Low-lying energy bands in a finite periodic multiple-well potential

We analyze the low-lying states for a one-dimensional potential consisting of $N$ identical wells, assuming that the wells are parabolic around the minima. Matching the exact wave functions around the minima and the WKB wave functions in the barriers, we find a quantization condition which is then solved to give a formula for the energy eigenvalues explicitly written in terms of the potential. In addition, constructing $N$ localized approximate eigenstates each of which matches on to that of the harmonic oscillator in one of the parabolic wells, and diagonalizing the Hamiltonian in the subspace spanned by the localized states on the assumption that the localized states form an orthogonal basis, we also find the same formula for the energy eigenvalues which the method of matching the wave functions gives. In the large-$N$ limit, the formula reproduces, at the leading order, the expression for the widths of the narrow energy bands of the Mathieu equation present in the mathematical literature. As there are differences between the $N$-well system in the large-$N$ limit and the fully periodic system, we include a two-dimensional model in which the quadratic minima are located on the vertices of a regular $N$-sided polygon with rotational symmetry of order $N$. We argue that the lowest band of the two-dimensional model closely resembles the tight-binding energy bands of the fully periodic one-dimensional system in that most of the eigenvalues are degenerate in the large-$N$ limit with the eigenfunctions satisfying the Bloch condition under the discrete rotations.