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Roman Bessonov and I just posted the paper “A spectral Szego theorem on the real line” on arxiv. You can also read it here. Given a probability measure $\sigma$ on the unit circle, one can ask when the analytic polynomials are NOT dense in $L^2(\sigma)$. The theorem of Szego claims that this is so iff any of the following conditions holds:

• $\int_{T}\log \sigma'd\theta>-\infty$
• The sequence of recurrence parameters $\{a_n\}$ (or Schur parameters) of polynomials orthogonal with respect to $\sigma$ belongs to $\ell^2$.

Given a measure on the real line $\mu$ that satisfies normalization

$\displaystyle \int_{R}\frac{d\mu(t)}{1+t^2}<\infty$,

we can ask the question when the set of functions

$\displaystyle \int_0^\infty e^{itx}f(x)dx, f\in C^\infty_c(0,\infty)$

is NOT dense in $L^2(\mu)$. The answer is given by the theorem of Kolmogorov-Krein-Wiener: it is iff

$\displaystyle \int_{R} \frac{\log \mu'}{1+t^2}dt>-\infty$

However, the spectral characterization of this condition has been missing. In the paper, we consider the Krein string, – the “mother of all non-negative self-adjoint operators with simple spectrum”. It is given by the formal differential operator

$\displaystyle S=-\frac{\partial}{\partial M} \left( \frac{\partial}{\partial t}\right)$

where $M$ is any non-decreasing function on $R^+$. The corresponding self-adjoint operator can be defined and its spectral measure $\mu$ along with one additional real parameter determines $M$ completely. In the paper, we characterize all strings $M$ for which the logarithmic integral of $\mu$ converges. This is done by proving analogous statement for diagonal De Branges canonical systems. The existence of the entropy is important for the prediction theory of stationary Gaussian processes with continuous time. It is likely that the obtained characterization will allow one to quantify some statements in this theory.

In the recent preprint, I study the wave equation for the elliptic operator in divergence form. In $\mathbb{R}^3$, define

$H=-{\rm div }(1+V)\nabla,$

where $V$ oscillates and decays at infinity. More precisely, $V={\rm div} \,Q$ where $\|V\|<\infty, \|Q\|<\infty$ and the norm $\|f\|$ is defined as

$\|f\|=\left(\sum_{n=0}\max_{|x|\in [n,n+1]}|f|^2\right)^{1/2}.$

I also assume $\|V\|_\infty<1$ to make sure that $H$ is non-negative operator. The wave equation for $H$ is

$u_{tt}+Hu=0, \, u(x,0)=f_1,\, u_t(x,0)=f_2.$

The main result of the paper states that the following wave operators

$\lim_{t\to\pm \infty} e^{it\sqrt{H}}e^{-it\sqrt{-\Delta}}f=W^{\pm}f$

exist for every $f\in L^2(\mathbb{R}^3)$ and the limit is understood in $L^2(\mathbb{R}^3)$ norm. The condition on $V$ is optimal in some sense, i.e., the rate of decay is sharp and the oscillation is necessary if the potential is not short-range.

The proof is based on the analysis of the asymptotical behavior of the Green’s function $G(x,y,k^2)$ where $k\in \mathbb{C}^+$, $y$ is fixed, and $|x|\to\infty$. The result about asymptotics is similar to that for the orthogonal polynomials on the circle in the Szego case. The main difference with the one-dimensional situation is that the resulting “Szego” function belongs to the vector-valued Hardy space. The operator $e^{it\sqrt{H}}$ can be written through the resolvent $(H-z)^{-1}$ by the contour integral and this is how the Green’s function enters the proof. The method is quite general and can be adapted to wave equations for the Schrodinger equation and other problems.

