On Double Eigenvalues of Schrِdinger Operators on Two-Dimensional Tori

An abstract result concerning double eigenvalues of seif-adjoint operators is presented. It is applied to Schrödinger operators −Δ + V on a generic flat torus R2/Γ with V∈C∞(R/Γ): Given N ≥ 1 and ϵ > 0 there exists a potential W in C∞(R2,Γ) such that (i) −Δ + V + W has at least N double eigenvalues and (ii) ||W||L∞ ≤ ϵ. It is also explained why a similar result for Schrödinger operators on S2 is unlikely to hold.