Remark 9.2. Admissible boundary conditions are precisely the ones that lead to self-adjoint operators. For example if \(\a \) is quasi-regular and \(\b \) is not quasi-regular, then (for the chosen \(v\)) the operator

\[ Au=Du,\qquad \dom (A)=\{u\in D_{\max }: W(u,v;p)(\a )=0\}, \]

is self-adjoint in \(L^2(\a ,\b ;w)\).

Compactness of the resolvent of \(A\) is generally difficult to check (it is true when both boundary points are regular). In our cases, it is "inherited" from the PDE and we do obtain the properties listed in Theorem 2.6 for our Sturm–Liouville problems.

Remark 9.3. In the case where both boundary points are quasi-regular, there are generally also self-adjoint operators where the boundary conditions are not separated (i.e. include function evaluations at both boundary points), which can also be explicitly described, but this description is a bit more complicated, so we did not give it.

Remark 9.4. If both boundary points are not quasi-regular, then the only self-adjoint operator we obtain is

\[ Au=Du,\qquad \dom (A)=D_{\max }, \]

i.e. the operator with no (explicit) boundary conditions. It that case we say that there are no admissible boundary conditions.

Remark 9.5. The idea behind the condition \(W(u,v;p)(\a )=0\) is that we force \(u\) to behave like the chosen \(v\) near \(\a \).

Example 9.6. We reconsider the second derivative operator from Example 7.6. Since both boundary points are regular, they are quasi-regular and we can omit the limits in the boundary conditions. We have that \(Dv=0\) is \(v''=0\) with general solution \(v(x)=C_1x+C_2\). The Wronskian is \(W(u,v;p)=u'v-uv'\). To obtain the Dirichlet condition \(u(\a )=0\), Remark 9.5 suggests taking \(v\) with \(v(\a )=0\), which gives (uniquely up to a multiplicative constant) \(v(x)=x-\a \). With this choice we have

\[ W(u,v;p)(x)=W(u,x-\a ;1)(x)=u'(x)(x-\a )-u(x), \]

so that \(W(u,v;p)(\a )=0\) gives \(u(\a )=0\) as expected.

The general separated admissible boundary condition at \(\a \) equals

\[ W(u,C_1x+C_2;p)(\a )=u'(\a )[C_1\a +C_2]-u(\a )C_1=0. \]

Choosing \(C_1=1\) and \(C_2=-\alpha \) we again obtain the Dirichlet condition \(u(\a )=0\). Choosing \(C_1=0\) and \(C_2=1\) (i.e. choosing \(v=1\)) we obtain the Neumann condition \(u'(\a )=0\).

Note that since we only considered separated boundary conditions, we do not obtain the periodic boundary conditions in this way.

Example 9.7. We reconsider Example 6.5. The Wronskian is \(W(u,v;p)=x(u'v-uv')\). The right boundary point is regular and therefore quasi-regular. In Example 7.8 we calculated the solutions of \(Du=0\). For \(\sigma =0\) we obtained \(1\), \(\ln (x)\) and for \(\sigma >0\) we obtained \(x^\sigma \) and \(x^{-\sigma }\).

From the PDE we obtained the boundary condition \(u(\b )=0\). We show how to obtain this from the Wronskian. If \(\sigma =0\), then we choose \(v(x)=\ln (\b )-\ln (x)\). We obtain \(W(u,v;p)(\b )=u(\b )\) so that the corresponding boundary condition is \(u(\b )=0\). For \(\sigma >0\) we choose \(v(x)=x^\sigma -\b ^{2\sigma }x^{-\sigma }\). Then \(W(u,v;p)(\b )=-u(\b )2\sigma \b ^\sigma \), so that the corresponding boundary condition again is \(u(\b )=0\).

For \(\sigma \geq 1\) the left boundary point is not quasi-regular and therefore we do not need an explicit boundary condition at \(\a \).

For \(\sigma <1\) the left boundary point is quasi-regular and therefore a boundary condition is needed. We first consider the case \(\sigma =0\). With the choice \(v=1\) we obtain \(W(u,v;p)(0)=\lim _{x\downarrow 0}xu'(x)\) so that we obtain the boundary condition

\[ \lim _{x\downarrow 0}xu'(x)=0. \]

Instead with the choice \(v=\ln (x)\) we obtain \(W(u,v;p)(0)=\lim _{x\downarrow 0}xu'(x)\ln (x)-u(x)\) so that we obtain the boundary condition

\[ \lim _{x\downarrow 0}xu'(x)\ln (x)-u(x)=0. \]

We now consider the case \(\sigma \in (0,1)\). Picking the solution \(x^\sigma \), we obtain

\[ W(u,x^\sigma )(\a )=\lim _{x\downarrow 0}W(u,x^\sigma )(x) =\lim _{x\downarrow 0}x\left (u'(x)x^\sigma -u(x)\sigma x^{\sigma -1}\right ) =\lim _{x\downarrow 0}u'(x)x^{\sigma +1}-u(x)\sigma x^\sigma , \]

so that the boundary condition is \(\lim _{x\downarrow 0}u'(x)x^{\sigma +1}-u(x)\sigma x^\sigma =0\). We could similarly have picked \(x^{-\sigma }\) to obtain another admissible boundary condition.