Mage's Rhizome

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====== Time-dependent potentials ======

If the Hamiltonian $H(t)=-\Delta+V(t)$ is time-dependent, the approach involving evolution semigroups has to be modified and below we give a proof for the existence of solutions to the Schrödinger equation with potential $V(t)$. An obvious restriction is $V(t) \in K$ from the class of Kato perturbations from [[classes_of_static_potentials|before]] for all times. By the Kato--Rellich theorem this guarantees that the family $\{H(t)\}_t$ of Hamiltonians has a joint domain $D(H)=D(-\Delta)$. But it remains open what time-related restriction on the potential we have to choose to get a well-defined solution, i.e. a family of evolution operators $U(t,s)$ forming an //evolution system//.

**Definition.** The unitary operators $U(t,s), 0\leq s,t \leq T$, form a evolution system if
  - $U(t,t) = \mathrm{id}$
  - $U(t,r)U(r,s) = U(t,s)$
  - $U(t,s)^{-1} = U(t,s)^* = U(s,t)$
  - $\partial_t U(t,s) = -i H(t) U(t,s)$ and $\partial_s U(t,s) = i U(t,s) H(s)$ on $D(H)$

Note that the members of the family $\{H(t)\}_t$ are in general not commuting among each other or with the evolution operator. We use a technique called the **stepwise static approximation**[(Reed M and Simon B, //Methods of Modern Mathematical Physics, Vol. II: Fourier Analysis, Self-Adjointness// (Academic Press, 1975), Theorems X.70 and 71; based on Kato T, //Integration of the equation of evolution in a Banach space//, J. Math. Soc. Japan **5** 208-234 (1953))] that divides the time interval $[0,T]$ into short subintervals and takes $H(t)$ constant over each such subinterval, then passing to smaller and smaller subintervals. The relevant potential classes, also with regard to regularity issues later, are the following:

**Definition.** ("Sobolev--Kato space") $K^m = \{ V \in K  \mid \partial^{\alpha}V \in K, |\alpha|\leq m \}$ with the respective norm where [[wp>multi-index notation]] is used for the (weak) partial derivative.

**Definition.** ("Sobolev--Kato--Lipschitz space") $\mathrm{Lip}([0,T], K^m)$ with norm
\[
\max_{t\in[0,T]} \|V(t)\|_{K^m} + \max_{s<t}\frac{\|V(t)-V(s)\|_{K^m}}{|t-s|}.
\]

**Proof sketch.** We show that potentials $V \in \mathrm{Lip}([0,T], K)$ give a well defined Schrödinger solutions but the proof omits a few important steps that would take up too much space here.[(The full proof and many things more on the subject can be found in my PhD thesis on [[https://arxiv.org/abs/1610.05552|arXiv]], §3.7.)]

1. Take the time interval $[0,t]$ with $t\leq T$ divided into $k$ subintervals $[t_j,t_{j+1}]$ with equal length $t/k$. We define the stepwise static approximation by using [[Stone's theorem]] for $k$ different static Hamiltonians:
\[
U_k(t,0) = \exp(-iH(t_{k-1})t/k) \ldots \exp(-iH(t_{0})t/k) =: U_k^{(k)} \ldots U_k^{(1)}.
\]
We show now that for $k\rightarrow \infty$ this so-defined evolution operator converges uniformly in $t$ in the strong topology of the Hilbert space $\mathcal{H}$.

2. So we show the Cauchy property of this sequence with a similar trick as in the proof of [[Stone's theorem]]:
\[
(U_k(t,0)-U_l(t,0))\psi_0 = i\int_0^t U_l(t,s) (V(s_{(l)}) - V(s_{(k)})) U_k(s,0)\psi_0 \,ds
\]
Here the time $s_{(k)}$ is the largest time step $t_k$ in the partitioning belonging to $U_k$ smaller than $s$. Clearly it holds $s_{(l)} - s_{(k)} \rightarrow 0$ if both indices increase. So it seems plausible that $V(s_{(l)}) - V(s_{(k)})$ in the $K$-norm, but we need also $U_k(s,0)\psi_0$ converging strongly in $D(H)$ to be able to even multiply with the potential difference and use the norm estimate ([[classes_of_static_potentials|here bottom]]), this we show in 4. onward.

