Instead of being posed on the real line as the spatial domain, we take for space the integer lattice $\mathbb{Z}$. So we are looking at functions $\phi: \mathbb{R}\times\mathbb{Z} \ni(t,n) \mapsto \phi(t,n) \in \mathbb{C}$. The Schroedinger equation is copied over, but with the usual spacial Laplacian replaced by the discrete Laplacian. So it is convenient to write the equation as \[ i \frac{d}{dt} \phi(t,n) = 2 \phi(t,n) - \phi(t,n+1) - \phi(t,n-1) .\] One can also add a potential term, but for this basic discussion we will omit it.
Since the equation is discrete translation invariant, and also invariant under time translation, we can hope to do some sort of "separation of variables". This motivates us to look at "eigenfunctions" of the discrete Laplacian. Namely, let's consider the equation \[ 2 \phi(n) - \phi(n+1) - \phi(n-1) = E \phi(n) \] for functions on $\mathbb{Z}$. It is convenient to treat this as a first order recurrence system in two unknowns: \[ \begin{pmatrix} \phi(n+1) \newline \phi(n) \end{pmatrix} = \underbrace{\begin{pmatrix} 2 - E & -1 \newline 1 & 0 \end{pmatrix}}_{S_E} \begin{pmatrix} \phi(n) \newline \phi(n-1) \end{pmatrix} \] We can understand the behavior of the solutions by looking at the eigenvalues of $S_E$.
Note first that $\det(S_E) = 1$, so the classification of eigenvalues depends only on the trace (thanks to the characteristic equation). We see that there are three regimes (when $E \in \mathbb{R}$)
- When $E\in (0,4)$, the trace is in $(-2,2)$, and hence the two eigenvalues are complex conjugates of the form $\exp(\pm \kappa i)$, with $2\cos(\kappa) = 2 - E$.
- When $E\not\in [0,4]$, both eigenvalues are real, and given by $\frac12 [ 2-E \pm \sqrt{E(E-4)}]$. Note that necessarily one has modulus larger than $1$ and the other less than $1$.
- When $E\in {0,4}$, there is an eigenvalue (either $1$ or $-1$) with algebraic multiplicity 2, but geometric multiplicity 1. In this case $S_E$ is conjugate to a shear transformation.
$E$-dependent asymptotics
We see therefore that when $E\in (0,4)$, the solutions to the eigenvalue equation are uniformly bounded.
When $E\not\in[0,4]$, the solutions to the eigenvalue equation are unbounded: they must grow exponentially as $n\to +\infty$ or $\to -\infty$.
When $E\in {0,4}$, generic solutions grow linearly as $n\to\pm\infty$, while there is an exceptional solution that remains uniformly bounded.
We can contrast this to the continuous case; in this case the ODE to solve is \[ -\phi''(x) = E \phi(x). \] And the classification is different: for $E < 0$ solutions are linear combinations of $\exp(\pm \sqrt{|E|}x)$ and grow exponentially at infinity. For $E > 0$ the solutions are linear combinations of $\exp( \pm \sqrt{|E|} i x)$ and remain oscillatory and uniformly bounded. While for $E = 0$ we have both the exceptional constant solution and the generic linear solutions.
This difference can be interpreted as follows: by imposing a discrete domain, we are forcing a minimum wavelength to oscillations, which forces a maximum frequency. This is an ultraviolet cutoff and hence caps the available (kinetic) energy of the wave functions. (This is also why lattice QCD succeeds as a theory.)
Fourier Transform
An alternative is to study the original problem using the Fourier transform. Where as for functions on the real line we have $\mathscr{F}\phi'' = - \xi^2 \hat{\phi}(\xi)$, the multiplier is different in the discrete case.
