## Optimal lower bound for the gap between the first two eigenvalues of one-dimensional Schrödinger operators with symmetric single-well potentials

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- by Mark S. Ashbaugh and Rafael Benguria
- Proc. Amer. Math. Soc.
**105**(1989), 419-424 - DOI: https://doi.org/10.1090/S0002-9939-1989-0942630-X
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## Abstract:

We prove the optimal lower bound ${\lambda _2} - {\lambda _1} \geq 3{\pi ^2}/{d^2}$ for the difference of the first two eigenvalues of a one-dimensional Schrödinger operator $- {d^2}/d{x^2} + V(x)$ with a symmetric single-well potential on an interval of length $d$ and with Dirichlet boundary conditions. Equality holds if and only if the potential is constant. More generally, we prove the inequality ${\lambda _2}[{V_1}] - {\lambda _1}[{V_1}] \geq {\lambda _2}[{V_0}] - {\lambda _1}[{V_0}]$ in the case where ${V_1}$ and ${V_0}$ are symmetric and ${V_1} - {V_0}$ is a single-well potential.## References

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## Bibliographic Information

- © Copyright 1989 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**105**(1989), 419-424 - MSC: Primary 81C05; Secondary 34B25
- DOI: https://doi.org/10.1090/S0002-9939-1989-0942630-X
- MathSciNet review: 942630