Explicit solution for the linear ODE system $p'_n = p_{n+1} + p_{n-1} - 2\,p_n$?

Question

For the one-dimensional heat equation $\partial_t \rho = \partial_{xx} \rho$ with initial condition $\rho(x,0) := \rho_0(x) \geq 0$ satisfying $\int_0^\infty \rho_0(x)\,\mathrm{d}x = 1$ and some moderate growth condition at infinity, it is well-known that the solution is given by $$\rho(x,t) = \frac{1}{4\pi t}\int_{\mathbb R} \mathrm{e}^{-\tfrac{(x-y)^2}{4t}}\rho_0(y)\,\mathrm{d}y.$$ I am wondering if there is such a nice explicit formula in the discrete analogue of the one-dimensional heat equation mentioned above. Specifically, if $\{p_n(t)\}_{n \in \mathbb Z}$ is the unique solution of $$p'_n = p_{n+1} + p_{n-1} - 2\,p_n$$ with initial data $\{p_n(0)\}_{n\in \mathbb Z}$ satisfying $p_n(0) \geq 0$ for all $n\in \mathbb Z$ and $\sum_{n \in \mathbb Z} p_n(0) = 1$ (plus some moderate growth condition so that the solution to this linear ODE system is indeed unique), does there exist an explicit representation of the solution $p_n(t)$ for all $n \in \mathbb Z$ and all $t > 0$? Thank you very much!

Answer 1

The finiteness condition you gave amounts to absolute summability, $\{p_n\}\in \ell^1(\Bbb Z)$. With that you can encode the $p_n$ as coefficients of a Fourier series, $P(s,t)=\sum_{n\in\Bbb Z} p_n(t)e^{ins}$ and the system diagonalizes/uncouples as $$\partial_tP(s,t)=(e^{is}+e^{-is}-2)P(s,t)=-2(1-\cos(s))P(s,t)$$ with solution $$P(s,t)=P(s,0)e^{-2(1-\cos(s))t}.$$

Extract the coefficient functions with the usual Fourier integrals $$ p_n(t)=\frac{1}{2\pi}\int_{-\pi}^\pi P(s,0)e^{-2(1-\cos(s))t}e^{-ins}\,ds. $$

Answer 2

Setting $S:l^1 \to l^1$, $S(x_k)_{k \in \mathbb{Z}}=(x_{k-1})_{k \in \mathbb{Z}}$ and $A:=S+S^{-1}-2I$ the solution of $p'(t)=Ap(t)$, $p(0)=q \in l^1$ is (hope I calculated the Cauchy product correctly) $$ p(t)=\exp(tA)q=\exp(-2t)\exp(tS)\exp(tS^{-1})q $$ $$ =\exp(-2t)\left(\sum_{n=0}^\infty \frac{t^n}{n!}\sum_{j=0}^n {n \choose j}q_{2j-n+k} \right)_{k \in \mathbb{Z}}. $$