A Second Eigenvalue Bound for the Dirichlet Schrodinger Equation with a Radially Symmetric Potential
Southwest Texas State University, Department of Mathematics
We study the time-independent Schrodinger equation with radially symmetric potential k|x|α, k ≥ 0, k ∈ ℝ, α ≥ 2 on a bounded domain Ω in ℝn, (n ≥ 2) with Dirichlet boundary conditions. In particular, we compare the eigenvalue λ2 (Ω) of the operator -Δ + k|x|α on Ω with the eigenvalue λ2(S1) of the same operator -Δ + krα on a ball S<sub>1</sub>, where S<sub>1</sub> has radius such that the first eigenvalues are the same (λ1(Ω) = λ1(S1)). The main result is to show λ2(Ω) ≤ λ2(S1). We also give an extension of the main result to the case of a more general elliptic eigenvalue problem on a bounded domain Ω with Dirichlet boundary conditions.
Schrodinger eigenvalue equation, Dirichlet boundary conditions, Eigenvalue bounds, Radially symmetric potential
Haile, C. (2000). A second eigenvalue bound for the Dirichlet Schrodinger equation wtih a radially symmetric potential. <i>Electronic Journal of Differential Equations, 2000</i>(10), pp. 1-19.