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Gauge transported propagator for the bounded-coefficient magnetic operator

Statement

Assume the Axiom of Choice and the Axiom of Countable Choice (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )). Fix $ℏ > 0$ , $m > 0$ , $q \in R$ , admissible magnetic data $(A, Φ)$ on $R^{3}$ and a gauge function $χ : R^{3} \times I \to R$ as in Gauge multiplication preserves H2 and transports the magnetic momentum, with underlying open interval $I \subseteq R$ . Put $A_{j}^{'} (\cdot, t) : = A_{j} + D_{j} χ (\cdot, t)$ , $Φ^{'} (\cdot, t) : = Φ - \partial_{t} χ (\cdot, t)$ and, for $t \in I$ , $H^{'} (t) : = K_{A^{'} (t), Φ^{'} (t)}, G (t) : = multiplication by e^{i \frac{q}{ℏ} χ (\cdot, t)},$ so that $(A^{'} (t), Φ^{'} (t))$ is admissible for every $t$ and $H^{'} (t)$ is self-adjoint with domain $W^{2, 2} (R^{3}; C)$ by Self-adjointness of the bounded-coefficient magnetic Schrodinger operator. Let $K : = K_{A, Φ}$ and $U (τ) : = e^{- i τ K / ℏ}$ for $τ \in R$ , the strongly continuous unitary group generated by $- i K / ℏ$ (A self-adjoint operator generates a strongly continuous unitary group, Integral of a measurable function against a projection-valued measure).

(a) For every $t \in I$ one has the operator identity on $W^{2, 2}$ $H^{'} (t) = G (t) K G (t)^{- 1} - q \partial_{t} χ (\cdot, t),$ where the last term is multiplication by $- q \partial_{t} χ (\cdot, t)$ ; explicitly, $G (t) K G (t)^{- 1} = K_{A^{'} (t), Φ}$ is the magnetic operator with the transported vector potential and the original scalar potential $Φ$ .

(b) For every $u_{0} \in W^{2, 2}$ and $s \in I$ the curve $u (t) : = G (t) U (t - s) G (s)^{- 1} u_{0} (t \in I)$ lies in $W^{2, 2}$ for every $t$ , is differentiable into $L^{2}$ on $I$ , and satisfies $u (s) = u_{0}$ and $i ℏ u^{'} (t) = H^{'} (t) u (t)$ for every $t \in I$ . It is the unique such curve: if $v : I \to W^{2, 2}$ is continuous into $W^{2, 2} (R^{3}; C)$ , differentiable into $L^{2}$ , with $i ℏ v^{'} (t) = H^{'} (t) v (t)$ for all $t$ and $v (s) = u_{0}$ , then $v = u$ .

(c) The two-parameter family $U^{'} (t, s) : = G (t) U (t - s) G (s)^{- 1} (t, s \in I)$ consists of unitary operators mapping $W^{2, 2}$ onto $W^{2, 2}$ , satisfies $U^{'} (t, t) = I$ and the composition law $U^{'} (t, r) U^{'} (r, s) = U^{'} (t, s)$ for all $t, r, s \in I$ , and $u (t) = U^{'} (t, s) u_{0}$ is the solution of (b).

The theorem provides dynamics for the transformed coefficient class only; it asserts no general well-posedness for arbitrary time-dependent unbounded generators.

Facts & Assumptions

Given: $ℏ, m > 0$ , $q \in R$ , admissible $(A, Φ)$ , the gauge function $χ$ , the transported data $A^{'} (t), Φ^{'} (t)$ , the operators $K = K_{A, Φ}$ , $H^{'} (t) = K_{A^{'} (t), Φ^{'} (t)}$ and the multipliers $G (t)$ .

[F1]

$K$ is self-adjoint with domain $D (K) = W^{2, 2}$ , and $U (τ) = e^{- i τ K / ℏ}$ is a strongly continuous unitary group with $U (τ) D (K) = D (K)$ , $K U (τ) = U (τ) K$ on $D (K)$ , and $\lim_{τ \to 0} \frac{1}{τ} (U (τ) x - x) = - \frac{i}{ℏ} K x$ exactly for $x \in D (K)$ (A self-adjoint operator generates a strongly continuous unitary group, Integral of a measurable function against a projection-valued measure).

[F2]

For each $t$ , $G (t)$ is unitary, preserves $W^{1, 2}$ and $W^{2, 2}$ , and $(P_{j} - q A_{j}^{'} (\cdot, t)) G (t) u = G (t) (P_{j} - q A_{j}) u$ for $u \in W^{1, 2}$ ; moreover $t \mapsto G (t) h$ is continuous for $h \in L^{2}$ and differentiable with $\frac{d}{d t} G (t) h = \frac{i q}{ℏ} (\partial_{t} χ (\cdot, t)) G (t) h$ (Gauge multiplication preserves H2 and transports the magnetic momentum). The same assertions hold for $- χ$ in place of $χ$ (Gauge multiplication preserves H2 and transports the magnetic momentum).

[F3]

For each $t$ , $(A^{'} (t), Φ^{'} (t))$ is admissible: $D_{k} A_{j}^{'} = D_{k} A_{j} + D_{k} D_{j} χ (\cdot, t)$ and $Φ^{'} = Φ - \partial_{t} χ (\cdot, t)$ are bounded with the uniform bounds of the hypotheses; and $H^{'} (t) = K_{A^{'} (t), Φ^{'} (t)}$ is self-adjoint with domain $W^{2, 2}$ (Self-adjointness of the bounded-coefficient magnetic Schrodinger operator, Bounded-coefficient magnetic Schrodinger operator).

