Let $\sigma>0$, $$\varphi(x):=\frac1{\sqrt{2\pi\sigma^2}}e^{-\frac{x^2}{2\sigma^2}}\;\;\;\text{for }x\in\mathbb R$$ and $$\psi(x):=\sum_{k\in\mathbb Z}\varphi(k+x)\;\;\;\text{for }x\in\mathbb R.$$
Can we give a sharp bound for $\psi$ on $(-1,1)$?
I've tried to derive a bound using the inequality $$\varphi(x)\le\sqrt{\frac 2\pi}\frac\sigma{2\sigma^2+x^2}\;\;\;\text{for all }x\in\mathbb R.\tag1$$ As shown here, $(1)$ yields \begin{equation}\begin{split}\psi(x)&\le\sqrt{\frac 2\pi}\sigma\sum_{k\in\mathbb Z}\frac1{2\sigma^2+(k+x)^2}\\&=\sqrt\pi\frac{\overbrace{\sinh(2\pi\sqrt{2\sigma^2})}^{\ge\:0}}{\underbrace{\cosh(2\pi\sqrt a)}_{>\:1}-\underbrace{\cos(2\pi x)}_{\in[-1,\:1]}}\le\sqrt\pi\frac{\sinh(2\pi\sqrt{2\sigma^2})}{\cosh(2\pi\sqrt{2\sigma^2})-1}.\end{split}\tag2\end{equation}
However $(2)$ is not satisfying, since the right-hand side tends to infinity as $\sigma\to0+$.
