Definition:Self-Adjoint Boundary Conditions

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Definition

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

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

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.


Sources