Studies on scattering of longitudinally and transversely incident beam of electrons by hollow cylindrical potential and coaxial cylindrical potentials have shown the presence of quasi bound “whispering” modes [1,2]. The realization of Levinson’s theorem [3,4] has been studied for some scattering potentials and results are widely available.
Roberts and Valluri  presented a geometric analytic technique, which utilizes conformal mapping W->Z=We^W between two complex domains to solve the 1-dimensional finite square well potential. The symmetry of the hollow cylindrical potential can be used to solve the Schrodinger equation as a 1-dimensional finite square well potential in the radial direction. This leads to the possible generalization to a concentric double walled cylindrical potential by considering it as a double well finite potential in the radial direction. The number of bound states of such a potential can be counted using the Lambert W formalism, as it is a geometric method, and the relation to the scattering phase shift can be established.
1) Vivishek Sudhir and P. C. Deshmukh
Scattering of electrons off hollow cylindrical potentials
J. Comput. Theor. Nanosci. 7, 2036 (2010)
2) Vivishek Sudhir and P. C. Deshmukh
Scattering of electrons by multi-walled cylindrical potentials
J. Comput. Theor. Nanosci. 8, 2321 - 2326 (2011)
3) N. Levinson
On the uniqueness of the potential in a Schrodinger equation for a given asymptotic phase
Danske Vid. Selsk. Mat.-Fys. Medd. 25:9 (1949)
Quantum Collision Theory
North Holland, Amsterdam (1975)
5) Roberts and S.R. Valluri
The quantum finite square well and the Lambert W Function
Can. J. Phys. 95: 105-110 (2017)