Trace Formulae and High Energy Asymptotics for the Stark Operator

Abstract

In L 2 (R 3 ), we consider the unperturbed Stark operator H 0 (i.e., the Schrödinger operator with a linear potential) and its perturbation H = H 0 + V by an infinitely smooth compactly supported potential V . The large energy asymptotic expansion for the modified perturbation determinant for the pair (H 0 , H) is obtained and explicit formulae for the coefficients in this expansion are given. By a standard procedure, this expansion yields trace formulae of the Buslaev-Faddeev type.

Generalized scattering phases for asymptotically hyperbolic manifolds

“…More precisely, these asymptotics are of the same type as those obtained in Euclidean scattering by [41,34,46,47,10,52,33] for q = 1 and [6,7] for q ≥ 2. See also [10,11,9,19] and [26] in more geometric frameworks.…”

Trace Formulas for Schrödinger Operators with Complex Potentials

“…The trace formulas similar to (1.24), (1.25) were proved by Buslaev [B66] for real potentials, see also [C81] and [G85, P82, R91]. Trace formulas for the case Stark operators and magnetic Schrödinger operators are discussed in [KP03], [KP04]. Trace formulas for Schrödinger operators on the lattice are considered in [IK12] and for the case of complex potentials in [K17], [KL16].…”

Resonances for 1d Stark operators

“…If V is, e.g. in the Schwartz class, then the expansion becomes much easier and higher order terms can be derived as well (see [20]), similar to the 3-dim case in [31]. Under Condition V we will obtain an analytic continuation of D + (λ), λ ∈ C + to the entire complex plane and information on its zeros and obtain upper bounds on the number of resonances of the operator H. We denote by (λ n ) ∞ 1 the sequence of zeros in C − of D + (counting multiplicities), arranged such that 0 < |λ 1 | |λ 2 | |λ 2 | .…”