In our final section we place stronger restrictions on the choice of potential, we assume that they are Gaussian. Although this appears to be restrictive, we may approximate general potentials by Gaussians. When the potentials are chosen to have this nice form the problem can be rephrased in terms of linear operators, which can be handled via discrete Fourier analysis.

In this section we consider Hamiltonians of the form

\begin{equation} H(u)=\sum_ja_1(u_j-u_{j-1}+x)^2+4a_2\left(\frac{u_{j-1}-u_{j+1}}{2}+x\right)^2,\end{equation}

where \(V_1(r)=a_1r^2\), \(V_2(r)=a_2r^2\) are Gaussian potentials, and the factor 4 is motivated by Taylor expanding a general potential.

Writing in matrix form and diagonalising, we obtain the following form for the Hamiltonian

\begin{equation} H = U^T Y^T\left( a_1 \Psi_1 + a_2\Psi_2 \right) Y U+ (a_1 + 4a_2)(N+1)x^2,\end{equation}

where \(\Psi_1\) and \(\Psi_2\) are diagonal matrices consisting of eigenvalues of the linear operators. Spectral analysis reveals

\begin{equation} H=\sum_{j=0}^N\left(a_1\lambda^{(1)}_j+a_2\lambda^{(2)}_j\right)w_j^2+(a_1+4a_2)(N+1)x^2 ,\end{equation}

where \(\lambda^{(1)}_j=sin^2\left(\pi y_j\right)\), \(\lambda^{(2)}_j=sin^2\left(2\pi y_j\right)\) and \(y_j=\frac{j}{N+1}\). Substituting into free energy and simplifying we obtain the following theorem.

Theorem
The mixed next nearest neighbour interaction model on the torus with potentials \(V_1(r)=a_1r^2, V_2(r)=a_2r^2\), where \(min\{a_1,a_1+4a_2\}>0\), has free energy \begin{equation}f(x)=(a_1+4a_2)x^2-\frac{1}{2\beta}\log\frac{\pi}{\beta}+\frac{1}{2\beta}\lim_{N\to\infty}\sum_{j=1}^N\log\left(4a_1\sin^2\left(\pi y_j\right)+4a_2\sin^2\left(2\pi y_j\right)\right)\Delta y .\end{equation} where we set \(y_j=\frac{j}{N+1}\) for \(0 \leq j \leq N\) and \(\Delta y=\frac{1}{N+1}\).

We may compute the summation explicitly by developing the summands with double angle formulae, so that we obtain quadrature forms for which we may establish convergence.

Theorem

For Gaussian potentials \(V_1(r)=a_1r^2\), \(V_2(r)=a_2r^2\), with \(\min\{a_1,a_1+4a_2\}>0\), the gradient model with periodic boundary conditions has free energy
\begin{equation*}
f(x) = (a_1+4a_2)x^2-\frac{1}{2\beta}\log\frac{\pi}{\beta}+\frac{1}{2\beta}\int_0^1\log\left(4a_1\sin^2\left(\pi t\right)+4a_2\sin^2\left(2\pi t\right)\right) dt.
\end{equation*}
Moreover, the rate of convergence is \(O ( N^{-1} \log N) \).