Let us consider $V(x) = 2-\sin(x) - \sin(\sqrt{2} x)$ on $x\in \mathbb{R}$ so that $V(x)>0$ everywhere. One can see that $$ \frac{C_1}{t^2} \leq \min_{|x|\leq t} V(x)\leq \frac{C_2}{t^2} $$ for all $t\in \mathbb{R}$. Can we get an asymptotic expansion of $\min_{|x|\leq t} V(x)$ as an exact form $$ \min_{|x|\leq t} V(x) \sim \frac{c}{t^2} \qquad \text{as}\; t\to\infty? $$

Let us consider $V(x) = 2-\sin(x) - \sin(\sqrt{2} x)$ on $x\in \mathbb{R}$ so that $V(x)>0$ everywhere. One can see that $$ \frac{C_1}{t^2} \leq \min_{|x|\leq t} V(x)\leq \frac{C_2}{t^2} $$ for all $t > 0$. Can we get an asymptotic expansion of $\min_{|x|\leq t} V(x)$ as an exact form $$ \min_{|x|\leq t} V(x) \sim \frac{c}{t^2} \qquad \text{as}\; t\to\infty? $$