The transmission eigenvalue problem arising from the inverse scattering theory is of great importance in the theory of qualitative methods and in the practical applications. In this paper, we study the transmission eigenvalue problem for anisotropic inhomogeneous media in $\Omega\subset \mathbb{R}^d$,(d=2,3). Using the T-coercivity and the spectral approximation theory, we derive an a posteriori estimator of residual type and prove its effectiveness and reliability for eigenfunctions. In addition, we also prove the reliability of the estimator for transmission eigenvalues. The numerical experiments indicate our method is efficient and can reach the optimal order $DoF^{-2m/d}$ by using piecewise polynomials of degree $m$ for real eigenvalues.

翻译：反散射理论引起的传导电子价值问题在定性方法理论和实际应用中非常重要。在本文中,我们用$\Omega\subset\mathbb{R ⁇ d$,(d=2,3)来研究异种异种异种介质的传导电子价值问题(d=2,3),利用T-curcity和光谱近似理论,我们得出一个剩余类型后继估计器,并证明其有效性和可靠性。此外,我们还证明了传播电子元值估计器的可靠性。数字实验表明,我们的方法效率高,并且能够通过对真实电子值使用纯度多数值,达到最高值$DoF ⁇ -2m/d}。