Physlib.QuantumMechanics.Operators.SpectralTheory.Symmetric

Spectral theory for symmetric operators

i. Overview

In this module we develop the spectral theory for symmetric operators.

The numerical range of an operator, `Θ T = {⟪x, T x⟫_ℂ | x ∈ T.domain ∧ ‖x‖ = 1}`, is a subset of ℂ. For symmetric operators the numerical range consists only of real numbers and it is meaningful to discuss its upper/lower bounds. To facilitate this, we define `LinearPMap.realNumericalRange` as the projection of `LinearPMap.numericalRange` onto the real axis. For symmetric operators this simply reinterprets the numerical range as a subset of ℝ.

ii. Key results

- `realNumericalRange` (`Θᵣₑ`) : The projection of the numerical range onto the real axis. - `compl_ofReal_subset_regularityDomain` : The regularity domain of a symmetric operator contains all complex numbers with non-zero imaginary part. - `regularityDomain_isConnected_iff` : The regularity domain of a symmetric operator is connected if and only if it contains a real number. - `IsSymmetric.isEssentiallySelfAdjoint_of_defectNumber_eq_zero` : The basic criterion for essential self-adjointness: symmetric + densely defined + vanishing defect numbers at `± i`.

iii. Table of contents

A. Numerical range
B. Regularity domain
C. Point spectrum
D. Essential self-adjointness

iv. References

A. Numerical range

B. Regularity domain

C. Point spectrum

D. Essential self-adjointness

1 declaration

theorem

A symmetric densely defined operator with vanishing defect numbers at $\pm i$ is essentially self-adjoint

#isEssentiallySelfAdjoint_of_defectNumber_eq_zero

Let $H$ be a complex Hilbert space and $T$ be a symmetric, densely defined linear operator on $H$ . If the defect numbers of $T$ at $i$ and $-i$ are both zero, then $T$ is essentially self-adjoint.