Spectral properties of non- self-adjoint elliptic differential operators in the Hilbert space
Resumen
Let $\Omega$ be a bounded domain in $R^{n}$ with smooth boundary $\partial\Omega$. In this article, we will investigate the spectral properties of a non-self adjoint elliptic differential operator\\ $(Au)(x)=-\sum^{n}_{i,j=1}\left(\omega^{2\alpha}(x)a_{ij}(x) \mu(x)u'_{x_{i}}(x)\right)'_{x_{j}}$, acting in the Hilbert space $H=L^{2}{(\Omega)}$. with Dirichlet-type boundary conditions. Here $a_{ij}(x)= \overline{a_{ji}(x)}\;\;\;(i,j=1,\ldots,n),\;\;\; a_{ij}(x)\in C^{2}(\overline{\Omega})$, and the functions $a_{ij}(x)$ satisfies the uniformly elliptic condition, and let $ 0 \leq \alpha < 1$. Furthermore, for $\forall x \in \overline{\Omega}$, the function $\mu(x)$ lie in the $\psi_{\theta_1\theta_2}$ , where ${\psi_{\theta_1\theta_2}}=\{z \in {\bf C}:\;\pi/2<\theta_1 \leq|arg\;z| \leq \theta_2<\pi\},$
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M. S.Agranovich, Elliptic operators on compact manifolds,I.Itogi Nauki I Tekhniki: Sovremennye Problemy Mat :Fundamental'nye Napravleniya Val.63, VINITI, Moskow.1990, PP.5-129 (Russian)
K. Kh. Boimatov and A. G. Kostyuchenko, Distribution of eigenvalues of second-order non-selfadjoint differential operators, Vest. Mosk. Gos. Univ., Ser. I, Mat. Mekh, No. 3, 1990, pp. 24-31 (Russian). https://doi.org/10.1007/BF01077920
K. Kh. Boimatov, Asymptotic behaviour of the spectra of second-order non-selfadjoint systems of differential operators, Mat. Zametki, Vol. 51, No. 4, 1992, pp. 6-16, (Russian). https://doi.org/10.1007/BF01250542
K. Kh. Boimvatov, Spectral asymptotics of nonselfadjoint degenerate elliptic systems of differential operators Dokl. Akad. Nauk. Rossyi, Vol. 330, No.6, 1993,(Russian) (English transl. In Russian Acad.Sci.Dokl. Math. Vol.47, 1993, N3, PP.545-553)
K. Kh. Boimvatov, Separation theorems, weighted spaces and there applications. Trudy Mat. Inst. Steklov. Vol.170,1984, P.37-76,(Russian) (English transl. in Pros.Steklov. Inst. Math. 1987, N1 (170)
K. Kh. Boimatov, Spectral asymptotics of differential and pseudo-differential operators Part.2, Trudy sem.Ptrosk.V.10.1984.P.78-106, Russian, (English transl. In Soviet Math.V.35, N.5, 1986) https://doi.org/10.1007/BF01119189
I. C. Gokhberg and M. G. Krein, Introduction to the Theory of linear non-selfadjoint operators in Hilbert space, English transl. Amer. Math. Soc., Providence, R. I. 1969.
T. Kato, Perturbation Theory for Linear Operators, Springer, New York, 1966. https://doi.org/10.1007/978-3-642-53393-8
M. A. Naymark. Linear differential operators, Moscow. Nauka, 1969.
A. Sameripour and K. Seddigh, Distribution of the eigenvalues non-selfadjoint elliptic systems that degenerated on the boundary of domain, (Russian)Mat. Zametki 61(1997), no,3, 463-467 translation in Math. Notes 61(1997) no,3-4. 379-384 (Reviewer: Gunter Berger) 35P20(35J55) https://doi.org/10.1007/BF02355425
A. Sameripour and K. Seddigh, On the spectral properties of generalized non-selfadjoint elliptic systems of differential operators degenerated on the boundary of domain, Bull.Iranian Math. Soc, 24(1998) , no,1,15-32.47F05(35JXX 35PXX)
A. A. Shkalikov, Tauberian type theorems on the distribution of zeros of holomorphic functions, Matem. Sbornik Vol. 123 (165) 1984, No. 3, pp. 317-347; English transl. in Math. USSR-sb. 51, 1985.
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