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Sharp bounds on the number of scattering poles in even-dimensional spaces. (English) Zbl 0813.35075

Scattering poles or resonances can be defined as the poles of the meromorphic continuation of the cut-off resolvent of say a Laplace operator under perturbation when such a continuation exists. For odd space dimension \((\geq 3)\) the cut off resolvent extends meromorphically to the entire complex plane while in the even dimensional case a meromorphic extension is possible only in the Riemann logarithmic surface \(\Lambda = \{Z : - \infty < \arg z < \infty\}\).
Upper bounds on the member of scattering poles in odd dimensions are derived using Jensen’s inequality on the zeros of holomorphic functions. In the odd case, given certain localization hypotheses on the operator \(G\) with resolvent \(R(Z)\), it is shown that the operator \(R_ \chi (Z) = \chi R(Z) \chi,\chi=1\) on \(B_ \rho=\{x:| x|\leq\rho\}\), extends meromorphically to \(\Lambda\) with poles \(\lambda_ j\). The scattering poles in \(\Lambda_ r = \{\Lambda_ n | z | < r\}\), with multiplicity, are among the zeros of a function \(h_ r(Z) = \text{det} (I-S_ r (Z))\), holomorphic in \(\Lambda_ r\), where \(S_ r(Z)\) is a function defined in a fairly complex way from \(R_ \chi (Z)\). Again the author establishes an analogue of Jensen’s inequality for functions holomorphic on the Riemann logarithmic surface using a generalised version of the classical Carleman theorem. As a consequence he is able to determine that the counting function \(N(r,a)\) of the scattering poles of \(G\) satisfies the bound \[ N(r,a) \leq Ca \bigl( \varphi (Cr) + (j \circ ga)^ n \bigr), \quad \forall r,\;a>1, \] with a constant \(C>0\) independent of \(r\) and \(a\), where \(\varphi\) is a function whose growth properties are related to the smoothing properties of \(G\).

MSC:

35P25 Scattering theory for PDEs
35J10 Schrödinger operator, Schrödinger equation
47A40 Scattering theory of linear operators
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