Estimates for periodic and Dirichlet eigenvalues of the Schrödinger operator

Abstract

Consider the Schrödinger equation -y '' +Vy=λy for a complex-valued potential V of period 1 in the weighted Sobolev space H w of 2-periodic functions f:ℝ→ℂ,
H w ≡H ℂ w :=f(x)=∑ k=-∞ ∞ f ^(k)e iπkx |∥f∥ w <∞,
where
∥f∥ w :=2∑ k w(k) 2 |f ^(k)| 2 1/2
and w=(w(k)) k∈ℤ denotes a symmetric, submultiplicative weight sequence. Denote by λ n =λ n (V)(n≥0) the periodic eigenvalues of -d 2 dx 2 +V when considered on the interval [0,2], listed in such a way that λ 2n ,λ 2n-1 =n 2 π 2 +0(1), and denote by μ n =μ n (V)(n≥1) the Dirichlet eigenvalues of -d 2 dx 2 +V considered on [0,1], listed in such a way that μ n =n 2 π 2 +0(1).
Theorem. There exist (absolute) constants K 1 ,K 2 >0, so that for any 1-periodic potential V in H w ,
∑ n≥N w(2n) 2 |λ 2n -λ 2n-1 | 2 ≤K 1 (1+∥V∥ w ) K 2
and ∑ n≥N w(2n) 2 |μ n -λ 2n | 2 ≤K 1 (1+∥V∥ w ) K 2 ,
where N:=K 1 (1+∥V∥ w ) 2 .