Setsuro Fujiie

Quantization conditions of eigenvalues for semiclassical Zakharov-Shabat systems on the circle

Bohr-Sommerfeld type quantization conditions of semiclassical eigenvalues for the non-selfadjoint Zakharov-Shabat operator on the circle are derived using an exact WKB method. The conditions are given in terms of the action associated with the circle or the action associated with turning points following the absence or presence of real turning points.

A Time-Independent Approach for the Study of the Spectral Shift Function and an Application to Stark Hamiltonians

In this paper we develop a time-independent approach for the study of the spectral shift function (SSF for short). We apply this method for the perturbed Stark Hamiltonian. We obtain a weak and a Weyl-type asymptotics with optimal remainder estimate of the SSF of the operator pair (P = P0 + V(x), P0 = − h2Δ +x1), x = (x1,…, xn) where V(x) ∈ 𝒞∞(ℝn, ℝ) decays sufficiently fast at infinity, and h is a small positive parameter. Near a non-trapping energy λ, we give a pointwise asymptotic expansions in powers of h of the derivative of the SSF, and we compute explicitly the two leading terms.