Resolvent Operator

The resolvent of a Hamiltonian $H$ is the operator-valued function

$$R(z) = (zI-H)^{-1},$$

defined for complex numbers $z$ where the inverse exists. Physicists often also write

$$G(z)=(z-H)^{-1},$$

leaving the identity operator implicit. The resolvent is the basic energy-domain Green-function object: it packages spectral information, boundary prescriptions, and inverse-operator intuition in one operator-valued function.

Definition

Let $H$ be a self-adjoint Hamiltonian on a Hilbert space. A complex number $z$ belongs to the resolvent set of $H$ if $zI-H$ has a bounded inverse defined on the Hilbert space. For such $z$,

$$R(z)=(zI-H)^{-1}.$$

The spectrum is the complement of the resolvent set:

$$\sigma(H) = \mathbb C\setminus\rho(H).$$

For self-adjoint $H$, the spectrum lies on the real axis. Therefore every nonreal $z$ lies in the resolvent set.

Finite-Dimensional Model

If

$$H=\sum_n E_nP_n$$

is a finite-dimensional spectral decomposition, then

$$R(z) = \sum_n \frac{P_n}{z-E_n}, \qquad z\ne E_n.$$

In an eigenbasis, the resolvent is diagonal:

$$R(z) = \begin{pmatrix} \dfrac{1}{z-E_1} & 0 & \cdots\\ 0 & \dfrac{1}{z-E_2} & \cdots\\ \vdots & \vdots & \ddots \end{pmatrix}.$$

This is the simplest way to remember the construction: the resolvent applies the scalar function $f_z(E)=1/(z-E)$ to the Hamiltonian.

Spectral-Theorem Form

For a general self-adjoint Hamiltonian, the spectral theorem writes

$$H=\int_{\mathbb R}\lambda\,dE_H(\lambda),$$

where $E_H$ is the projection-valued spectral measure. The resolvent is then

$$R(z) = \int_{\mathbb R} \frac{1}{z-\lambda}\,dE_H(\lambda), \qquad z\notin\sigma(H).$$

This is the functional calculus applied to the function

$$f_z(\lambda)=\frac{1}{z-\lambda}.$$

For states $|\phi\rangle$ and $|\psi\rangle$, the matrix element is

$$\langle\phi|R(z)|\psi\rangle = \int_{\mathbb R} \frac{1}{z-\lambda}\, d\mu_{\phi,\psi}(\lambda),$$

where

$$d\mu_{\phi,\psi}(\lambda) = \langle\phi|dE_H(\lambda)|\psi\rangle.$$

This is why the resolvent is a spectral probe: it is a transform of the spectral measure.

Distance from the Spectrum

For self-adjoint $H$ and nonreal $z=E+i\eta$,

$$\|R(z)\| \le \frac{1}{|\eta|}.$$

More generally,

$$\|R(z)\| = \frac{1}{\operatorname{dist}(z,\sigma(H))}$$

for self-adjoint $H$. The resolvent becomes large when $z$ approaches the spectrum. This is the analytic version of the physical statement that spectral values are where the inverse becomes singular.

Poles and Bound States

If $E_k$ is an isolated eigenvalue with spectral projector $P_k$, then near $z=E_k$,

$$R(z) = \frac{P_k}{z-E_k} +R_{\rm reg}(z),$$

where $R_{\rm reg}(z)$ is regular near $E_k$. The pole locates the bound-state energy, and the residue is the projector onto the bound-state subspace.

For a nondegenerate bound state,

$$P_k=|k\rangle\langle k|.$$

For a degenerate bound state, $P_k$ projects onto the whole degenerate eigenspace. This residue viewpoint is the operator version of extracting states from Green-function poles.

Continuous Spectrum and Boundary Values

For continuous spectrum, the resolvent usually has no isolated pole at a real continuum energy. Instead, one studies boundary values from the upper or lower half-plane:

$$R^\pm(E) = \lim_{\epsilon\to0^+} (E-H\pm i\epsilon)^{-1}.$$

These limits may exist only in a weak, distributional, or weighted-space sense, depending on the problem. The sign of $i\epsilon$ selects a boundary prescription, such as outgoing or incoming scattering behavior. Taking coordinate matrix elements produces the kernels studied in Energy Green Function.

For a continuous spectral density, the imaginary part of a boundary value often carries density-of-states information. A schematic finite-volume version is

$$-\frac{1}{\pi}\operatorname{Im} \operatorname{Tr}R(E+i0) = \sum_n\delta(E-E_n),$$

with continuum versions requiring the correct trace, volume normalization, and limiting procedure.

