Spectral Theory - Examples and Constructions

The resolvent and resolvent equation provide powerful tools for studying spectra and developing perturbation theory.

DefinitionResolvent Operator

For an operator $T$ and $\lambda \in \rho(T)$ (the resolvent set), the resolvent is $R_\lambda(T) = (T - \lambda I)^{-1}$

The resolvent is a bounded operator for all $\lambda \in \rho(T)$ .

TheoremResolvent Equation

For $\lambda, \mu \in \rho(T)$ , $R_\lambda - R_\mu = (\mu - \lambda) R_\lambda R_\mu$

This identity is fundamental to analytic function theory of operators.

Proof

$R_\lambda - R_\mu = R_\lambda[(T - \mu I) - (T - \lambda I)]R_\mu = (\mu - \lambda)R_\lambda R_\mu$

■

TheoremAnalytic Dependence

The resolvent $\lambda \mapsto R_\lambda(T)$ is analytic on $\rho(T)$ as an operator-valued function. Near any point $\lambda_0 \in \rho(T)$ , it has a convergent power series expansion.

ExampleComputing Resolvents

Multiplication Operator: For $(M_\phi f)(x) = \phi(x) f(x)$ on $L^2$ , $R_\lambda(M_\phi) = M_{1/(\phi - \lambda)}$ defined for $\lambda \notin \text{Range}(\phi)$
Shift Operator: On $\ell^2$ , for $(Sx)_n = x_{n+1}$ , $R_\lambda(S) = \sum_{n=0}^\infty \frac{S^n}{\lambda^{n+1}}$ for $|\lambda| > 1$
Harmonic Oscillator: For $H = -\frac{d^2}{dx^2} + x^2$ , the resolvent can be expressed using Hermite functions

DefinitionResolvent Set and Spectrum

The resolvent set $\rho(T) = \{\lambda \in \mathbb{C} : (T - \lambda I) \text{ is bijective with bounded inverse}\}$ .

The spectrum $\sigma(T) = \mathbb{C} \setminus \rho(T)$ decomposes into:

Point spectrum $\sigma_p(T)$ : eigenvalues
Continuous spectrum $\sigma_c(T)$ : $T - \lambda I$ injective, range dense but not closed
Residual spectrum $\sigma_r(T)$ : $T - \lambda I$ injective, range not dense

TheoremStone's Formula

The spectral projections can be recovered from the resolvent via $E((a,b]) = \lim_{\varepsilon \to 0^+} \frac{1}{2\pi i} \int_a^b [R_{\lambda + i\varepsilon} - R_{\lambda - i\varepsilon}] \, d\lambda$

This formula connects the resolvent (analytic object) to the spectral measure (geometric object).

Remark

The resolvent approach to spectral theory is particularly powerful for perturbation theory. If $T_0$ is well-understood and $V$ is a "small" perturbation, one can analyze $\sigma(T_0 + V)$ by studying how the resolvent $(T_0 + V - \lambda I)^{-1}$ differs from $R_\lambda(T_0)$ .

Resolvent methods are indispensable in scattering theory, where one analyzes continuous spectra of Schrödinger operators using analytic continuation of resolvents.