Mathematical Methods for Physicists: A concise introduction - Site Map

SIMPLE LINEAR INTEGRAL EQUATIONS
A Hermitian kernel has at least one eigenvalue and it may have an in®nite number.
The proof will be omitted and we refer interested readers to the book by
Courant and Hibert mentioned earlier (Chapter 3).
The eigenvalues of a Hermitian kernel are real, and eigenfunctions belonging to
di€erent eigenvalues are orthogonal; two functions f …x† and g…x† are said to be
orthogonal if
Z
f *…x†g…x†dx ˆ 0:
To prove the reality of the eigenvalue, we multiply the homogeneous Fredholm
equation by u*…x†, then integrating with respect to x, we obtain
Z b
a
u*…x†u…x†dx ˆ
Z b Z b
a
a
K…x; t†u*…x†u…t†dtdx:
…11:20†
Now, multiplying the complex conjugate of the Fredholm equation by u…x† and
then integrating with respect to x, weget
Z b
a
u*…x†u…x†dx ˆ *
Z b Z b
a
a
K*…x; t†u*…t†u…x†dtdx:
Interchanging x and t on the right hand side of the last equation and remembering
that the kernel is Hermitian K*…t; x† ˆK…x; t†, we obtain
Z b
a
u*…x†u…x†dx ˆ *
Z b Z b
a
a
K…x; t†u…t†u*…x†dtdx:
Comparing this equation with Eq. (11.2), we see that ˆ *, that is, is real.
We now prove the orthogonality. Let i , j be two di€erent eigenvalues and
u i …x†; u j …x†, the corresponding eigenfunctions. Then we have
Z b
u i …x† ˆ i K…x; t†u i …t†dt;
a
Z b
u j …x† ˆ j K…x; t†u j …t†dt:
Now multiplying the ®rst equation by u j …x†, the second by i u i …x†, and then
integrating with respect to x, we obtain
Z b
Z b
j u i …x†u j …x†dx ˆ i j
a
Z b
Z b
i u i …x†u j …x†dx ˆ i j
a
a
a
Z b
a
Z b
a
K…x; t†u i …t†u j …x†dtdx;
K…x; t†u j …t†u i …x†dtdx:
a
…11:21†
Now we interchange x and t on the right hand side of the last integral and because
of the symmetry of the kernel, we have
Z b
Z b
i u i …x†u j …x†dx ˆ i j
a
a
Z b
a
K…x; t†u i …t†u j …x†dtdx:
…11:22†
422