On a Riesz basis of exponentials related to the eigenvalues of an analytic operator and application to a non-selfadjoint problem deduced from a perturbation method for sound radiation. (English) Zbl 1302.42016
Summary: We prove that the family of exponentials associated to the eigenvalues of the perturbed operator \(T(\varepsilon):= T_0+\varepsilon T_1+\varepsilon^2 T_2+\dots+\varepsilon^k T_k+\dots\) forms a Riesz basis in \(L^2(0,T)\), \(T>0\), where \(\varepsilon \in\mathbb C\), \(T_0\) is a closed densely defined linear operator on a separable Hilbert space \(\mathcal H\) with domain \(\mathcal D(T_0)\) having isolated eigenvalues with multiplicity one, while \(T_1,T_2, \dots\) are linear operators on \(\mathcal H\) having the same domain \(\mathcal D\supset \mathcal D(T_0)\) and satisfying a specific growing inequality. After that, we generalize this result using a \(H\)-Lipschitz function. As application, we consider a non-selfadjoint problem deduced from a perturbation method for sound radiation.{
©2013 American Institute of Physics}
©2013 American Institute of Physics}
MSC:
| 42B10 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
| 47A10 | Spectrum, resolvent |
| 81Q12 | Nonselfadjoint operator theory in quantum theory including creation and destruction operators |
| 81Q15 | Perturbation theories for operators and differential equations in quantum theory |
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