Physlib.Mathematics.ForMathlib.OneParameterSubgroups.Unitary

Unitary one-parameter groups

A norm-continuous unitary one-parameter group on a complex Hilbert space has a unique bounded self-adjoint generator. With the convention used here, the group generated by `A` is `U(t) = exp (-itA)`.

The same correspondence describes bounded generators of time translations, spatial translations, rotations, and other norm-continuous unitary one-parameter groups. The general theorem for strongly continuous groups and unbounded generators requires spectral theory that is not yet available in Physlib.

Main results

* `UnitaryOneParameterGroup.generator`: The canonical bounded self-adjoint generator. * `UnitaryOneParameterGroup.apply_eq_exp_generator`: The formula `U(t) = exp (-itA)`. * `UnitaryOneParameterGroup.ofSelfAdjoint`: The group generated by a bounded self-adjoint operator. * `UnitaryOneParameterGroup.stoneEquiv`: Stone's correspondence for bounded generators.

References

* M. H. Stone, Linear Transformations in Hilbert Space III. Operational Methods and Group Theory, Proc. Natl. Acad. Sci. 16 (1930), 172-175. [ref: stone_1930]

3 declarations

theorem

The generator of a unitary one-parameter group commutes with the group elements $U(t)$

#commute_generator

Let $H$ be a complex Hilbert space and $U: \mathbb{R} \to \mathcal{B}(H)$ be a norm-continuous unitary one-parameter group. Let $A$ denote the generator of the group, which is a bounded self-adjoint operator such that $U(t) = e^{-itA}$ . Then, for every $t \in \mathbb{R}$ , the operator $U(t)$ commutes with the generator $A$ , i.e., $U(t)A = AU(t)$ .

theorem

The derivative of a unitary one-parameter group $U(t)$ is $-i U(t) A$

#hasDerivAt

Let $H$ be a complex Hilbert space and $U: \mathbb{R} \to \mathcal{B}(H)$ be a norm-continuous unitary one-parameter group with a bounded self-adjoint generator $A$ . For any $t \in \mathbb{R}$ , the map $t \mapsto U(t)$ is differentiable at $t$ with derivative given by $-i U(t) A$ , where $i$ is the imaginary unit.

theorem

The derivative of $U(t)$ is $-i U(t) A$

#hasDerivAt_star

Let $H$ be a complex Hilbert space and let $U: \mathbb{R} \to \mathcal{B}(H)$ be a norm-continuous unitary one-parameter group. Let $A$ be the bounded self-adjoint generator of $U$ (defined such that $U(t) = e^{-itA}$ ). Then for any $t \in \mathbb{R}$ , the derivative of $U$ at $t$ is given by $\frac{d}{dt} U(t) = -i U(t) A.$

Physlib.Mathematics.ForMathlib.OneParameterSubgroups.Unitary

Unitary one-parameter groups

Main results

References

The generator of a unitary one-parameter group commutes with the group elements U(t)U(t)U(t)

The derivative of a unitary one-parameter group U(t)U(t)U(t) is −iU(t)A-i U(t) A−iU(t)A

The derivative of U(t)U(t)U(t) is −iU(t)A-i U(t) A−iU(t)A

The generator of a unitary one-parameter group commutes with the group elements $U(t)$

The derivative of a unitary one-parameter group $U(t)$ is $-i U(t) A$

The derivative of $U(t)$ is $-i U(t) A$