## Existence of Stark ladder resonances.(English)Zbl 0651.47006

This paper deals with a Hamiltonian of the form: $H(k,t)u(x)=- u''(x)+q(x-it)u(x)+k(x-it)u(x)$ acting in $$L^ 2(R)$$, where k, $$t\in R$$ and $$q(x)=\sum^{N}_{n=-N}c_ ne^{inx}$$ with $$c_ 0=0$$ and $$c_ n\neq 0$$ for at least one n. It is known that $$\tau_{ess}(H(k,t))\subset R-ikt$$ and that $$\tau_{disc}(H(k,t))\subset \{z:\quad -ikt\leq Im(z)\leq 0\}.$$ The present contribution shows that there exists $$k_ 0>0$$ such that for each $$k>k_ 0$$ there exists $$t\in R$$ such that H(k,t) has an eigenvalue (and by translation an infinity of eigenvalues) not on the line R-ikt. The authors conjecture that $$k_ 0=0$$.
Reviewer: C.A.Stuart

### MSC:

 47A55 Perturbation theory of linear operators 47E05 General theory of ordinary differential operators 81Q15 Perturbation theories for operators and differential equations in quantum theory