Spectral Theory - Resolvent Operator

Resolvent Operator

Main article: Resolvent formalism See also: Green's function and Dirac delta function

Using spectral theory, the resolvent operator R:

can be evaluated in terms of the eigenfunctions and eigenvalues of L, and the Green's function corresponding to L can be found.

Applying R to some arbitrary function in the space, say φ,

This function has poles in the complex λ-plane at each eigenvalue of L. Thus, using the calculus of residues:

where the line integral is over a contour C that includes all the eigenvalues of L.

Suppose our functions are defined over some coordinates { xj }, that is:

where the bra-kets corresponding to { xj } satisfy:

and where δ (x − y) = δ (x1 − y1, x2 − y2, x3 − y3, ...) is the Dirac delta function.

Then:

\begin{align} \left\langle x,\ \frac{1}{2\pi i }\ \oint_C \ d \lambda (\lambda - L)^{-1}\varphi \right\rangle &= \frac{1}{2\pi i }\ \oint_C \ d \lambda \ \langle x,\ (\lambda - L)^{-1} \ \varphi \rangle\\ &= \frac{1}{2\pi i }\ \oint_C \ d \lambda \int \ dy\ \ \langle x,\ (\lambda - L)^{-1}\ y\rangle \ \langle y, \ \varphi \rangle \end{align}

The function G(x, y; λ) defined by:

\begin{align} G(x,\ y;\ \lambda) &= \langle x,\ (\lambda - L)^{-1}\ y\rangle \\ &= \Sigma_{i=1}^n \Sigma_{j=1}^n \langle x,\ e_i \rangle \langle f_i,\ (\lambda - L)^{-1}e_j \rangle \langle f_j, \ y\rangle \\ &= \Sigma_{i=1}^n \frac{ \langle x,\ e_i \rangle \langle f_i, \ y\rangle }{\lambda - \lambda_i} \\ &= \Sigma_{i=1}^n \frac{ e_i (x) f_i^*(y) }{\lambda - \lambda_i}, \end{align}

is called the Green's function for operator L, and satisfies:

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