We consider the radial differential operator for a fixed angular mode \(n\),
\[ L_n=\frac{d^2}{d r^2}+\frac{1}{r} \frac{d}{d r}-\frac{n^2}{r^2}, \quad r \in(0, \infty) \]
acting on \(L^2((0, \infty) ; r d r)\) with inner product
\[ \langle u, v\rangle=\int_0^{\infty} u(r) v(r) r d r \]
Our objective is to verify the formal symmetry of \(L_n\) and to record boundary conditions that yield a self-adjoint realization.
Formal symmetry (integration by parts)
Using \(\frac{d}{d r}\left(r v^{\prime}\right)=v^{\prime}+r v^{\prime \prime}\), we write \[ \left\langle u, L_n v\right\rangle=\int_0^{\infty} u \frac{d}{d r}\left(r v^{\prime}\right) d r-\int_0^{\infty} \frac{n^2}{r} u v d r \]
Two integrations by parts give
\[ \left\langle u, L_n v\right\rangle-\left\langle L_n u, v\right\rangle=\left[r\left(u v^{\prime}-v u^{\prime}\right)\right]_0^{\infty} . \]
Hence the boundary form associated with \(L_n\) is
\[ B[u, v]:=\left[r\left(u v^{\prime}-v u^{\prime}\right)\right]_0^{\infty} \]
The operator is formally symmetric on any domain for which \(B[u, v]=0\) for all \(u, v\) in the domain.
Boundary behaviour and self-adjointness
To enforce \(B[u, v]=0\), we require \[ \lim _{r \rightarrow 0^{+}} r\left(u v^{\prime}-v u^{\prime}\right)=0, \quad \lim _{r \rightarrow \infty} r\left(u v^{\prime}-v u^{\prime}\right)=0 \]
This is achieved by standard endpoint conditions, for example: - Dirichlet: \(u(0)=0\) and suitable decay at \(\infty\); - Neumann: \(u^{\prime}(0)=0\) and \(u^{\prime}(\infty)=0\); - Robin (mixed): \(a u+b r u^{\prime}=0\) at an endpoint, with compatible conditions for \(v\); - Regularity at \(r=0\) : for \(|n| \geq 1\), regular solutions satisfy \(u(r) \sim r^{|n|}\) as \(r \rightarrow 0\); for \(n=0\), boundedness with non-singular derivative ensures \(r u^{\prime} \rightarrow 0\); - Decay at \(\infty\) : sufficient decay of \(u, u^{\prime}\) so that \(r u^{\prime} v \rightarrow 0\) and \(r u v^{\prime} \rightarrow 0\).
On a finite interval \((0, R)\), the boundary form reduces to \(\left[r\left(u v^{\prime}-v u^{\prime}\right)\right]_0^R\); separated Dirichlet/Neumann/Robin conditions at 0 and \(R\) render \(L_n\) self-adjoint in the Sturm-Liouville sense.
Functional-analytic remarks
We regard \(L_n\) as a densely defined operator on \(L^2((0, \infty) ; r d r)\). Formal symmetry together with an appropriate choice of domain-treating 0 and \(\infty\) via the limit-point/limit-circle classification-yields a selfadjoint realization, i.e. \(L_n=L_n^*\).
Consequences of self-adjointness
- The spectrum is real.
- Eigenfunctions corresponding to distinct eigenvalues are orthogonal in \(\langle\cdot, \cdot\rangle\) and typically form a complete system.
- The induced dynamics preserve the inner product (unitarity in relevant settings); in quantum mechanics, self-adjoint operators represent observables.
Examples
Self-adjoint with suitable domains/BCs
- Sturm-Liouville operators \(L=-\frac{d}{d x}\left(p(x) \frac{d}{d x}\right)+q(x)\).
- The Laplacian with Dirichlet/Neumann/Robin boundary conditions.
- Hamiltonians of conservative quantum systems.
Not self-adjoint as stated (without special BCs) 1. Generic first-order differential operators. 2. Non-Hermitian operators (e.g., PT-symmetric settings) absent a modified inner product. 3. Generators of dissipative dynamics.
Clarification of the "vanishing boundary term"
What must vanish is the boundary form \[ B[u, v]=\left[r\left(u v^{\prime}-v u^{\prime}\right)\right]_0^{\infty}=0 \]
not necessarily the pointwise boundary values of \(u\) or \(v\). Dirichlet, Neumann, Robin conditions, regularity at \(r=0\), and sufficient decay at \(\infty\) are standard mechanisms ensuring \(B[u, v]=0\).