Existence of non-trivial solutions to Dirichlet problem with a potential lying between eigenvalues.

Question

I'm consideirng the example of $-\Delta u + V(x) u = 0$ in $\Omega$ with $u = 0$ on $\partial \Omega$. I'm trying to see if it's true that if $-\lambda_1 < V(x) < -\lambda_2 < 0$ on $\overline{\Omega}$ that we do not have existence of non-trivial solutions to this equation. Here $\lambda_1$ and $\lambda_2$ are two distinct eigenvalues of $-\Delta u$, both of which are of course positive.

In the special case that $V$ is constant, this is certainly true since otherwise $V$ would itself be an eigenvalue. I have tried to see if I can use a sort of comparison principle to sohw that $u \equiv 0$ if $-\Delta u + V(x)u = 0$ in the case that $V$ is not constant but I can't seem to establish this.

For simplicity of course assume that $V$ is as smooth and as bounded as you'd like.

Answer 1

I assume $\Omega$ is a bounded open subset of $\mathbb{R}^n$ and, say, $V\in L^\infty(\Omega)$. What you say is correct, but of course you need to assume that the eigenvalues are also consecutive (otherwise e.g. $V$ itself could be another eigenvalue in between [edit: as you actually said]). The comparison principle you are seeking comes from the Courant–Fischer–Weyl variational characterization of the eigenvalues of $-\Delta + V,$ by which the $k$-th eigenvalue $\lambda_k(-\Delta + V\\ )$ in the increasing order is expressed as a certain min-max of the Rayleigh quotient, which is monotone wrto $V:$ $$\frac{\int_\Omega \left(|\nabla u|^2+V(x)u^2 \right)dx }{\int_\Omega u^2 dx }$$ (for a precise statement see e.g. Courant-Hilbert, or Reed-Simon, or Gilbarg-Trudinger, &c).

So if we denote $\lambda_k:=\lambda_k(-\Delta)$ and assume $-\lambda_{k+1} < V(x) < -\lambda_{k}\\ ,$ it follows by the monotonicity $$\lambda_k(-\Delta+V)<\lambda_k(-\Delta - \lambda_k)=0 =\lambda_{k+1}(-\Delta - \lambda_{k+1})< \lambda_{k+1}(-\Delta +V),$$ so that $0$ is not an eigenvalue of $-\Delta +V.$