The gap between the first two eigenvalues of Schrödinger operators with single-well potential

Optimal bounds on the fundamental spectral gap with single-well potentials

“…Later, in 2002, Horváth [10] returned with Lavine's methods to the problem of single-well potentials, without symmetry assumptions, but assuming a transition point at π 2 , and showed that the gap for the Dirichlet problem is minimized when the potential is constant. In 2015 Yu and Yang [13] extended Horváth's result by allowing other transition points (under a technical condition) and both Dirichlet and Neumann conditions. In this article, we provide lower bounds for the gap between the first two eigenvalues of the problems (1) with general single-well potential V (x) with a transition point a ∈ [0, π], without any restriction on a, and also for the case where the potential is convex.…”

“…In contrast to the earlier studies of single-well potentials, which restrict the transition point in one way or other, the minimizing potentials we find are not in general constant, although if extra conditions are imposed locating the transition point sufficiently far from 0 or π, our methods can lead to constant potentials. Although we do not pursue this idea here in detail, this remark is a way of understanding the results in [1,10,13].…”

The eigenvalue ratio of the vibrating strings with mixed boundary condition

“…The eigenvalue problems of Schrödinger operator and Sturm-Liouville operator, including the eigenvalue gap and the eigenvalue ratio problems, have been developed for many years. For the discussion for eigenvalue gap, we can refer to [2][3][4][5][6]. Especially, for the eigenvalue ratio problem of one-dimensional Schrödinger operator, Ashbaugh and Benguria [7] considered the ratio problem under Dirichlet boundary conditions; the optimal bound…”

“…The eigenvalue problems of Schrödinger operator and Sturm–Liouville operator, including the eigenvalue gap and the eigenvalue ratio problems, have been developed for many years. For the discussion for eigenvalue gap, we can refer to [2–6]. Especially, for the eigenvalue ratio problem of one‐dimensional Schrödinger operator, Ashbaugh and Benguria [7] considered the ratio problem under Dirichlet boundary conditions; the optimal bound $$$$ \frac{\lambda_n}{\lambda_1}\le {n}^2,n=2,3,4,\cdots $$$$ is obtained by utilizing the modified Prüfer transformation, and later they applied the same method to the regular Sturm–Liouville problem $$$$ -{\left[p(x){y}^{\prime}\right]}^{\prime }+q(x)y=\lambda w(x)y,0<k\le p(x)w(x)\le K, $$$$ and obtained $$$ \frac{\lambda_n}{\lambda_1}\le \frac{K}{k}{n}^2 $$$; the equality holds if and only if $$$ q(x)=0 $$$ and $$$ p(x)w(x)=k=K $$$ [8].…”

The fundamental gap of a kind of sub-elliptic operator

“…In particular, note that the fundamental gap bound related to single-well potential with transition point as midpoint is considered in [16], and the symmetric potential is imposed by various constraints in [17]. Yu [36] introduced the bound for the fundamental gap under Dirichlet and Neumann boundary condition, where the single-well potential with transition point to be not midpoint. Andrews and Clutterbuck [3] solved the gap conjecture via an ingenious way, where is the diameter of domain and the potential is weakly convex.…”