The dual eigenvalue problems for the Sturm–Liouville system

Abstract

a b s t r a c tIn this paper, we find the minimizer of the eigenvalue gap for the Schrödinger equation and vibrating string equation. In the first part, we show the first two Neumann eigenvalue gap of the Schrödinger equation with single-well potentials is not less than 1 and the equality holds if and only if the potential is constant. In the second part, since the first Neumann eigenvalue of the vibrating string equation is 0, we turn to show that the minimizing density function of the second Neumann eigenvalue is of the form hχ (a,π−a) +Hχ [0,π]\(a,π−a) for some a.

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

“…Cheng, et al [3] proves that if q is single-well (not necessarily symmetric) potential on [0, π ] with transition point a = π 2 , then the first two eigenvalues of the problem (1.1) still satisfy Eq. (1.3).…”

“…A number of studies on the optimal estimates of eigenvalue gap for the Schrödinger operators was obtained (see, e.g., [1][2][3][4][5]). In particular, Lavine [5] proves that if q is a symmetric single-well potential (q is symmetric at x = π 2 on [0, π ] and monotone decreasing on [0, π 2 ]), then the first two eigenvalues of the problem (1.1) satisfy…”

On the Neumann eigenvalues for second-order Sturm–Liouville difference equations

“…By the similar argument as the differential equations (see [3,22]), Wang and Shi showed that the periodic problem (1), (5) and the antiperiodic problem (1), (6) have exactly N real eigenvalues, while the Dirichlet problem (1), (7) has exactly N -1 real eigenvalues. Furthermore, they denoted by…”

“…and 1 0 q = M} giving the minimal Dirichlet eigenvalue gap [8]. Later on, Cheng et al [6,7] showed that if the potential function q is single-well with transition point a = 1/2, then ν 1 ≥ μ 1 and ν 1ν 0 ≥ π 2 . Equality holds if and only if q is constant.…”

The gap between the first two eigenvalues for Schrödinger operators with Dirichlet–Neumann boundary condition

“…In particular, Horváth [2] proves that if q is a single-well potential with transition point a D 2 , then the first two eigenvalues of Schrödinger operator with Dirichlet boundary conditions satisfy 2 1 3, where 1 and 2 are the first and second eigenvalues of Schrödinger operator with Dirichlet boundary conditions, respectively. Cheng et al [1] prove that if q is a single-well potential with transition point a D 2 , then the first two eigenvalues of Schrödinger operator with Neumma boundary conditions satisfy Inspired by the works of Cheng et al and Horváth [1,2], we try to calculate the gap between the first two eigenvalues for Schrödinger operators with single-well potential and Dirichlet-Neumann boundary condition. We give the bound for the gap between the first two eigenvalues of (1.1) (Theorem 1.1).…”