Assume a sequence of operators $A_n:L_2(\mathbb{R})→L_2(\mathbb{R}), n\in \mathbb{N}$, with the following expression:
$$A_nx(t)=e^{−(t−n)^2}x(t)$$
I'm trying to analyze its convergence, starting from the uniform convergence through
$$\|A_n - A_m \| = \sup_{\|x\| \leq 1} \|A_nx - A_mx \| = \sup_{\|x\| \leq 1} \left(\int_{\mathbb{R}}\left|e^{-(t-n)^2}x(t) - e^{-(t-m)^2} x(t)\right|^{2}dt \right)^{1/2} = \sup_{\|x\| \leq 1} \left(\int_{\mathbb{R}}x^2(t)\left[e^{-(t-n)^2} - e^{-(t-m)^2}\right]^2 dt \right)^{1/2}$$
However, I'm not sure what's the proper way to tackle the exponents in order to find a bound and see whether it converges. Drilling down and expanding the powers didn't seem to result in an obvious solution.
Would appreciate any hints or observations!