May 2011. Consider $H_0=-\Delta, x\in \mathbb{R}^d$. Going on the Fourier side one can see that $H_0$ is equivalent to multiplication by $|\omega|^2$ and so the spectrum of $H_0$ is purely absolutely continuous (a.c.). From the physics perspective, the presence of a.c. spectrum is an indication that the wave propagation governed by

$\displaystyle i\psi_t=H_0\psi$

does have a transport effect (though without much specifics).

Perturb the Laplacian as follows $H=H_0+V(x)$ where $V(x)$ is potential and ask the question what is a minimal assumptions on $V$ to guarantee that the a.c. spectrum is preserved. Make it a perturbation theory question, assume that $V$ is in some weighted Lebesgue space $L^p_w$. Then, what are the critical $p$ and $w$? In one-dimensional case, one answer is $L^2(\mathbb{R})$. This is a critical space. This result was proved for Schrodinger in a great paper by Deift and Killip but for Dirac it was known for at least half a century and dates back to the works by Mark Krein. Krein’s result on Dirac, however, is only a continuous analog of the classical results for polynomials orthogonal on the unit circle (Szego case).

For $d>1$, the conjecture by Barry Simon is that

$\displaystyle \int_{\mathbb{R}^d} \frac{V^2(x)}{|x|^{d-1}+1}dx <\infty$

is sufficient for the preservation of a.c. spectrum. Very little is known so far. Only the case of Schrodinger on the Cayley tree is well-understood. Take a rooted Cayley tree $\mathbb{B}$ with the origin at $O$ and assume that each vertex has exactly three neighbors while $O$ has only two. Consider the Laplacian on $\mathbb{B}$ defined at each point as the sum over the neighbors and then a simple calculation shows that the spectrum of $H_0$ is $[-2\sqrt 2, 2\sqrt 2]$ and it is purely a.c. Then, perturb by $V$. The multidimensional $L^2$ result reads as follows. Consider all paths that go from $O$ to infinity without self-intersections (rays). Put the probability measure on them by tossing a Bernoulli coin at each vertex. Then, the claim is that the a.c. spectrum contains the a.c. spectrum of unperturbed operator if with positive probability the potential $V(X_n)\in \ell^2$ where $X_n$ denotes the path from the origin. There is more quantitative version, of course, which implies Simon’s conjecture if the Jensen inequality is applied. The condition we have here is more general and more physically appealing: it says that we only need enough directions were the potential is small for the particle to propagate.

For $d>1$, sparse or slowly decaying and oscillating potentials can be handled. If the potential does not oscillate, then the scattering process is quite complicated, it is governed by very intricate evolution equation (that captures semiclassical WKB correction as very special case). This evolution equation is poorly studied and much work is needed in this direction. Soft one-dimensional methods seem to be of little help.

In Euclidean case, what would be the analog of the probability space on the set of paths that escape to infinity? This question was addressed here. It turns out that there is a natural Ito’s stochastic equation that describes these paths. The statement we have is somewhat weak though. It says that the a.c. spectrum contains the positive half-line if $V(X_t)\in L^1(\mathbb{R}^+)$ with positive probability and we can not yet replace summability by the square summability over the path $X_t$. Nevertheless, even this result gives rise to interesting questions like how one computes probabilities given by this Ito’s calculus? That conventionally can be reduced to the analysis of the corresponding potential theory and the modified harmonic measure. The potential theory one encounters in this case is somewhat in between elliptic and the parabolic one: on the large scale it is parabolic and on the small scale it is elliptic. The estimates on the harmonic measure in terms of the geometric properties of support can be found in my paper with Kupin.

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May 2012. I recently finished writing a survey for the Nikolskii conference volume, it contains more details.