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.

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

This problem is motivated by the multidimensional $L^2$ conjecture posed by Barry Simon. It deserves a post on its own but the crux of the matter is this. Take a model case:

$iu(x,t)=k\partial^2_{xx} u(x,t)+V(x,t)u(x,t)$

where $x\in \mathbb{T}, t\in [0,\infty)$ and the parameter $k$ is real as well as the potential $V(x,t)$. We define the Cauchy problem by taking $u(x,0)=1$. Then, assuming $V$ is sufficiently regular, we can define the unique solution. Its $L^2(\mathbb{T})$ norm is preserved in time. However, how fast do the Sobolev norms can grow? Even for fixed $k$, this is an interesting problem but we want to know what happens for Lebesgue generic $k$. My conjecture is that for $V\in L^2(\mathbb{T},[0,\infty))$ the $H^1(\mathbb{T})$ Sobolev norm is bounded in time. This is true for a.e. $k$. The partial results were recently obtained here but more analysis is needed. For example, for the transport equation the solution can be explicitly written and the Carleson theorem on the Fourier series yields the sharp result. Here is Harmonic analysis in action. For the general case the combination of soft-analysis methods and the Christ-Kiselev type perturbation arguments is needed. The quantity most relevant to multidimensional case is

$\displaystyle \int\limits_{-1}^1 \|u\|^2_{\dot{H}^1}dk$

One wants to control it by $L^2$ norm of $V$. The small values of $k$ correspond to the small gaps in the spectrum of the diagonal operator and this regime is the most difficult for analysis.

The difficulty of the problem is this– if one takes $V$ complex-valued then the $L^2$ norm of the solution is not preserved in time anymore. So, whatever analytical machinery one applies, it must be sensitive enough to account for that fact. A good classical example is function $\sin(x)$. Its Taylor expansion is hardly useful for the analysis of this function for large real $x$.

Let $\mathbb{T}$ denote the unit circle and let $d\sigma$ be a probability measure on $\mathbb{T}$. Take the sequence $\{1, z ,z^2, \ldots\}$ and orthonormalize in the Hilbert space $L^2(d\sigma)$ to produce $\{\phi_0(z),\phi_1(z),\phi_2(z),\ldots\}$.
We have

$\displaystyle \int_\mathbb{T} |\phi_n(z)|^2d\sigma=1$

by definition. The question is what is the size of $\|\phi_n\|_{L^p(d\sigma)}$ if $p> 2$ and we assume some additional information on the measure $\sigma$? This is a classical question in approximation theory.

The Steklov’s conjecture was: assume that $d\sigma$ is purely a.c. and has the weight uniformly bounded away from zero, i.e.

$d\sigma=w(\theta)d\theta, \, w(\theta)>\delta>0$

Is it true that $|\phi_n(z)| uniformly in $n$ and $z\in \mathbb{T}$? The negative answer to this question was given by Rakhmanov. He proved that the possible growth is essentially $\sqrt{n}$ up to an inverse logarithmic factor. Can one drop it? This is a nice problem.

Another important class of measures extensively studied in the literature is the so-called Szego class. It is defined as follows:
$d\sigma=d\sigma_s+w(\theta)d\theta$ where the singular component is arbitrary and for $w(\theta)$ one has

$\displaystyle \int_\mathbb{T} \log w(\theta)d\theta>-\infty$

How large the polynomials can be in this case? Well, for a while the conjecture was that the polynomial entropy is bounded, i.e.

$\displaystyle \int_{\mathbb{T}} \log|\phi_n(z)| |\phi_n (z)|^2d\sigma(z)

uniformly in $n$. That, however, again tuned out to be wrong with the possible growth like $\sqrt n$. This result is sharp.

Then, what restriction on the size of $\phi_n(z)$ do we get from the orthogonality? Not clear to me.

The 2D Euler equation can be written in the following form

$\dot \theta=\nabla \theta\cdot \nabla^\perp \Delta^{-1}\theta \quad\quad (1)$

where $\theta(x,y,t)$ is $2\pi$– periodic in both $x$ and $y$ and it denotes the vorticity of the velocity field. The symbol $\nabla^\perp$ stands for $(\partial_y, -\partial_x)$. To define the Cauchy problem, one needs to specify $\theta(x,y,0)=\theta_0(x,y)$

This is a transport equation with the divergence-free vector field and so all $L^p$ norms of $\theta$ are preserved in time. The global regularity is known for many functional spaces, e.g. $C^\infty$ or $H^s$ with large $s$ will do. The quantitative version is a bound

$\|\theta(t)\|_{H^s}\lesssim (1+\|\theta(0)\|_{H^s})^{e^{Ct}}, s\geq 2$

Is this estimate sharp? I do not know but one can have double exponential growth for arbitrarily large but finite time. The lower bound valid for all time is only superlinear.

Another interesting problem is dynamics of patches. Assume that $\theta_0(x,y)$ is the characteristic function of a domain $\Omega(0)$ and the problem is considered on the whole plane, not on the 2d torus. Then, $\theta(x,y,t)=\chi_{\Omega(t)}$ and the boundary of $\Omega(t)$ is smooth provided that the boundary of $\Omega(0)$ was smooth. However, one can study the growth of curvature and the rate of merging (if there are several patches).

The merging mechanism was studied recently and the INFINITE double exponential rate is possible at least if the regular strain is present. This result is sharp.

If one modifies equation $(1)$ for vorticity by writing $\Delta^{-1/2}$ instead of $\Delta^{-1}$, then the resulting equation is called SQG (surface quasi-geostrophic). The outstanding conjecture in the field is that the singularity forms in finite time. No proof so far.

*** I recently updated the paper on the double exponential growth of vorticity gradient over the finite time. Check my webpage or arXiv. Not too many changes: a couple of typos fixed, literature added. New rather interesting angle: the statement happens to be equivalent to Euler evolution being linearly unbounded in the Lipschitz norm. ***

*** I have completely rewritten the paper on the sharp corner formation and infinite double exponential merging. The main result is the same but the exposition was improved (hopefully) a lot. I also did a self-similarity analysis. ***

*** I have finished one more revision of the paper on the corner formation and you can check either my webpage or arxiv. The estimates are now sharp and the regularity of the strain is as optimal as the method can give. I think I approach the stage when the paper can finally be read by the students. ***