[F4]

A real function continuous on a closed interval and differentiable in its interior with zero derivative is constant (The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ ). For an $L^{2}$ -differentiable curve $z$ , expansion of the inner product gives $(∥ z ∥^{2})^{'} = 2 Re ⟨ z^{'}, z ⟩$ . If $T$ is self-adjoint and $z \in D (T)$ , then $⟨ T z, z ⟩$ is real, so $2 Re ⟨ - i T z / ℏ, z ⟩ = 0$ .

Proof

1.1givenF2F3algebra

[Transported magnetic square] Let $w \in W^{2, 2}$ and put $u : = G (t) w$ , so that $u \in W^{2, 2}$ by [F2] and $w = G (t)^{- 1} u$ . By Self-adjointness of the bounded-coefficient magnetic Schrodinger operator clause (a) and [F2], $P_{A, j} w \in W^{1, 2}$ and $P_{A^{'}, j} (t) G (t) P_{A, j} w = G (t) P_{A, j} (P_{A, j} w)$ for the transported momentum $P_{A^{'}, j} (t) = P_{j} - q A_{j}^{'} (\cdot, t)$ ; here the covariance identity is applied twice, the first time to $w \in W^{1, 2}$ and the second to the class $P_{A, j} w \in W^{1, 2}$ . Summing over $j$ , $\frac{1}{2 m} \sum_{j} P_{A^{'}, j} (t)^{2} u = G (t) \frac{1}{2 m} \sum_{j} P_{A, j}^{2} w$ , and adding the scalar potential term, which is multiplication by $q Φ$ commuting with $G (t)$ , gives $K_{A^{'} (t), Φ} u = G (t) K_{A, Φ} w = G (t) K G (t)^{- 1} u$ . Since $u \in W^{2, 2}$ was arbitrary, $G (t) K G (t)^{- 1} = K_{A^{'} (t), Φ}$ on $W^{2, 2}$ .

2.1givenstep 1.1F3algebra

[Conclusion of (a)] By the definition of $Φ^{'} (t)$ and the linearity of $K$ in the scalar potential, $H^{'} (t) = K_{A^{'} (t), Φ^{'} (t)} = K_{A^{'} (t), Φ} - q \partial_{t} χ (\cdot, t)$ as operators on $W^{2, 2}$ ; combining with step 1.1 gives (a).

3.1givenstep 2.1F1F2algebra

[The curve solves the equation] Let $u_{0} \in W^{2, 2}$ and $s \in I$ , and put $w_{0} : = G (s)^{- 1} u_{0}$ , $w (t) : = U (t - s) w_{0}$ and $u (t) : = G (t) w (t)$ . By [F1]-[F2], both $w (t)$ and $u (t)$ lie in $W^{2, 2}$ at each time, and $w^{'} = - i K w / ℏ$ in $L^{2}$ . Decompose the difference quotient of $u$ as $G (t + h) (w (t + h) - w (t)) / h + (G (t + h) - G (t)) w (t) / h$ . The first term tends to $G (t) w^{'} (t)$ by strong $L^{2}$ continuity and unitarity; the second tends to $(i q / ℏ) (\partial_{t} χ) G (t) w (t)$ by [F2] on the fixed vector $w (t)$ . Thus $i ℏ u^{'} = - q (\partial_{t} χ) u + G (t) K w = H^{'} (t) u$ by step 2.1, and $u (s) = G (s) w_{0} = u_{0}$ . No time continuity in the $W^{2, 2}$ norm is needed. This proves existence in (b).

4.1givenstep 3.1F3F4algebra

[Uniqueness] Let $v$ be as in (b), and put $z (t) : = v (t) - u (t)$ . At every $t$ , $z (t) \in W^{2, 2} = D (H^{'} (t))$ , and $z$ is differentiable into $L^{2}$ with $i ℏ z^{'} = H^{'} (t) z$ and $z (s) = 0$ . By self-adjointness from [F3] and the norm derivative identity of [F4], $(∥ z (t) ∥^{2})^{'} = 2 Re ⟨ - i H^{'} (t) z (t) / ℏ, z (t) ⟩ = 0$ . Apply [F4] on the closed interval joining $s$ and $t$ , which lies in the open interval $I$ : the norm squared is constant and equals zero. Hence $v (t) = u (t)$ for every $t$ , proving uniqueness without any assertion of $W^{2, 2}$ continuity for $G (t)^{- 1} v (t)$ .

5.1step 3.1step 4.1F1F2algebra∎

[Propagator properties] Each $U^{'} (t, s) = G (t) U (t - s) G (s)^{- 1}$ is unitary as a product of unitaries and maps $W^{2, 2}$ onto $W^{2, 2}$ by [F1] and [F2]. The identity $U^{'} (t, t) = G (t) G (t)^{- 1} = I$ is immediate; the composition law follows from $G (r)^{- 1} G (r) = I$ : $U^{'} (t, r) U^{'} (r, s) = G (t) U (t - r) G (r)^{- 1} G (r) U (r - s) G (s)^{- 1} = G (t) U (t - r + r - s) G (s)^{- 1} = U^{'} (t, s)$ . By step 3.1, $u (t) = G (t) U (t - s) G (s)^{- 1} u_{0} = U^{'} (t, s) u_{0}$ solves the equation, and by step 4.1 it is the unique solution. This is (c).

Gauge transported propagator for the bounded-coefficient magnetic operator

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

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