On the spectrum of a matrix model for the D = 11 supermembrane compactified on a torus with non-trivial winding

The spectrum of the Hamiltonian of the double compactified D = 11 supermembrane with non-trivial central charge or, equivalently, the non-commutative symplectic super Maxwell theory, is analysed. In distinction to what occurs for the D = 11 supermembrane in Minkowski target space where the bosonic potential presents string-like spikes which render the spectrum of the supersymmetric model continuous, we prove that the potential of the bosonic compactified membrane with non-trivial central charge is strictly positive definite and becomes infinity in all directions when the norm of the configuration space goes to infinity. This ensures that the resolvent of the bosonic Hamiltonian is compact. We find an upper bound for the asymptotic distribution of the eigenvalues.