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# Green's Function Theory

Lecture 26

We shall now direct our efforts towards finding what in linear algebra corresponds to the inverse of the linear operator . This means that we are going to find a linear operator which satisfies the equation

Once we have found this inverse operator , it is easy to solve the inhomogeneous problem

for . This is so because the solution is simply

If the vector space arena is an infinite-dimensional Hilbert space, the inverse operator

is usually called the Green's function of , although in the context of integral equations the expression

is sometimes called the resolvent of . Its singularities yield the eigenvalues of , while integration in the complex -plane yields, as we shall see, the corresponding eigenvectors. It is therefore difficult to overstate the importance of the operator .

Subsections

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Ulrich Gerlach 2010-12-09