Perturbation of resonances in quantum mechanics
Let V(x) be a real-valued function on \(R^ 3\) such that \[ | V|^ 2_{\gamma}\equiv (4\pi)^{-2}\iint | V(x)V(y)| | x-y|^{-2}\exp \{2\gamma (| x| +| y|)\}dxdy<\infty \] for some \(\gamma >0\). The resolvent kernel of the Schrödinger operator \(H=-\Delta +V(x)\) with the potential V(x) is meromorphic in the half- plane Im K\(>-\gamma\). Its poles in the region Im \(K\leq 0\) are called resonances. The authors discuss the dependence of the resonances on the parameter \(\lambda\) for the operator \(H_{\lambda}=-\Delta +V(x,\lambda)\), where V(x,\(\lambda)\) is a real analytic function of \(\lambda\) in the Banach space with the norm \(| V|_{\gamma}\). They study the case when \(H_{\epsilon}=-\Delta +\sum^{n}_{i=1}\epsilon^{-2}\lambda_ i(\epsilon)V_ i(\epsilon^{-1}(x-x_ i)),\) where \(| V_ i|_ 0<\infty\), \(\lambda_ i(\epsilon)\) are differentiable in \(\epsilon\), and \(\lambda_ i(0)=1\). In this case \(H_{\epsilon}\to H_ 0\) strongly as \(\epsilon \to +0\). The cases \(n=1,2,3\) are studied in detail, the resolvent kernel of \(H_ 0\) is calculated explicitly, and the resonances are computed. Perturbation of resonances is discussed. The study of \(H_{\epsilon}\) is part of the extensive program initiated by the authors and their collaborators. A book on this topic is in preparation. Reviewer's remark: In J. Math. Phys. 21, 2052-2053 (1980; Zbl 0452.35096), and ibid. 23, 1112-1114 (1982), a variational principle for resonances was given by the reviewer; in articles by the same author [see J. Math. Anal. Appl. 86, 562-591 (1982; Zbl 0486.35065) and J. Math. Phys. 26, 1012-1020 (1985)], perturbation of resonances was studied, methods for calculating the resonances were given, and the relation between resonances and the singularity expansion method was discussed; in the reviewer, Dokl. Akad. Nauk SSSR 166, 1319-1322 (1966), a sharp logarithmic bound on the location of resonances was given.
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