A continuous geometric Borg theorem on flat tori

Let V ∈ C ∞ ( T d ; R ) V\in C^\infty (\mathbb {T}^d;\mathbb {R}) , d ≥ 2 d\geq 2 , where T d = R d / ( 2 π Z ) d \mathbb {T}^d=\mathbb {R}^d/(2\pi \mathbb {Z})^d , and let \[ H ( k ) = ( − i ∇ + k ) 2 + V ( x ) , k ∈ C d , H(k)=(-i\nabla +k)^2+V(x),\qquad k\in \mathbb {C}^d, \] act on periodic functions on T d \mathbb {T}^d . We prove that the Bloch variety of H H contains the graph of an entire function f : C d → C