A note on the absence of resonances for Schrödinger operators. (English) Zbl 0688.35073

It is well-known that the Schrödinger operator \(H:=-\Delta +V\) in \(L^ 2({\mathbb{R}}^ n)\) has no eigenvalues larger than a given number \(\Lambda\) if \[ (*)\quad 2(V(x)-\Lambda)+x\cdot (\nabla V)(x)\leq 0\quad (x\in {\mathbb{R}}^ n) \] [cf., for example, M. S. P. Eastham and the reviewer, Schrödinger-type operators with continuous spectra (1982; Zbl 0491.35003)], and this condition admits of an interpretation in terms of notions from classical mechanics. M. Klein [Commun. Math. Phys. 106, 485-494 (1986; Zbl 0651.47007)] showed that an assumption similar to (*) can be used to rule out resonances of H in a certain energy interval. In the present paper Klein’s result is generalized by introducing a further modification of (*). To obtain the required resolvent estimate, a special case of Hörmander’s construction of a parametrix for elliptic pseudodifferential operators is used. As an additional reference to the circle of questions involved we mention a paper by H. L. Cycon and C. Erdmann [Rep. Math. Phys. 23, 169-176 (1986; Zbl 0646.35017)].
Reviewer: H.Kalf


35P99 Spectral theory and eigenvalue problems for partial differential equations
35J10 Schrödinger operator, Schrödinger equation
47B25 Linear symmetric and selfadjoint operators (unbounded)
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