Schrödinger Equation in Hilbert Space

The equation $i\hbar\dot\psi=H\psi$ can mean several different analytic statements. For a self-adjoint autonomous Hamiltonian, every Hilbert-space initial vector has a continuous unitary evolution, while only domain vectors have a Hilbert-space derivative. The distinction determines what an initial-value problem asserts. The general physical formulation is developed in Time-Dependent Schrödinger Equation; here the operator $H$ is independent of time.

Helpful background. Stone’s Theorem supplies the unitary group and its exact derivative domain. Quadratic Forms distinguishes finite form energy from membership in the operator domain.

Strong Schrödinger solutions

Let $H$ be self-adjoint on $D(H)$ in a complex Hilbert space, and set $U(t)=e^{-itH/\hbar}$. A strong solution on an interval $I$ containing zero is a function

$$\psi\in C^1(I;\mathcal H),\qquad \psi(t)\in D(H),$$

satisfying the equation in $\mathcal H$ at every time and the prescribed initial condition. The equation then also makes $t\mapsto H\psi(t)$ continuous.

For $\psi_0\in D(H)$, Stone’s theorem gives the strong solution $\psi(t)=U(t)\psi_0$. Functional calculus gives

$$U(t)D(H)=D(H),\qquad HU(t)\psi_0=U(t)H\psi_0.$$

In particular, the solution is continuous in the graph norm as well as in the Hilbert norm. These assertions include the boundary conditions encoded in $D(H)$; a formal pointwise derivative of a wavefunction cannot replace them.

For an initial vector outside $D(H)$, the orbit has no strong derivative at any time. Indeed, a derivative at time $t_*$ would, after multiplication by $U(-t_*)$, give a derivative at zero. This does not mean that the orbit has no physical or mathematical meaning.

Weak solutions test the equation against domain vectors

A weak solution here means a norm-continuous $\psi:I\to\mathcal H$ such that, for every fixed $\phi\in D(H)$, the scalar function $\langle\phi,\psi(t)\rangle$ is continuously differentiable and

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

The inner product is antilinear in its first argument. The Hamiltonian acts on the test vector, so the right side is meaningful without assuming $\psi(t)\in D(H)$.

Every unitary orbit is a weak solution. To check this, write

$$\langle\phi,U(t)\psi_0\rangle =\langle U(-t)\phi,\psi_0\rangle.$$

The vector in the first slot is differentiable because $\phi\in D(H)$. Differentiating and using antilinearity produces exactly the displayed weak equation. No derivative of $U(t)\psi_0$ is needed.

The test domain matters. Testing only against functions supported away from a boundary can lose the boundary condition that selects the Hamiltonian. Different self-adjoint extensions may solve the same interior differential equation. The weak formulation above refers to the complete specified operator domain, not just to its formal expression.

The integrated equation has a domain statement too

For every $\psi_0\in\mathcal H$, the norm-continuous orbit has a Bochner integral

$$w(t)=\int_0^t U(s)\psi_0\,ds.$$

Although $U(s)\psi_0$ need not belong to $D(H)$, the integral does. Spectral calculus proves this using the bounded multiplier

$$g_t(\lambda)=\int_0^t e^{-is\lambda/\hbar}\,ds, \qquad \lambda g_t(\lambda) =i\hbar(e^{-it\lambda/\hbar}-1).$$

The second multiplier has absolute value at most $2\hbar$, so $g_t(H)\psi_0\in D(H)$. Consequently

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

This is an integrated solution identity. It is generally invalid to replace $H\int\psi$ by $\int H\psi$ without ensuring that the latter integrand exists. The autonomous orbit is also called the mild solution. Existence and Uniqueness of Time Evolution develops the corresponding uniqueness and forced-evolution statements.

A discontinuous initial state still evolves

Use dimensionless units and let $H=-d^2/dx^2$ on $D(H)=H^2(\mathbb R)$. The normalized square pulse $\psi_0=\mathbf1_{[0,1]}$ belongs to $L^2(\mathbb R)$ but not to $H^1(\mathbb R)$. With the unitary Fourier convention its momentum-space wavefunction is

$$\widehat\psi_0(k) =\frac{1}{\sqrt{2\pi}}\frac{1-e^{-ik}}{ik},$$

with its continuous value at $k=0$. Multiplying this function by $k^2$ does not give an $L^2$ function, so $\psi_0\notin D(H)$. In fact its mean kinetic energy is infinite. The square pulse is therefore an idealized state, while smooth approximations can have finite energies that grow as the edges sharpen.

Its free evolution is a perfectly defined weak and mild solution but is never a strong solution in $L^2$. Unitary Schrödinger evolution preserves the spectral Sobolev regularity; it does not acquire the global smoothing of a heat semigroup. Pointwise or local dispersive regularity statements are different questions and do not contradict this domain invariance.

Norm is conserved for every initial vector. Likewise, its entire energy spectral probability distribution is conserved, so every initially finite energy moment remains finite and unchanged. Energy conservation should not be written as an inner product involving $H\psi$ when only the form expectation exists. See Conserved Quantities for the spectral formulation.

Exercises

1. Determine $g_t(0)$ in the integrated equation and explain what happens to a vector in $\ker H$.

Solution

The integral definition gives $g_t(0)=t$, also the continuous limit of the quotient away from zero. For $H\psi_0=0$, $w(t)=t\psi_0$ and $Hw(t)=0$, so the integrated equation gives the stationary vector $\psi(t)=\psi_0$.

2. For $\psi_0\in D(H^2)$, compute the second time derivative and show that $\|H\psi(t)\|$ is conserved.

Solution

Domain invariance permits a second differentiation, giving $\ddot\psi(t)=-H^2U(t)\psi_0/\hbar^2$. Since $HU(t)\psi_0=U(t)H\psi_0$ and $U(t)$ is unitary, $\|H\psi(t)\|=\|H\psi_0\|$. The latter equality only requires $\psi_0\in D(H)$; the second derivative needs the stronger hypothesis.

References

• B. C. Hall, Quantum Theory for Mathematicians, Graduate Texts in Mathematics 267, Springer, 2013, publisher edition — unbounded operators and quantum evolution.
• 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 and Chapter 7, author’s text — strong autonomous solutions and the free Schrödinger operator.