3. By adding a big enough constant one first makes all $H(t) \geq 1$ (i.e. $\langle \varphi,H(t)\varphi \rangle \geq  1$ for all $\varphi \in D(H)$) and thus has a well-defined inverse $H(t)^{-1}:\mathcal{H} \rightarrow D(H)$ bounded by 1.

4. We add $D(H)$-identities $H(t)^{-1}H(t)$ to the definition of $U_k$:
\begin{align*}
U_k(t,0) &= U_k^{(k)} \ldots U_k^{(1)} \\
&= H(t_{k-1})^{-1}H(t_{k-1}) U_k^{(k)} H(t_{k-2})^{-1} \ldots H(t_{1})H(t_{0})^{-1}H(t_{0}) U_k^{(1)} \\
&= H(t_{k-1})^{-1} U_k^{(k)} H(t_{k-1})H(t_{k-2})^{-1} \ldots H(t_{1})H(t_{0})^{-1} U_k^{(1)} H(t_{0}).
\end{align*}
We used the fact that the Hamitonians commute with the short-stime evolution operators at the same time-step in this.

5. Now we rewrite the $H(t)H(s)^{-1}$ encounters as $K(t,s) + \mathrm{id} : \mathcal{H} \rightarrow \mathcal{H}$ where the operator $K(t,s)$ is bounded by $L |t-s|$ (not shown here), $L$ the Lipschitz constant of $V$ in the Sobolev--Kato--Lipschitz space. We have now:
\[
U_k(t,0) = H(t_{k-1})^{-1} U_k^{(k)} \prod_{j=1}^{k-1} (K(t_j,t_{j-1}) + \mathrm{id}) U_k^{(j)} H(0)
\]
with a time-ordered product.

6. The leading $H(t_{k-1})^{-1}$ is $\mathcal{H} \rightarrow D(H)$ with bound 1. So an estimate in the $D(H) = H^2$ Sobolev norm gives:
\begin{align*}
\|U_k(t,0)\psi_0\|_{H^2} &\leq \prod_{j=1}^{k-1} \left( \frac{Lt}{k} + 1 \right) \|H(0)\psi_0\|_\mathcal{H} \\
&= \left( \frac{Lt}{k} + 1 \right)^{k-1} \|H(0)\psi_0\|_\mathcal{H} \longrightarrow e^{Lt} \|H(0)\psi_0\|_\mathcal{H}.
\end{align*}
This shows convergence for $\psi_0 \in D(H)$ and also the $H^2$ regularity of such solutions.

7. Finally we show the Cauchy property from (2).
\begin{align*}
\|(U_k(t,0)-U_l(t,0))\psi_0\|_\mathcal{H} &\leq \int_0^t \|V(s_{(l)}) - V(s_{(k)})\|_K \|U_k(s,0)\psi_0\|_{H^2} ds\\
&\leq \max_{s \in [0,t]} \|V(s_{(l)}) - V(s_{(k)})\|_K  \|H(0)\psi_0\|_\mathcal{H} \int_0^t e^{Ls} ds \\
&= \max_{s \in [0,t]} \|V(s_{(l)}) - V(s_{(k)})\|_K  \|H(0)\psi_0\|_\mathcal{H} \left( e^{Lt} -1 \right) \longrightarrow 0
\end{align*}
This convergence is uniform because $t$ is from a bounded set $[0,T]$. Finally already (6) showed that the evolution operators are uniformly bounded on $D(H)$ and thus can extended to the whole Hilbert space.

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