Kato Theory for Time-Dependent Evolution

Pointwise self-adjointness of $H(t)$ does not by itself construct a time-dependent quantum evolution. The domains and the time regularity must support a propagator theorem. Kato’s common-domain theory gives one precise route; common-form-domain results allow some operator domains to vary. This page states useful sufficient versions and explains their proof structure, rather than claiming the most general possible hypotheses.

Helpful background. Time-Dependent Hamiltonians proves the bounded case. Domains of Hamiltonians supplies static Kato–Rellich perturbation theory. Quadratic Forms defines closed forms and their associated operators.

A common-domain evolution theorem

Let $I=[0,T]$. Suppose each $H(t)$ is self-adjoint on the same dense domain $D\subset\mathcal H$, and suppose

$$t\longmapsto H(t)\psi\quad\text{belongs to }C^1(I;\mathcal H) \quad\text{for every }\psi\in D.$$

Endpoint derivatives are interpreted one-sidedly. Then there is a unique strongly continuous unitary propagator $U(t,s)$ with

$$U(t,s)D=D$$

such that for every $\psi\in D$, $t\mapsto U(t,s)\psi$ is continuously differentiable in $\mathcal H$ and

$$i\hbar\partial_t U(t,s)\psi=H(t)U(t,s)\psi.$$

It has the usual composition and inverse laws. Initial vectors in $\mathcal H\setminus D$ also evolve continuously, but the theorem does not declare their trajectories strongly differentiable.

This is the self-adjoint specialization of Kato’s common-domain contraction-generator theorem, using $A(t)=-iH(t)/\hbar$. Schmid and Griesemer explain the equivalence between its original regularity hypotheses and the displayed $C^1$ condition. The result concerns differentiability of $H(t)\psi$ on a fixed domain; it does not ask for the unbounded operators themselves to have finite operator norm on $\mathcal H$.

Why graph norms enter the construction

Equip $D$ with the graph norm of one reference Hamiltonian. Each $H(t)$ is bounded from this Banach space into $\mathcal H$. Closedness and the common domain make the graph norms equivalent, while the time regularity controls how their equivalence constants change over a compact interval.

The construction approximates the evolution by products of frozen self-adjoint evolutions. These products are unitary in $\mathcal H$, but that alone is insufficient: one must also control them on $D$. Graph-norm estimates prevent successive factors from losing the regularity needed by the next generator.

Those estimates yield convergence of the approximating propagators, retention of the common domain, and passage to the differential equation on $D$. Backward evolution supplies inverses, and the equation gives uniqueness. This is the proof architecture; the convergence and domain-invariance steps require Kato’s estimates and are not consequences of a merely formal time-ordered exponential.

A checkable driven perturbation

Let $H_0$ be self-adjoint on $D$, and let $V$ be symmetric with

$$\|V\psi\|\leq a\|H_0\psi\|+b\|\psi\|, \qquad \psi\in D.$$

Take real $g\in C^1(I)$ and suppose

$$a\sup_{t\in I}|g(t)|<1.$$

Kato–Rellich makes $H(t)=H_0+g(t)V$ self-adjoint on $D$ at every time. For fixed $\psi\in D$,

$$\frac{d}{dt}H(t)\psi=g'(t)V\psi,$$

which is norm-continuous. The common-domain theorem therefore applies.

For an infinitesimally $H_0$-bounded perturbation, the coefficient $a$ can be chosen arbitrarily small. Any fixed bounded $C^1$ protocol then passes this test. The Coulomb estimate in Domains of Hamiltonians provides such an example in three dimensions. This conclusion concerns existence and regularity, not an adiabatic approximation: no assumption that the drive is slow has been made.

Constant form domains allow changing operator domains

A distinct sufficient theorem starts from closed symmetric forms $q_t$ with a common dense domain $Q$. Choose a Hilbert norm $\|\cdot\|_Q$ for which $Q$ is continuously embedded in $\mathcal H$. Assume constants $c\geq0$ and $0<m\leq M<\infty$ exist such that

$$m\|\psi\|_Q^2 \leq q_t[\psi]+(c+1)\|\psi\|^2 \leq M\|\psi\|_Q^2$$

for all $t\in I$ and $\psi\in Q$. Assume also that $t\mapsto q_t$ is $C^2$ in the norm of bounded sesquilinear forms on $Q$.

Let $H(t)$ be the associated self-adjoint operators. A sufficient version of Kisyński’s theorem gives a unique unitary propagator preserving $Q$ and carrying

$$U(t,s)D(H(s))=D(H(t)).$$

For initial vectors in $D(H(s))$, it gives the strong Schrödinger equation in $\mathcal H$. The stated norm-$C^2$ condition is a convenient stronger hypothesis than the pointwise form conditions in the cited theorem. The fixed positive shift can be removed by a scalar phase.

With only the corresponding $C^1$ form regularity, the theorem still constructs form-domain evolution and an equation in the dual space $Q^*$:

$$i\hbar\frac{d}{dt}\langle\phi,\psi(t)\rangle =q_t(\phi,\psi(t)),\qquad \phi\in Q.$$

One must not automatically promote this to a Hilbert-space derivative or operator-domain invariance. The regularity level determines the conclusion.

A time-dependent point interaction illustrates the distinction. Its form may have fixed domain $H^1(\mathbb R)$, while its derivative-jump condition, and therefore $D(H(t))$, changes with the coupling. The static construction is developed in Quadratic Form Methods; the evolving problem belongs to the form-domain theorem, not the fixed-operator-domain statement above.

Exercises

1. If $g$ is continuous but nowhere differentiable, does the common-domain theorem above prove evolution for $H_0+g(t)V$? Does its failure prove nonexistence?

Solution

It generally fails the stated $C^1$ hypothesis, so this theorem cannot be invoked. That is not a nonexistence proof. More general evolution theorems may apply; in particular a bounded norm-continuous Hamiltonian family is covered by the preceding bounded theorem without differentiability of $g$.

2. Why does an arbitrary $C^1$ bounded protocol work for a perturbation of relative bound zero, even when its amplitude is large?

Solution

On the compact interval let $G=\sup|g|$. Choose an admissible relative coefficient $a<1/(G+1)$, allowing its companion constant $b$ to grow. Then $aG<1$, so all Hamiltonians are self-adjoint on the same domain. Large amplitude may worsen estimates, but it does not obstruct this existence criterion.

References

• A. Balmaseda, D. Lonigro, and J. M. Pérez-Pardo, “On the Schrödinger equation for time-dependent Hamiltonians with a constant form domain,” Mathematics 10, 218, 2022, doi:10.3390/math10020218, author text — Assumption 3.8 and Theorem 3.10 distinguish form and strong evolution.
• T. Kato, “Integration of the equation of evolution in a Banach space,” Journal of the Mathematical Society of Japan 5, 208–234, 1953, doi:10.2969/jmsj/00520208 — original common-domain construction.
• J. Schmid and M. Griesemer, “Kato’s theorem on the integration of non-autonomous linear evolution equations,” Mathematical Physics, Analysis and Geometry 17, 265–271, 2014, doi:10.1007/s11040-014-9154-5, author text — precise common-domain regularity hypotheses.