×

Estimates of holomorphic functions in zero-free domains. (English) Zbl 1182.30066

The authors study scalar holomorphic functions in zero-free domains and obtain some lower bounds on functions holomorphic in \(\mathbb C_+\) without zeros in the strip \(\{z\in\mathbb C : 0 < \text{Im}\,z <1\}\). The main motivation comes from the scattering theory for the wave equation in the exterior of a bounded connected domain \(K\subset\mathbb R^n\), \(n\geq 3\) odd, with smooth boundary \(\partial K\).
A lower bound for \(|f(z)|\) is obtained, which is very close to an optimal one as it is shown by an example. For functions \(f(z)\) growing as \(\mathcal O(e^{{|z|}^\beta}\)), \(1 <\beta < 2\), the result is different, and the authors study this class of functions. As the examples show, the lower bounds cannot be improved if we have zeros \(z_k\) with multiplicities \(m(z_k)\to +\infty\). In the physically important examples, the resonances and the conjugated zeros are simple. It is important to search conditions leading to lower bounds \(|f(z)|\geq e^{-a|z|}\) in zero-free domains, and this problem is also treated.
The authors examine the case \(I + B(z)\), where \(B(z)\) is a finite rank operator valued function holomorphic in \(\mathbb C_+\) such that \((I + B(z))^{-1}\) exists for \(0 \leq\text{Im}\, z \leq \delta\) and \(Image\, B(z)\subset V\) with a finite dimensional space \(V\) independent of \(z\). In particular, the authors cover the case of matrix valued functions \(a(z) :\mathbb C^m\to\mathbb C^m\) holomorphic in \(\mathbb C_+\) with \(\text{det}\,a(z)\neq 0\) for \(0 \leq\text{Im}\,z \leq\delta\). In this generality it seems that this is the first result leading to an estimate on the norm of the inverse matrix and some applications in numerical analysis could be interesting. Next the authors examine an operator valued function \(A(z)\) holomorphic in \(\mathbb C_+\), assuming that \(A(z)\) is a trace class operator for \(z\in\mathbb C_+\) and \(I + A(x)\) is unitary for \(x\in\mathbb R\). An estimate for \(\|(I +A(z))^{-1}\|\) is obtained provided that \(I + A(z)\) is invertible for \(0 < \text{Im}\, z < 1\).

MSC:

30E99 Miscellaneous topics of analysis in the complex plane
47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
35P25 Scattering theory for PDEs
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Bony J.F., Petkov V. (2006) Resolvent estimates and local energy decay for hyperbolic equations. Annali dell’Università di Ferrara - Sez. VII - Sci. Math. 52: 233–246 · Zbl 1142.35059
[2] N. V. Govorov, Riemann’s boundary problem with infinite index, Operator Theory: Advances and Applications, 67, Birkhäuser Verlag, Basel, 1994. · Zbl 0790.30027
[3] Ikawa M. (1988) Decay of solutions of the wave equation in the exterior of several convex bodies. Ann. Inst. Fourier 2: 113–146 · Zbl 0636.35045
[4] Klopp F., Zworski M. (1995) Generic simplicity of resonances. Helv. Phys. Acta 68: 531–538 · Zbl 0844.47040
[5] Lax P.D., Phillips R.S. (1989) Scattering Theory, 2nd Edition. Academic Press, New York · Zbl 0697.35004
[6] Petkov V., Stoyanov L. (1995) Sojourn times of trapping rays and the behavior of the modified resolvent of the Laplacian. Ann. Inst. H. Poincaré (Physique théorique) 62: 17–45 · Zbl 0838.35093
[7] Petkov V., Zworski M. (2001) Semi-classical estimates on the scattering determinant. Ann. Henri Poincaré 2: 675–711 · Zbl 1041.81041 · doi:10.1007/PL00001049
[8] V. Petkov, L. Stoyanov, Singularities of the scattering kernel related to trapping rays, to appear in Advances in Phase Space Analysis of Partial Differential Equations, Birkhäuser, Boston. · Zbl 1197.35183
[9] Reed M., Simon B. (1978) Methods of Modern Mathematical Physics, IV, Analysis of Operators. Academic Press, New York · Zbl 0401.47001
[10] Rudin W. (1986) Real and Complex Analysis. McGraw-Hill Book Company, New York · Zbl 0954.26001
[11] Tang S.H., Zworski M. (2000) Resonances expansions of scattering poles. Comm. Pure Appl. Math. 53: 1305–1334 · Zbl 1032.35148 · doi:10.1002/1097-0312(200010)53:10<1305::AID-CPA4>3.0.CO;2-#
[12] Titchmarsh E.C. (1976) The Theory of Functions, Second Edition. Oxford University Press, Oxford
[13] B. R. Vainberg, Asymptotic Methods in Equations of Mathematical Physics, Gordon and Breach, 1989. · Zbl 0743.35001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.