Resolvent formalism

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In mathematics, the resolvent formalism is a technique for applying concepts from complex analysis to the study of the spectrum of operators on Hilbert spaces and more general spaces.

The resolvent captures the spectral properties of an operator in the analytic structure of the resolvent. Given an operator A, the resolvent may be defined as

R (z; A) = (A - z I) - 1 .

Among other uses, the resolvent may be used to solve the inhomogeneous Fredholm integral equations; a commonly used approach is a series solution, the Liouville-Neumann series.

The residue of a closed contour integral may be understood to be a projection operator

$\operatorname{res} R(z;A)\vert_{z=\lambda} = \frac{-1}{2\pi i} \oint_{C_\lambda} R(\zeta;A) d\zeta = P_\lambda$

where λ corresponds to an eigenvalue of A

$A\vert \psi_\lambda \rangle = \lambda \vert \psi_\lambda \rangle$

and $C λ$ is a contour in the positive direction around the eigenvalue λ.

The Hille-Yosida theorem relates the resolvent to an integral over the one-parameter group of transformations generated by A. Thus, for example, if A is Hermitian, then $U (t) = exp(i t A)$ is a one-parameter group of unitary operators. The resolvent can be expressed as the integral

$R(z;A)= \int_0^\infty e^{-zt}U(t) dt.$

[edit] History

The first major use of the resolvent operator was by Ivar Fredholm, in a landmark 1903 paper in Acta Mathematica that helped establish modern operator theory. The name resolvent was given by David Hilbert.