Lecturer: 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*G(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

_{x}**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*U(x)=F(x). It was shown that G

_{x}^{+}(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.

Written by Philippe Dennery,Andre Krzywicki, Chapter 1, 2, 3 and 4