “We stress that in the purely local regime, the following lemma has been proved in [57] without any restriction on δ.”
- A unified approach to universal inequalities for eigenvalues of elliptic operators (2004, background): “This fact (and more) was noted by Ashbaugh and Benguria [7] for the Hile–Protter type inequality and was first observed in the case of the Payne–Pólya–Weinberger inequality by Allegretto [1], Harrell [unpublished] (in [16] the PPW type inequality (4.1) is interpreted as a family of pointwise bounds ”
- A constant rank theorem for solutions of fully nonlinear elliptic equations (2007, methodology): “In [7], the existence of convex solutions for semilinear elliptic equations in two dimensions was proved by a form of deformation lemma using the strong maximum principle (see also the work of Singer, Wong, Yau, and Yau [17] for a similar approach).”
- Isoperimetric Inequalities for Eigenvalues of the Laplacian (2007, background): “Since 1983, many authors have obtained lower bounds on this eigenvalue gap [165, 187, 123, 166].”
- The Cauchy process and the Steklov problem (2004, background): “This result has had many interesting applications in the literature and in particular it can be used to prove that for general convex domain, μ2 − μ1 > π(2)/d(2)D, [44] and [37].”