One Problem for the Bessel Equation with a Spectral Parameter in the Boundary Condition

In this paper, we consider the spectral problem for thesemi-integer Bessel equation with a boundary condition containingthe square of the spectral parameter and a complex physicalparameter. The system of eigenfunctions of the problem and thecharacteristic equation for the eigenvalues are derived. Theequation for multiple roots of the characteristic equation isderived. The results on the basis properties (Riesz basis) of thesystem of eigenfunctions at different values of the parameter areobtained. For each case a biorthogonally conjugate system isconstructed. At the end of the paper there is an example for theorder of Bessel functions equal to \(1/2\) .