Optimal bounds on the fundamental spectral gap with single-well potentials
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- by Zakaria El Allali and Evans M. Harrell II PDF
- Proc. Amer. Math. Soc. 150 (2022), 575-587
Abstract:
We characterize the potential-energy functions $V(x)$ that minimize the gap $\Gamma$ between the two lowest Sturm-Liouville eigenvalues for \[ H(p,V) u ≔-\frac {d}{dx} \left (p(x)\frac {du}{dx}\right )+V(x) u = \lambda u, \quad \quad x\in [0,\pi ], \] where separated self-adjoint boundary conditions are imposed at end points, and $V$ is subject to various assumptions, especially convexity or having a “single-well” form. In the classic case where $p=1$ we recover with different arguments the result of Lavine that $\Gamma$ is uniquely minimized among convex $V$ by constant potentials, and in the case of single-well potentials, with no restrictions on the position of the minimum, we obtain a new, sharp bound, that $\Gamma > 2.04575\dots$.References
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Additional Information
- Zakaria El Allali
- Affiliation: Team of Modeling and Scientific Computing, Department of Mathematics and Computer, Faculty Multidisciplinary of Nador, University Mohammed Premier, Morocco
- MR Author ID: 656462
- Email: z.elallali@ump.ma
- Evans M. Harrell II
- Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160
- MR Author ID: 81525
- Email: harrell@math.gatech.edu
- Received by editor(s): August 9, 2018
- Published electronically: November 4, 2021
- Communicated by: Michael Hitrik
- © Copyright 2021 by the authors
- Journal: Proc. Amer. Math. Soc. 150 (2022), 575-587
- MSC (2020): Primary 34B27, 35J60, 35B05
- DOI: https://doi.org/10.1090/proc/14501
- MathSciNet review: 4356169