Physlib

Physlib.QuantumMechanics.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`. - `mem_resolventSet_of_im_ne_zero` : every non-real `z` lies in the resolvent set of a self-adjoint operator. - `sub_smul_surjective` : A self-adjoint `T` has `T - z • 1` surjective for every non-real `z` (in particular `T ± i • 1` are onto). - `spectrum_real` : The spectrum of a self-adjoint unbounded operator is real. - `unitaryConj_isSelfAdjoint` : Unitary conjugation preserves self-adjointness.

iii. Table of contents

  • A. Resolvent set
  • B. Spectrum
  • C. Unitary conjugation

iv. References

A. Resolvent set

B. Spectrum

C. Unitary conjugation

3 declarations

theorem

Non-real complex numbers are in the resolvent set of a self-adjoint operator

Let TT be a self-adjoint operator on a complex Hilbert space. For any complex number z∈Cz \in \mathbb{C}, if its imaginary part is non-zero (Im(z)≠0\text{Im}(z) \neq 0), then zz belongs to the resolvent set ρ(T)\rho(T) of TT.

theorem

Surjectivity of T−zIT - zI for Non-real zz and Self-Adjoint TT

Let TT be a self-adjoint operator. For any complex number z∈Cz \in \mathbb{C} with a non-zero imaginary part (Im(z)≠0\text{Im}(z) \neq 0), the operator T−zIT - zI is surjective, where II denotes the identity operator.

theorem

Unitary conjugation preserves self-adjointness

Let HH and H′H' be complex Hilbert spaces. Let AA be a partially defined linear operator on HH and let u:H→H′u : H \to H' be a unitary operator. If AA is self-adjoint, then its unitary conjugation uAu−1u A u^{-1} (defined on the domain u(dom(A))u(\text{dom}(A))) is a self-adjoint operator on H′H'.