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In the talk we will discuss Sturm-Liouville problems with a boundary condition
depending affinely or quadratically on an eigenparameter:

−y′′ + q(x)y = λ, 0 < x < 1 (1)
y′(0) sin β = y(0) cos β , 0 ≤ β < π (2)
y′(1) = (aλ2 + b + c)y(1), (3)

where λλ is the spectral parameter, q(x) is a real valued and continuous function on the interval
[0,1], and a, b, c are real.
In the papers by Binding, Browne, Watson, and Code the existence and asymptotics of

eigenvalues of (1)-(3) were studied. It was proved that the eigenvalues of (1)-(3) form an infi-
nite sequence, accumulating only at +∞, and the following cases are possible:

(a) All the eigenvalues are real and simple;
(b) All the eigenvalues are simple and all, except a conjugate pair of non-real, are real;
(c) All the eigenvalues are real and all, except one double, are simple;
(d) All the eigenvalues are real and all, except one triple, are simple.

The necessary and sufficient conditions for minimality and completeness of the chosen sys-
tem of root functions of the corresponding operator were previously given in the form, which

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uses some special associated functions. In the current talk another method with the direct use
of characteristic functions will be discussed. This direct method is known for the affine case
and was extensively discussed in the literature. The aim of the present paper is to develop this
direct method for the quadratic case and to consider the affine and quadratic cases together
in a unified way.