A polynomial bound on the number of the scattering poles for a potential in even dimensional spaces \({\mathbb{R}}^ n\). (English) Zbl 0607.35069
This paper is devoted to the proof of the following estimate: Let n be even, \(n\geq 4\) and let \(V\in {\mathfrak C}_ 0^{\infty}({\mathbb{R}}^ n)\). Let S(\(\lambda)\) be the scattering matrix associated with the Schrödinger operator \(H=-\Delta +V\). It is known that S(\(\lambda)\) is a meromorphic function of Log \(\lambda\) on the Riemann surface of the logarithm. Then the number \(N(T^{\epsilon})\) of poles of S(\(\lambda)\) in the region \(T^{-\epsilon}\leq | \lambda | \leq T^{\epsilon},| \arg \lambda | \leq Log T^{\epsilon}\) is estimated, for any \(\epsilon \in [0,2^{-1/2})\) by \(N(T^{\epsilon})\leq C_{\epsilon}(1+T)^{n+1}\). That result extends an earlier result of a similar nature proved by Melrose for the odd dimensional case. The proof consists in relating the poles of S(\(\lambda)\) to those of \((\mathbf{1} -K(\lambda))^{-1}\) where K(\(\lambda)\) is a modified Lippmann Schwinger operator, by standard scattering theory arguments, of relating the poles of the latter to the zeroes of a suitable Fredholm determinant h(Log \(\lambda)\), of showing that h(z) is an entire function of z and of estimating its growth rate at infinity by the use of asymptotic estimates of the characteristic values of K(\(\lambda)\) for large \(\lambda\). The continuation of K(\(\lambda)\) and the latter estimates make use of the Hankel function representation of the free resolvent.
Reviewer: J.Ginibre
MSC:
35P25 | Scattering theory for PDEs |
35J10 | Schrödinger operator, Schrödinger equation |
81U05 | \(2\)-body potential quantum scattering theory |
47A40 | Scattering theory of linear operators |
Keywords:
scattering matrix; Schrödinger operator; Riemann surface; Lippmann Schwinger operator; Hankel function representation; free resolventReferences:
[1] | Calderón A. P., Proc. Nat. Acad. Sci 69 (1972) |
[2] | Dunford N., Interscience (1963) |
[3] | Gohberg I. C., Transl. Math. Monogr 18 (1968) |
[4] | Hörmander L., Comm. Pure Appl. Math 24 (1971) |
[5] | Jensen J. L. W. V., Acta Mathema-tica 22 (1899) |
[6] | DOI: 10.1002/cpa.3160120302 · Zbl 0091.09502 · doi:10.1002/cpa.3160120302 |
[7] | Lax P. D., Indiana J. of Math (1972) |
[8] | DOI: 10.1016/0022-1236(83)90036-8 · Zbl 0535.35067 · doi:10.1016/0022-1236(83)90036-8 |
[9] | Growth estimate for the poles in potential scattering ,priprint |
[10] | Nikiforov, A. and Duvarov, V. 1976. ”Elements de la theorie des fonctions speciales”. |
[11] | DOI: 10.1016/0022-247X(72)90289-2 · Zbl 0229.35072 · doi:10.1016/0022-247X(72)90289-2 |
[12] | DOI: 10.1016/0022-247X(71)90007-2 · Zbl 0249.47004 · doi:10.1016/0022-247X(71)90007-2 |
[13] | DOI: 10.1007/BF00251419 · Zbl 0167.43002 · doi:10.1007/BF00251419 |
[14] | Watson G., A treatise on the theory of Bessel functions (1966) · Zbl 0174.36202 |
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