K.S. Narain , ICTP

This lesson continues with the study of the second order differential operator L and its adjoint operator L⁺. Special attention was given in the beginning of the lesson to the boundary conditions and the definition of homogeneous boundary condition, this definition implies that the set of functions that satisfy this type of boundary conditions form a vector space.

Then the definition for a Hermitian or self-adjoint operator was given, that is L=L⁺, and some examples of self-adjoint operators studied. It was shown that given a second order linear differential operator L with homogeneous boundary condition, then it is always possible to define an adjoint operator L⁺ such that L=L⁺.In the examples different boundary conditions were used the Dirichlet boundary condition, Neumann boundary condition and periodic boundary conditions.

The idea of introducing the Green's function is to have a function G(x,y) with the propertie that L_x acting in G(x,y) give the identity function in function space, that is the Dirac delta function ?(x-y)/W(x) where W(x) is a weight function called the wave function, then we have L_xG(x,y)=?(x-y)/W(x). It is stated that G(x,y) as a function of x must satisfy the boundary conditions. It was also defined the adjoin Green's function G⁺(x,y) such that the equation L_x⁺G⁺(x,y)=?(x-y)/W(x) is satisfied. Then it was demonstrated that if the Green's function is known then you can always find the solution to the equation L_xU(x)=F(x). It was shown that G⁺(x,y)=G(y,x) where G is the complex conjugate of G(x,y) this result has two implications first is says that if G(x,y) exist then G⁺(x,y) also exist an that G(y,x) satisfy the adjoin boundary conditions as a function of x. Finally and as an example the derivation of the Green's function for a second order differential operator was started.

Mathematics for Physicists (Text Book from Google e-books preview)Written by<span class="addmd"> Philippe Dennery,Andre Krzywicki, Chapter 1, 2, 3 and 4</span>