= High-energy shift from a central delta barrier
{title2=$E_n-E_n^{(0)}\to\kappa/a$}
Put $c=m\kappa/\hbar^2$, $k_n=(n+1/2)\pi/a$, and $\delta_n=a(\lambda_n-k_n)$. The quantization equation gives $\delta_n=\arctan(c/\lambda_n)$, so $\lambda_n\delta_n\to c$ and $\delta_n\to0$. Therefore $\lambda_n^2-k_n^2\to2c/a$, proving the displayed absolute <energy> shift. Its ratio to the unperturbed energy tends to zero. The normalized unperturbed even state has probability density $1/a$ at the origin, so its first-order delta-barrier expectation is also $\kappa/a$.
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