Resolvent Identity

The resolvent satisfies the identity

$$R(z)-R(w) = (w-z)R(z)R(w),$$

whenever $z$ and $w$ lie in the resolvent set. In scalar form, this is just

$$\frac{1}{z-\lambda} - \frac{1}{w-\lambda} = \frac{w-z}{(z-\lambda)(w-\lambda)}.$$

The identity is a workhorse in spectral theory and perturbation theory because it compares inverses at different spectral parameters.

A related identity for two Hamiltonians $H$ and $H_0$ is

$$R(z)-R_0(z) = R(z)(H-H_0)R_0(z),$$

where

$$R(z)=(z-H)^{-1}, \qquad R_0(z)=(z-H_0)^{-1}.$$

This is the algebraic ancestor of the scattering-state construction in Lippmann–Schwinger Equation Preview.

Relation to Time Evolution

The resolvent is related to the time-evolution operator by a one-sided transform. For $\operatorname{Im}z\gt0$,

$$R(z) = \frac{1}{i\hbar} \int_0^\infty dt\, e^{izt/\hbar} e^{-iHt/\hbar}.$$

The convergence comes from the positive imaginary part of $z$. Boundary values such as $E+i0$ retain a memory of this time direction and prescription.

This formula explains why energy-domain Green functions and time-domain propagators are related while still being different objects. The resolvent is not the time-evolution operator; it is a transformed inverse with a spectral parameter.

Common Mistakes

• Forgetting the identity operator in $zI-H$.
• Evaluating the resolvent directly on the spectrum without an $i0$ or other prescription.
• Treating continuous-spectrum boundary values as ordinary bounded operators without checking the functional setting.
• Confusing resolvent poles with arbitrary peaks in a plotted response function.
• Dropping degeneracy: the residue at a degenerate eigenvalue is a projector onto a subspace, not a single vector.
• Replacing the time-evolution operator by the resolvent without performing the appropriate transform.

Cross-Links

• What Is a Green Function?
• Energy Green Function
• Lippmann–Schwinger Equation Preview
• Spectral Theorem, Practical Version
• Functions of Operators
• Continuous Spectra
• Anderson Localization uses typical off-diagonal resolvent decay and local spectral distributions to define localization lengths.
• Formula Sheet
• Green Function Table
• QFT Bridge: Green Functions

References

• M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, revised and enlarged ed., Academic Press, 1980.
• G. Teschl, Mathematical Methods in Quantum Mechanics, 2nd ed., American Mathematical Society, 2014.
• B. C. Hall, Quantum Theory for Mathematicians, Springer, 2013.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• J. R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover, 2006.

Exercises

1. For a two-level Hamiltonian $H=E_1P_1+E_2P_2$, compute $R(z)$.

Solution

Apply the scalar function $f_z(E)=1/(z-E)$ to each spectral value:

$$R(z) = \frac{P_1}{z-E_1} + \frac{P_2}{z-E_2}, \qquad z\ne E_1,E_2.$$

2. Prove the resolvent identity $R(z)-R(w)=(w-z)R(z)R(w)$.

Solution

Use $R(z)=(z-H)^{-1}$ and $R(w)=(w-H)^{-1}$. Then

$$R(z)-R(w) = R(z)\bigl[(w-H)-(z-H)\bigr]R(w).$$

The bracket equals $w-z$, so

$$R(z)-R(w) = (w-z)R(z)R(w).$$

3. Show from the spectral representation that an isolated eigenvalue produces a pole in the resolvent.

Solution

If $E_k$ is isolated, split the spectral decomposition into its projector $P_k$ and the rest:

$$R(z) = \frac{P_k}{z-E_k} + \sum_{n\ne k}\frac{P_n}{z-E_n}$$

in the finite-dimensional model, or the analogous spectral-integral decomposition in the general case. The second term is regular near $E_k$ if no other spectrum lies there, while the first term has a simple pole with residue $P_k$.

4. Verify the time-evolution transform for a single energy eigenstate with energy $E$ and $\operatorname{Im}z\gt0$.

Solution

On an eigenstate, the integral becomes

$$\frac{1}{i\hbar} \int_0^\infty dt\, e^{i(z-E)t/\hbar}.$$

Since $\operatorname{Im}z\gt0$, the exponential decays. The integral is

$$\int_0^\infty dt\, e^{i(z-E)t/\hbar} = \frac{i\hbar}{z-E}.$$

Multiplying by $1/(i\hbar)$ gives

$$\frac{1}{z-E},$$

which is the scalar resolvent eigenvalue.