= Open-chain endpoint corrections for an exponential Ising kernel
{title2=$m^TJ_N^{-1}m=D\sum_{j=1}^{N-1}(m_{j+1}-m_j)^2+A\sum_{j=1}^Nm_j^2+B(m_1^2+m_N^2)$}
For $N\ge2$, the open-chain <Ising model> coupling $[J_N]_{ij}=J\rho^{|i-j|}$, $0<\rho<1$, has a tridiagonal inverse. Its endpoint diagonal entries are $1/[J(1-\rho^2)]$, its interior diagonal entries are $(1+\rho^2)/[J(1-\rho^2)]$, and adjacent off-diagonal entries are $-\rho/[J(1-\rho^2)]$. Direct <matrix multiplication> verifies the inverse because the geometric sequences cancel in every interior row. The displayed <quadratic form> follows with $D=\rho/[J(1-\rho^2)]$, $A=(1-\rho)/[J(1+\rho)]$ and $B=\rho/[J(1+\rho)]$. For $\rho=e^{-\kappa}$, $D=(2J\sinh\kappa)^{-1}$ and $A=\tanh(\kappa/2)/J$. Omitting the endpoint term requires an infinite chain, a correctly periodized coupling, or a bulk <thermodynamic limit>.
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