adjoint operator
For a matrix A, the transpose A^T is the partner that lets you slide A from one side of a dot product to the other: the dot product of A x with y equals the dot product of x with A^T y. The adjoint operator is the same idea for a differential operator: it is the partner L* that lets you move L off the first function and onto the second inside an integral inner product, paying only a boundary-term toll along the way.
Concretely, for a differential operator L acting on functions on [a, b], the formal adjoint L* is defined by Lagrange's identity: the integral of (L u) v minus the integral of u (L* v) equals a boundary term evaluated at a and b. Integration by parts produces both L* and that boundary term. When you also fix the function spaces by adjoint boundary conditions chosen to kill the boundary term, you get the full adjoint boundary-value problem. The special, central case is when L equals L* and the boundary conditions match their own adjoints — then the problem is self-adjoint, the boundary term vanishes by design, and you inherit real eigenvalues, orthogonal eigenfunctions, and a symmetric Green's function G(x, s) = G(s, x). Sturm-Liouville form exists precisely to make second-order problems self-adjoint.
The adjoint is the gateway to solvability. Whether the inhomogeneous problem L u = f can be solved, and whether its solution is unique, is decided by the adjoint problem L* v = 0 through the Fredholm alternative: a solution exists only if f is orthogonal to every solution of the homogeneous adjoint problem. So even when you only care about L, you cannot avoid L* — it controls existence, uniqueness, and the reciprocity symmetry of Green's functions.
For L u = u'' on [0, 1], integrating by parts twice gives integral of u'' v - integral of u v'' = [u' v - u v'] from 0 to 1, so L* v = v'' — the operator is formally self-adjoint. With matching boundary conditions u(0) = u(1) = 0 the boundary term vanishes and L is genuinely self-adjoint, forcing G to be symmetric.
Self-adjointness needs both the right operator form and the right boundary conditions — the boundary term must die.
An operator can be formally self-adjoint (L = L* as a symbol) yet not self-adjoint as a problem if the boundary conditions fail to kill the boundary term. Self-adjointness is a property of the operator together with its boundary conditions, never of the differential expression alone.