In this paper we study the spectra of bounded self-adjoint linear operators that are related to finite Hilbert transforms \(\mathcal {H}_L:L^2([b_L,0])\rightarrow L^2([0,b_R])\) and \(\mathcal {H}_R:L^2([0,b_R])\rightarrow L^2([b_L,0])\). These operators arise when one studies the interior problem of tomography. The diagonalization of \(\mathcal {H}_R,\mathcal {H}_L\) has been previously obtained, but only asymptotically when \(b_L\ne -b_R\). We implement a novel approach based on the method of matrix Riemann–Hilbert problems (RHP) which diagonalizes \(\mathcal {H}_R,\mathcal {H}_L\) explicitly. We also find the asymptotics of the solution to a related RHP and obtain error estimates.

中文翻译：

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两个相邻区间上的有限希尔伯特变换的对角化：黎曼-希尔伯特方法。


在本文中，我们研究与有限希尔伯特变换相关的有界自伴线性算子的谱\(\mathcal {H}_L:L^2([b_L,0])\rightarrow L^2([0,b_R ])\)和\(\mathcal {H}_R:L^2([0,b_R])\rightarrow L^2([b_L,0])\) 。当人们研究层析成像的内部问题时，就会出现这些算子。 \(\mathcal {H}_R,\mathcal {H}_L\)的对角化先前已获得，但仅在\(b_L\ne -b_R\)时渐近。我们实现了一种基于矩阵黎曼-希尔伯特问题（RHP）方法的新颖方法，该方法显式对角化\(\mathcal {H}_R,\mathcal {H}_L\) 。我们还找到相关 RHP 解的渐进性并获得误差估计。