Spectral Theory - Key Properties

Unbounded operators arise naturally in quantum mechanics and PDEs, requiring careful treatment of domains and self-adjointness.

DefinitionUnbounded Operator

A linear operator $T : D(T) \to H$ on a Hilbert space $H$ with domain $D(T) \subset H$ is unbounded if it is not bounded. The domain $D(T)$ is typically a dense subspace of $H$ .

The operator $T$ is closed if its graph $\Gamma(T) = \{(x, Tx) : x \in D(T)\}$ is closed in $H \times H$ .

DefinitionSelf-Adjoint vs Symmetric

An unbounded operator $T$ is symmetric if $\langle Tx, y \rangle = \langle x, Ty \rangle$ for all $x, y \in D(T)$ .

The operator $T$ is self-adjoint if $T = T^*$ , meaning:

$D(T) = D(T^*)$
$\langle Tx, y \rangle = \langle x, Ty \rangle$ for all $x, y \in D(T)$

where $D(T^*) = \{y \in H : x \mapsto \langle Tx, y \rangle \text{ is bounded on } D(T)\}$ .

Self-adjointness is stronger than symmetry for unbounded operators. Many differential operators are symmetric but not self-adjoint until appropriate boundary conditions are specified.

ExampleDifferential Operators

Momentum: $\hat{p} = -i\hbar \frac{d}{dx}$ on $L^2(\mathbb{R})$ with domain $H^1(\mathbb{R})$ is self-adjoint
Harmonic Oscillator: $H = -\frac{d^2}{dx^2} + x^2$ on $L^2(\mathbb{R})$ with domain $\{f \in H^2 : x^2 f \in L^2\}$ is self-adjoint
Laplacian: $-\Delta$ on $L^2(\Omega)$ with Dirichlet boundary conditions $u|_{\partial\Omega} = 0$ and domain $H^2(\Omega) \cap H^1_0(\Omega)$ is self-adjoint

TheoremSpectral Theorem for Unbounded Self-Adjoint Operators

Let $T$ be an unbounded self-adjoint operator on a Hilbert space $H$ . Then there exists a unique spectral measure $E$ on $\sigma(T) \subset \mathbb{R}$ such that $T = \int_{\sigma(T)} \lambda \, dE(\lambda)$ where the integral is understood in the sense that for $x \in D(T)$ , $Tx = \int_{\sigma(T)} \lambda \, dE(\lambda) x$ and $D(T) = \{x : \int_{\sigma(T)} |\lambda|^2 \, d\|E(\lambda)x\|^2 < \infty\}$ .

ExampleSpectra of Differential Operators

Free Particle: $-\Delta$ on $L^2(\mathbb{R}^n)$ has spectrum $\sigma(-\Delta) = [0, \infty)$ (pure continuous spectrum)
Hydrogen Atom: The Hamiltonian has discrete spectrum $E_n = -\frac{13.6 \text{ eV}}{n^2}$ for bound states and continuous spectrum $[0, \infty)$ for scattering states
Periodic Potential: Schrödinger operators $-\Delta + V$ with periodic $V$ have band spectrum (unions of intervals)

Remark

For unbounded operators, the spectral theorem requires self-adjointness, not just symmetry. Determining whether a symmetric operator is self-adjoint often requires sophisticated criteria (von Neumann's theory of deficiency indices, Weyl's limit point/limit circle criterion).

Understanding self-adjointness is crucial for quantum mechanics and PDE theory, where physical observables and evolution operators must be self-adjoint to ensure unitarity and energy conservation.