Remark 7.3. The idea behind quasi-regularity is as follows (we refer to Section 1.1 for an earlier relevant discussion). For all \(\lambda \in \mC \), the space of solutions of \(Du=\lambda u\) is two-dimensional. If \(\a \) is not quasi-regular, then the condition that \(u\) should be square integrable (with weight \(w\)) near \(\a \) already excludes some of these solutions and therefore de facto serves as a boundary condition. Hence no explicit/additional boundary condition at \(\a \) is needed. On the other hand, if \(\a \) is quasi-regular, then an explicit boundary condition at \(\a \) is needed to whittle down the number of solutions.

Remark. The reasoning behind theorem 7.5 is that if the coefficients are “nice” at the boundary point (encapsulated in the definition of regular), then the solution is “nice” at the boundary point (encapsulated in the definition of quasi-regular).

Example 7.6. The Sturm–Liouville problem with \(\a ,\b \in (-\infty ,\infty )\) and \(\a <\b \) and

\[ w=1,\quad p=1,\quad q=0, \]

(which includes Examples 6.3 and 6.4 as special cases) has both boundary points regular.

Example 7.7. For example 6.5 the left boundary point \(\a =0\) is singular for any of the three reasons: \(w(\a )=0\), \(p(\a )=0\), and \(q\) is not continuous at \(\a \) (this third reason if \(\sigma >0\)). The right boundary point \(\b =b\) is regular.

Example 7.8. We investigate quasi-regularity of the left boundary point of Example 6.5.

We solve \(Du=0\) which in this case is

\[ (xu')'-\frac {\sigma ^2}{x}u=0. \]

We separately consider the cases \(\sigma =0\) and \(\sigma >0\). With \(\sigma =0\) we obtain

\[ (xu')'=0. \]

Integrating once gives

\[ xu'=C_1, \]

where \(C_1\) is a constant. Dividing by \(x\) and integrating again gives

\[ u=C_1\ln (x)+C_2, \]

where \(C_2\) is a constant. Since \(x\) and \(x\ln ^2(x)\) are continuous at zero (using standard limits for the second of these functions), we have (choosing \(\delta =b/2\))

\[ \int _0^{b/2} |1|^2\,x\,dx,~\int _0^{b/2} |\ln (x)|^2\,x\,dx<\infty , \]

so that we have two linearly independent solutions in \(L^2(0,b/2;x)\) and therefore we have quasi-regularity. Now consider the case \(\sigma >0\). We make the Ansatz \(u(x)=x^r\). Then \(u'=rx^{r-1}\) so that \(xu'=rx^r\), which gives \((xu')'=r^2x^{r-1}\). Substituting this in the differential equation gives

\[ r^2x^{r-1}-\sigma ^2x^{r-1}=0. \]

We conclude that \(r^2=\sigma ^2\) so that \(r=\pm \sigma \) and we obtain the general solution \(u(x)=C_1x^\sigma +C_2x^{-\sigma }\). We have that \(x^\sigma \) is continuous and therefore belongs to \(L^2(0,b/2;x)\). We have

\[ \int _0^{b/2} |x^{-\sigma }|^2\,x\,dx=\int _0^{b/2} x^{1-2\sigma }\,dx, \]

and from standard integrals we know that this is finite if and only if \(1-2\sigma >-1\), i.e. \(\sigma <1\). Therefore we obtain quasi-regularity if and only if \(\sigma <1\).

Remark 7.9. We comment further on Example 7.8 in the context of the heat equation on the disc (and therefore \(\sigma \in \mN _0\)). For \(\sigma \in \mN \) we obtain that the left boundary point of the \(R\)-equation is not quasi-regular and therefore the condition that the solution of the heat equation for each \(t\) should be in \(L^2(\Omega )\) already serves at a boundary condition at the origin. However, for \(\sigma =0\) we have that the left boundary point of the \(R\)-equation is quasi-regular and therefore we need an explicit boundary condition at \(r=0\) (which does not arise from the given boundary condition for the heat equation).