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