Existence and Uniqueness of Time Evolution

Well-posed evolution requires existence, uniqueness, and continuous dependence on the data. Self-adjointness provides these properties for an autonomous Hamiltonian, even when an initial vector is not in its domain. Duhamel’s formula extends the result to a prescribed source and gives a useful bound on approximation errors. The homogeneous generator theorem belongs to Stone’s Theorem.

Helpful background. Schrödinger Equation in Hilbert Space defines strong, weak, and mild solutions. Spectral Projections supplies the bounded-energy cutoffs used in the uniqueness proof.

Duhamel’s formula for a self-adjoint Hamiltonian

Let $H$ be self-adjoint, let $U(t)=e^{-itH/\hbar}$, and fix a finite interval $0\leq t\leq T$. For $\psi_0\in\mathcal H$ and $f\in C([0,T];\mathcal H)$, consider

$$i\hbar\dot\psi(t)=H\psi(t)+f(t),\qquad \psi(0)=\psi_0.$$

The source is an additive vector-valued term. It is not the same as a time-dependent Hamiltonian multiplying the unknown state, and it need not preserve normalization.

There is a unique norm-continuous weak solution, given by

$$\psi(t)=U(t)\psi_0 -\frac{i}{\hbar}\int_0^t U(t-s)f(s)\,ds.$$

The integral is a Bochner integral in $\mathcal H$. This expression defines the mild solution without assuming that $H\psi(t)$ exists. Strong continuity of $U$ and continuity of $f$ give continuity of the integral and its norm estimate.

For each $\phi\in D(H)$, testing the formula and differentiating the scalar integral gives

$$i\hbar\frac{d}{dt}\langle\phi,\psi(t)\rangle =\langle H\phi,\psi(t)\rangle+\langle\phi,f(t)\rangle.$$

The derivative falls on a domain test vector through $U(s-t)\phi$, and its norm is controlled by $\|H\phi\|$. This justifies differentiating without asserting that the unknown solution lies in $D(H)$.

The formula also works for Bochner-integrable $f$. In that case scalar pairings are absolutely continuous and the weak equation holds almost everywhere, rather than necessarily at every time.

Spectral cutoffs prove uniqueness

Suppose $\psi$ is a norm-continuous weak solution with the stated initial data. Let $P_R=E_H([-R,R])$, where $R>0$ has energy units. The operator $HP_R$ is bounded with norm at most $R$, and $P_R\mathcal H\subset D(H)$.

Testing the weak equation with $P_R\phi$ for arbitrary $\phi\in\mathcal H$ yields a bounded evolution equation for $\psi_R=P_R\psi$:

$$i\hbar\dot\psi_R=HP_R\psi_R+P_Rf, \qquad \psi_R(0)=P_R\psi_0.$$

Initially this identity is weak. Its right side is norm-continuous, so integrating scalar pairings identifies $\psi_R$ with a continuously differentiable vector-valued integral. Thus it is also a strong bounded-operator equation.

The bounded equation has the unique variation-of-constants solution. Since its evolution on $P_R\mathcal H$ is the restriction of $U(t)$,

$$P_R\psi(t)=U(t)P_R\psi_0 -\frac{i}{\hbar}\int_0^t U(t-s)P_Rf(s)\,ds.$$

Now $P_R\to I$ strongly. Dominated convergence passes the cutoff through the integral because $\|P_Rf(s)\|\leq\|f(s)\|$. The full Duhamel formula follows, proving uniqueness. This argument avoids differentiating the norm of a weak solution when its vector derivative may not exist.

Stability and a residual error bound

Let $\psi$ and $\chi$ solve equations with the same $H$ but initial data $\psi_0,\chi_0$ and sources $f,g$. Unitarity and the triangle inequality give

$$\|\psi(t)-\chi(t)\| \leq\|\psi_0-\chi_0\| +\frac{1}{\hbar}\int_0^t\|f(s)-g(s)\|\,ds.$$

For homogeneous evolution, the initial distance is exactly preserved. For sources, the estimate controls accumulated forcing error without an exponential growth factor.

If a proposed approximation $\chi$ is a strong path in $D(H)$, define its residual by

$$r(t)=i\hbar\dot\chi(t)-H\chi(t)-f(t).$$

Then the approximation satisfies the exact equation with source $f+r$. Its error is bounded by its initial mismatch plus $\hbar^{-1}\int\|r\|$. A small residual is useful only in a norm and domain where this expression is defined. A spatial discretization needs an appropriate lift and consistency estimate before this continuum bound applies.

When the mild solution is strong

One convenient sufficient condition is

$$\psi_0\in D(H),\qquad f\in C([0,T];D(H)),$$

where the source is continuous in the graph norm. The unitary group preserves $D(H)$ and is strongly continuous there. The integral can therefore be taken in the graph-norm space, giving a vector in $D(H)$; differentiation then verifies the strong equation.

This is a sufficient condition, not a necessary one. A continuous Hilbert-space source alone guarantees the mild and weak conclusions above, but should not be silently used as a general strong-solution theorem. A changing unbounded generator requires additional hypotheses developed in Time-Dependent Hamiltonians and Kato Theory.

Exercises

1. Let $H\phi=E\phi$ with $\|\phi\|=1$, take $\psi_0=0$, and use the resonant source $f(t)=F e^{-itE/\hbar}\phi$ for constant $F$. Find the solution.

Solution

In the integral, $U(t-s)f(s)=F e^{-itE/\hbar}\phi$ is independent of $s$ apart from the fixed final phase. Therefore $\psi(t)=-(iFt/\hbar)e^{-itE/\hbar}\phi$. Its norm is $|F|t/\hbar$, saturating the source estimate and illustrating why an additive source is not norm-preserving closed evolution.

2. Suppose a strong approximate path has the correct initial state and $\|r(t)\|\leq\varepsilon$ on $[0,T]$. What error estimate follows, and what does renormalizing the path fail to establish?

Solution

The estimate gives $\|\psi(t)-\chi(t)\|\leq\varepsilon t/\hbar$. Renormalization checks or changes only the norm; it does not bound the vector residual, phase error, or distance from the exact solution. The residual bound must be established independently.

References

• A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences 44, Springer, 1983, Chapter 4, publisher edition — inhomogeneous abstract Cauchy problems and mild solutions.
• G. Teschl, Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators, 2nd ed., Graduate Studies in Mathematics 157, American Mathematical Society, 2014, Section 5.1, author’s text — autonomous unitary evolution and uniqueness.