On Minima of Sum of Theta Functions and Application to Mueller-Ho Conjecture

Let z = x + iy is an element of H := {z = x + iy. C : y > 0} and theta(s; z) = Sigma((m,n)is an element of Z2) e(-s pi/y |mz+n|2) be the theta function associated with the lattice Lambda = Z circle plus zZ. In this paper we consider minimization problems (min)(H) theta (2; z + 1/2) + rho theta(1; z), rho is an element of [0,infinity), (min)(H) theta (1; z + 1/2) + rho theta(2; z), rho is an element of [0,infinity), where the parameter rho is an element of [0,8) represents the competition of two intertwining lattices, and the particular selection of the parameters s = 1, 2 is determined by the physical model, which can be generalized by our strategy and method proposed here. We find that as. varies, the optimal lattices admit a novel pattern: they move from rectangular (the ratio of long and short sides changes from root 3 to 1 continuously), square and rhombus (the angle changes from pi/2 to pi/3 continuously) to hexagonal continuously; geometrically, up to an invariant group (a subgroup of the classical modular group), they move continuously on a special curve; furthermore, there exists a closed interval of rho such that the optimal lattices is always a square lattice. This is the first, novel and also the complete result on the minimizer problem for theta functions with parameter rho. This is in sharp contrast to optimal lattice shapes for a single theta function (rho = 8 case), for which the hexagonal lattice prevails. As a consequence, we give a partial and positive answer to optimal lattice arrangements of vortices in competing systems of Bose-Einstein condensates as conjectured (and numerically and experimentally verified) by Mueller and Ho (Phys Rev Lett 88:180403, 2002); this is the first progress on the Mueller-Ho conjecture. Lastly, we mention that the strategy and method we propose here is general, and can be used in much more general minimization problems on the lattices.