Correct spaces for quantum mechanics

The general formulation of quantum mechanics is done by describing quantum mechanical states by vectors $|\psi_t(x)\rangle$ in some Hilbert space $\mathcal{H}$ and describes their time evolution by the Schrödinger equation $$i\hbar\frac{\partial}{\partial t}|\psi_t\rangle = H|\psi_t\rangle$$ where $H$ is the Hamilton operator (for the free particle we have $H=-\frac{\hbar^2}{2m}\Delta$).

Now I have often seen used spaces like $\mathcal{H}=L^2(\mathbb{R}^3)$ (in the case of a single particle), but I was wondering whether this is correct or not. In fact shouldn't we require to be able to derivate $\left|\psi_t\right>$ twice in $x$ and thus choose something like $\mathcal{H} = H^2(\mathbb{R}^3)$?

If we treat directly $\psi(t,x) := \psi_t(x)$, shouldn't we require them to be in something like $H^1(\mathbb{R};H^2(\mathbb{R}^3))$? i.e., functions in $H^1(\mathbb{R})$ with values in $H^2(\mathbb{R}^3)$, e.g. the function $t\mapsto\psi_t$.

Solution 1:

Unfortunately, taking $\mathcal{H}=H^2(\mathbb{R}^3)$ won't work. If $f\in H^2(\mathbb{R}^3)$ then, in general, it is not true that $\Delta f\in H^2(\mathbb{R}^3)$. This is a problem, because the Hamiltonian must be an operator from $\mathcal{H}$ to $\mathcal{H}$.

The only mathematically sound way out is taking $\mathcal{H}=L^2(\mathbb{R}^3)$ and considering the Laplacian as an unbounded operator, meaning that it is not defined on the whole $L^2(\mathbb{R}^3)$ space but only on a dense subspace. This idea dates back to von Neumann.

Solution 2:

For the linear Schrödinger equation on $\mathbb{R}^n$ $$ \begin{cases} \partial_t u = i\Delta u \\ u(x,0)=u_0(x) \end{cases} $$ the solution propagator $e^{it\Delta}u_0=(e^{-4\pi it|\xi|^2}\hat{u_0})^{\vee}$ forms a unitary group on $H^s$ for every $s \in \mathbb{R}$ as it's a Fourier multiplier of modulus 1. Thus we wouldn't have a solution in $H^1(\mathbb{R}:H^s(\mathbb{R}^n))$ (no decay in time of $H^s$-norm), but $C(\mathbb{R}:H^s(\mathbb{R}^n))$.

For conjugate exponents $p,p'$ with $p' \in [1,2]$ we have that the propagator $e^{it\Delta} : L^{p'}(\mathbb{R}^n) \rightarrow L^p(\mathbb{R}^n)$ is continuous with $$ \|e^{it\Delta}f\|_p \leq c|t|^{-n/2(1/p'-1/p)}\|f\|_{p'}. $$ This can be proved by interpolating the $L^2$ isometry result along with the case $p=1,p'=\infty$ (which is handled by Young's inequality). This leads to the so called global smoothing effects or Strichartz estimates. For example, the above result along with Hardy-Littlewood-Sobolev inequality implies $$ \left(\int_\mathbb{R} \|e^{it\Delta}f\|_p^q \; dt \right)^{1/q} \leq c\|f\|_2 $$ given some relations on $n,p$ and $q$.

Now you can attack nonlinear problems like the semilinear Schrödinger equation $$ \begin{cases} i\partial_t u = -\Delta u - \lambda |u|^{\alpha-1}u \\ u(x,0)=u_0(x) \end{cases} $$ for $\alpha>1$. One approach is to use Duhamel's principle and solve the (weak) integral formulation $$ u(t) = e^{it\Delta}u_0 + i\lambda \int_0^t e^{i(t-t')\Delta}(|u|^{\alpha-1}u)(t') \; dt' $$ via the contraction mapping principle on an appropriate space. The Strichartz estimates play a key role in deriving the contraction property. For a result about local existence for $H^1$ data, see this paper of Kato in which he produces a solution in $C(I:H^1) \cap C^1(I:H^{-1})$ for some interval $I$ given assumptions on the nonlinear term. More references may be found on the Dispersive Wiki. For more mathematical details on Strichartz estimates and the contraction argument, check out Cazenave's Semilinear Schrödinger Equations.