A bead threaded on a circular wire of circumference (in metres) has its transverse…
PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Linear algebra and vector spaces · Linear algebra and vector spaces
Conditions of Validity, Ch Inner products, orthogonality and the adjoint
Problem
A bead threaded on a circular wire of circumference L (in metres) has its transverse displacement modelled by the eigenvalue problem for Ly=−y′′ on [0,L], with domain D(T)={y:y(0)=y(L),y′(0)=y′(L)} (periodic boundary conditions, matching position and slope round the loop). Someone argues: "the only self-adjoint domain for −d2/dx2 on an interval is the Dirichlet one, y(0)=y(L)=0, because that is the case where the boundary form visibly vanishes term by term. Since the periodic conditions do not set y or y′ to zero at either endpoint separately, this operator cannot be self-adjoint."
((a)) Write down the boundary form B(f,g)=[f∗g′−f∗′g]0L for two functions f,g∈D(T), and evaluate it using the periodic conditions.
((b)) State whether the quoted argument is correct, and give the correct classification of (L,D(T)).