A Second Eigenvalue Bound for the Dirichlet Schrodinger Equation with a Radially Symmetric Potential
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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 S1, where S1 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.