Resonances for 1d Stark operators

Abstract

Abstract. We consider the Stark operator perturbed by a compactly supported potential (of a certain class) on the real line. We prove the following results: (a) upper and lower bounds on the number of resonances in complex discs with large radii, (b) the trace formula in terms of resonances only, (c) all resonances determine the potential uniquely.

Asymptotics of resonances for 1D Stark operators

“…The operators H 0 and H are self-adjoint on the same domain since the operator V (H 0 − i) −1 is compact, see e.g. [28,12,24,21]. The spectrum of both H 0 and H is purely absolutely continuous and covers the real line R (see [28,12,24]).…”

“…Introduce the space L p (R) equipped by the norm f p = ( R |f (x)| p dx) 1 p 0 and let f 2 = f 2 2 . Recall results from [21]. Let V satisfy (1.1).…”

“…Resonances for the operator H were discussed in [21], where a central role was played by the Fredholm determinant. We recall some results from [21]. Under condition (1.1) each operator Y 0 (λ), Im λ = 0, is trace class and thus we can define the determinant:…”

“…The zeros of S(λ), λ ∈ C + coincide with the zeros of D − and the poles of S(λ), λ ∈ C − are precisely the zeros of D + . In [21] we defined the resonances of the perturbed Stark operator H as the zeros of the analytic continuation of the determinant D + (λ), λ ∈ C + in the lower half-plane C − . Due to (2.14) the sets of zeros of D ± in C ∓ are symmetric with respect to the real line.…”

Sharp bounds for finitely many embedded eigenvalues of perturbed Stark type operators

“…The Stark operator describes a charged quantum particle in a constant electric field with an additional electric potential q . It has attracted a lot of attentions from both mathematics and physics [1, 2, 4, 6, 8, 9, 11, 13, 20, 21, 23, 24, 40, 42].…”