Definition:Self-Adjoint Boundary Conditions

Definition
Consider the functional $J \sqbrk {\mathbf y}$, such that:


 * $\ds J \sqbrk {\mathbf y} = \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$

Let the momenta of $J$ be:


 * $\mathbf p = \nabla_{\mathbf y'} \map F {x, \mathbf y, \mathbf y'}$

Let the following boundary conditions hold:


 * $\map {\mathbf y'} a = \bigvalueat {\map {\boldsymbol \psi} {\mathbf y} } {x \mathop = a}$

If:


 * $\exists \map g {x, \mathbf y}: \bigvalueat {\map {\mathbf p} {x, \mathbf y, \map {\boldsymbol \psi} {\mathbf y} } } {x \mathop = a} = \bigvalueat {\nabla_{\mathbf y'} \map g {x, \mathbf y} } {x \mathop = a}$

then the boundary conditions are called self-adjoint.