For functions on $\mathbb{Z}$, its Fourier transform is a function on $\mathbb{T}$. Our Fourier bases are functions of the form $\exp(i \theta n)$, acting on which we find the discrete Laplacian yields \[ 2 \exp(i \theta n) - \exp(i \theta(n+1) - \exp(i\theta(n-1)) = \exp(i\theta n) \cdot [ 2 - 2 \cos\theta ] = 4\sin^2(\theta/2) \exp(i \theta n). \] So we see that the (positive) discrete Laplacian becomes multiplication by $4\sin^2(\theta/2)$ on the Fourier side.
The half frequency phenomenon is the same as Shannon-Nyquist.
Dispersion
An interesting fact is that this means that discrete Schroedinger equation has different dispersive behavior from the continuous case. In the continuous case, let $G$ be the Green's function obtained from solving the linear Schroedinger equation with $\delta_0$ initial data. For each $t \neq 0$ the object $G(t,\cdot)$ is a tempered distribution that is uniformly bounded, and by scaling homogeneity we see this means that generic solutions to the linear free Schroedinger equation on the line decays like $t^{-1/2}$.
(We have slightly more. If the initial data is a wave packet that is spatially concentrated near 0 and frequency concentrated near $\xi_0$, then the solution will be concentrated in a region moving with speed $2\xi_0$ that also broadens. Outside the region the solution decays much faster.)
Looking at the proof again, we used scaling homogeneity to reduce to finding an $L^\infty$ bound of $G(1,\cdot)$, which is defined as the oscillatory integral \[ \int e^{-i\xi^2} e^{ix\xi} ~d\xi. \]
In the discrete case, we have no scaling, since there is a fixed spatial scale. The solution operator is of the form \[ \phi_0(n) \mapsto \int_0^{2\pi} \sum_m e^{-i 4 t \sin^2(\theta/2)} e^{i (n-m)\theta} \phi_0(m) ~d\theta. \] So the analogous statement would be established by looking at the large $t$ asymptotics of the oscillatory integral \[ t \mapsto \sup_m \int_0^{2\pi} \exp[ -i (4t\sin^2(\theta / 2) - m \theta) ] ~d\theta \] If we try to apply van der Corput, we see that the phase function $\eta(\theta; t,m) = t\sin^2(\theta/2) - m \theta$ satisfies \[ \begin{gathered} \eta' = 2t \sin(\theta) - m \newline \eta'' = 2t \cos(\theta) \newline \eta''' = 2 t \sin(\theta) \end{gathered} \] We see that whenever $t = m/2$ is a half integer, the point $\theta = \pi/2$ is a second-order critical point of $\eta$, so we have to apply the third order van der Corput lemma, which gives uniform decay at rate $t^{-1/3}$.
On the other hand, observe also that:
- When $|m| > (2+\epsilon) |t|$, we have $|\eta'| > \epsilon |t|$, giving superpolynomial decay rates by integration by parts. So that for initial data with bounded support (say $|n| \leq N$), at time $t$ the solution is concentrated on the interval $|n| \leq N + 2|t|$. So we have something similar to finite speed of propagation. This is very different from the continuum case, where wave packets can propagate at essentially arbitrary speed.
- When $|m| < (2-\epsilon) |t|$, we have $|\eta'|^2 + |\eta''|^2 = 4t^2 + m^2 - 4tm \sin(\theta) \geq (2|t| - |m|)^2 > \epsilon^2 |t|^2$. So for initial data with bounded support on $|n| \leq N$, restricted to the region $|n| \leq N + (2-\epsilon)|t|$ we have uniform decay of rate $t^{-1/2}$ from second order van der Corput.
- The slow rate of decay is concentrated near the wave-front, which propagates at speed $2$. The uniform decay rate of $t^{-1/3}$ implies, together with $L^2$ conservation, that the wave-front region has a width that grows at most like $|t|^{2/3}$ (sublinearly, in concordance with the previous two bullet points).
In some ways this indicates that the discrete Schroedinger equation behaves qualitatively similar to the continuous Klein-Gordon equation (with nonzero mass).
