Physlib.QuantumMechanics.DDimensions.Operators.SpectralTheory.SelfAdjoint

Spectral theory for self-adjoint operators

i. Overview

In this module we develop the spectral theory for self-adjoint operators.

ii. Key results

- `resolventSet_eq_regularityDomain` : The resolvent set and regularity domain coincide. That is, if `T - z • 1` has a continuous (equivalently, bounded) inverse then its range is all of `H`. - `spectrum_real` : The spectrum of a self-adjoint unbounded operator is real.

iii. Table of contents

A. Resolvent set
B. Spectrum

iv. References

A. Resolvent set

B. Spectrum

4 declarations

theorem

$\rho(T) = \text{regularityDomain}(T)$ for self-adjoint operators $T$

#resolventSet_eq_regularityDomain

Let $H$ be a complex Hilbert space and $T$ be a self-adjoint partially defined linear operator on $H$ . The resolvent set $\rho(T)$ of $T$ is equal to its regularity domain. That is, a complex number $z \in \mathbb{C}$ is in the resolvent set if and only if there exists a constant $c > 0$ such that for all $x$ in the domain of $T$ , $\|Tx - zx\| \geq c \|x\|$ .

theorem

For self-adjoint $T$ , $\text{ran}(T - zI) = H \implies z \in \rho(T)$

#mem_resolventSet_of_range_eq_top

Let $T$ be a self-adjoint partially defined linear operator on a complex Hilbert space $H$ . For any complex number $z \in \mathbb{C}$ , if the range of the operator $T - zI$ (where $I$ is the identity operator) is the entire space $H$ , then $z$ belongs to the resolvent set $\rho(T)$ .

theorem

The spectrum of a self-adjoint operator is real ( $\sigma(T) \subseteq \mathbb{R}$ )

#spectrum_real

Let $H$ be a complex Hilbert space and $T$ be a self-adjoint partially defined linear operator on $H$ . The spectrum of $T$ , denoted by $\sigma(T)$ , is a subset of the real numbers, i.e., $\sigma(T) \subseteq \mathbb{R}$ .

theorem

The residual spectrum of a self-adjoint operator is empty ( $\sigma^r(T) = \emptyset$ )

#residualSpectrum_eq_empty

Let $T$ be a self-adjoint partial linear operator on a complex Hilbert space $H$ . The residual spectrum of $T$ , denoted by $\sigma^r(T)$ , is empty, i.e., $\sigma^r(T) = \emptyset$ .

Physlib.QuantumMechanics.DDimensions.Operators.SpectralTheory.SelfAdjoint

Spectral theory for self-adjoint operators

i. Overview

ii. Key results

iii. Table of contents

iv. References

A. Resolvent set

B. Spectrum

ρ(T)=regularityDomain(T)\rho(T) = \text{regularityDomain}(T)ρ(T)=regularityDomain(T) for self-adjoint operators TTT

For self-adjoint TTT, ran(T−zI)=H ⟹ z∈ρ(T)\text{ran}(T - zI) = H \implies z \in \rho(T)ran(T−zI)=H⟹z∈ρ(T)

The spectrum of a self-adjoint operator is real (σ(T)⊆R\sigma(T) \subseteq \mathbb{R}σ(T)⊆R)

The residual spectrum of a self-adjoint operator is empty (σr(T)=∅\sigma^r(T) = \emptysetσr(T)=∅)

$\rho(T) = \text{regularityDomain}(T)$ for self-adjoint operators $T$

For self-adjoint $T$ , $\text{ran}(T - zI) = H \implies z \in \rho(T)$

The spectrum of a self-adjoint operator is real ( $\sigma(T) \subseteq \mathbb{R}$ )

The residual spectrum of a self-adjoint operator is empty ( $\sigma^r(T) = \emptyset$ )