The spectrum of the torus profile to a geometric variational problem with long range interaction

The profile problem for the Ohta-Kawasaki diblock copolymer theory is a geometric variational problem. The energy functional is defined on sets in R3 of prescribed volume and the energy of an admissible set is its perimeter plus a long range interaction term related to the Newtonian potential of the set. This problem admits a solution, called a torus profile, that is a set enclosed by an approximate torus of the major radius 1 and the minor radius q. The torus profile is both axially symmetric about the z axis and reflexively symmetric about the xy-plane. There is a way to set up the profile problem in a function space as a partial differential-integro equation. The linearized operator L of the problem at the torus profile is decomposed into a family of linear ordinary differential-integro operators Lm where the index m=0,1,2,… is called a mode. The spectrum of L is the union of the spectra of the Lm's. It is proved that for each m, when q is sufficiently small, Lm is positive definite. (0 is an eigenvalue for both L0 and L1, due to the translation and rotation invariance.) As q tends to 0, more and more Lm's become positive definite. However no matter how small q is, there is always a mode m of which Lm has a negative eigenvalue. This mode grows to infinity like q−3/4 as q→0.