Find a sequence $(b_n)_n$ such that $\sum_n b_n = 0$ and $g(t) \equiv \sum_n e^{-2\lambda_n t}a_n^2 + \sum_n e^{-\lambda_n t} a_n b_n + \sum_n b_n^2$

Question

Fix a sequence of real numbers $(a_n)_n \in \ell^2$ and a sequence of positive numbers $(\lambda_n)_n \in \ell^2$.

Question. What is a crisp characterization of functions $g:[0,\infty) \to [0,\infty)$, is it always possible to construct a sequence of real numbers $(b_n)_n \in \ell^2$ such $\sum_n b_n = 0$ and $$ g(t) = \sum_n e^{-2\lambda_n t}a_n^2 + \sum_n e^{-\lambda_n t} a_n b_n + \sum_n b_n^2\,\,\forall t \in [0,\infty) ? $$