Estimates for Periodic and Dirichlet Eigenvalues of the Schrِdinger Operator with Singular Potentials

In this paper, the periodic and the Dirichlet problems for the Schrödinger operator −(d2/dx2)+V are studied for singular, complex-valued potentials V in the Sobolev space H−αper[0, 1] (0⩽α<1). The following results are shown: (1)The periodic spectrum consists of a sequence (λk)k⩾0 of complex eigenvalues satisfying the asymptotics (for any ε>0)λ2n−1, λ2n=n2π2+V(0)±V(−2n) V(2n)+O(n3α/2−1/2+ε), where V(k) denote the Fourier coefficients of V. (2)The Dirichlet spectrum consists of a sequence (μn)n⩾1 of complex eigenvalues satisfying the asymptotics (for any ε>0)μn=n2π2+V(0)−V(−2n)+V(2n)2+O(n2α−1+ε).