Consider a Schrödinger operator
$$H=-\Delta+V$$
on a *nice* bounded domain $\Omega\subset\mathbb R^d$ (say, a ball or a cube), and assume for simplicity that $V$ is smooth.

Let $\lambda_D,\lambda_N$ and $\lambda_P$ denote the ground state eigenvalue of $H$ with Dirichlet, Neumann and periodic boundary conditions respectively. By using the variational characterization of the eigenvalues, it is sometimes possible to get trivial one-sided inequalities such as $$\lambda_P\leq\lambda_D.$$ I'm interested in more detailed comparisons between these eigenvalues:

Question.Does there exist results giving bounds on the distances $$|\lambda_P-\lambda_D|,~|\lambda_N-\lambda_D|,~|\lambda_N-\lambda_P|,$$ probably depending on $\Omega$ and $V$?

My interest in this question stems from the following kind of problem: Suppose we have an asymptotic result on the magnitude of $\lambda_D$ on the open ball $B(0,r)$ as $r\to\infty$. Is it sometimes possible to infer from this that $\lambda_N$ and $\lambda_P$ have the same asymptotic behavior?