In the recent preprint with Jen Beichman we considered the 2D Euler evolution on the tube $S=\mathbb{R}\times \mathbb{T}$. Each rectangle $\Omega_L=[-L,L]\times\mathbb{T}$ is a steady state. We proved that if $L$ is sufficiently large, then these steady states are stable for all time. For example, if one takes a patch $\Omega$ such that $|\Omega\Delta\Omega_L|$ is small, then the Euler evolution $\Omega(t)$ of this patch will  have $|\Omega(t)\Delta \Omega_L|$ small for arbitrary $t$. This result generalizes analogous statement for the stability of the disc on the plane (proved by Sideris-Vega). As Sideris and Vega, we used the method of V. Arnold. The idea of this method is to study the variational problem associated to the conserved quantities. In our case, these are

$I_0=|\Omega(t)|,\,\, I_1=\int_{\Omega(t)} xdxdy,\,\, I_3=\int_{\Omega(t)}\psi dxdy$

where $\psi$ is a stream function given by $\psi=\Delta^{-1}\chi_{\Omega(t)}.$ Then, we set up a variational problem with constraint

$I_3\to\min,\, I_2=0,\, I_1=4\pi L$

We proved that the global minimizer is $\Omega_L$ and if $I_3(E)$ is close to the minimum value for some patch $E$, then $E$ is close to $\Omega_L$ in a weak topology. This essentially gives the required stability.

March 3, 2014.

If one considers the 2d Euler equation of incompressible inviscid fluids on the plane in the vorticity form and takes the initial data as the characteristic function of a certain domain, then the Yudovich theory guarantees that the solution will exist globally and will be equal to the characteristic function of a time-dependent domain which is homeomorphic to the original one for all times. The numerical experiments dating back to the works of P. Saffman and Zabusky et al. indicate the existence of the centrally-symmetric V-states, i.e. a symmetric pair of patches that rotates with constant angular velocity around the origin without changing shape. If the distance between the patches in the pair equals to $\lambda$ and $\lambda>0$, then the boundary of the V-state seems to be smooth. However, when $\lambda=0$, the both patches form a 90 degrees angle at the point of contact. The analytical proof for the existence of these V-states has never been obtained and this is an interesting problem. In the recent preprint, I considered the analogous equation with the cut-off. Loosely speaking, this corresponds to looking at the window around the origin where the contact of the patches is supposed to happen. Mathematically, the model with cut-off is important as it possesses the explicit singular solution: $y_0(x)=|x|$. Then, I addressed the problem of existence of the curve of smooth solutions that converge to $y_0$ in the uniform metric when the parameter $\lambda\to 0$.  Technically, this boils down to application of the implicit function theorem and is somewhat tedious. This technique might be important to better understand the mechanism of the merging and the sharp corner formation in the Euler dynamics. Another important problem is to prove that the merging in finite time is possible for the $\alpha$-model when $\alpha<1$ and is close to $1$.

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.

January 2012. 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 $x$.

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June 2013. I recently posted the new preprint on my webpage in which, in particular, the following statement is proved:

Consider $iu(x,t)=k\partial^2_{xx} u(x,t)+V(x,t)u(x,t), \, u(x,0,k)=1$ in which $V$ is only bounded in both $x\in T$ and $t\in [0,1]$. Then, for a.e. $k$ the solution $u(x,t,k)$ has Sobolev norm $H^\alpha(T)$, $\alpha<1$ uniformly bounded for all $t\in [0,1]$.

This result is based on a recent generalization of Carleson theorem on Fourier maximal function (through the variational norm). The statement is rather striking as it gives more regularity for the solution than one can guess as long as the generic coupling constant is taken.

January  2012. 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. !!! SOLVED, see below. 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. _______________________________________________________________________ August 2013. In the joint paper with A. Aptekarev and D. Tulyakov which you can find on my webpage, we proved the sharp estimates for the uniform norms of the orthogonal polynomials in the Steklov class. This class is defined by the condition that

$\sigma'>\delta$

for a.e. point on the circle where $\delta$ is some small positive constant. The world record so far was made by Rakhmanov about 30 years ago. The method we used turned out to be powerful enough to also give the sharp bounds for the polynomial entropy in this class. We hope to iterate the construction to prove the lower bounds over the subsequence. If we succeed, this will give the full solution to the famous Steklov’s conjecture. ______________________________________________________________________________ February 2014. Building on the previous paper (discussed above), we were able to settle a problem by Steklov. That involved a lot of technical work. Here is the main result: suppose $\sigma$ is a probability measure on the circle which satisfies

$\sigma'(\theta)>\delta/(2\pi),\quad {\rm a.e.} \theta$

and $\delta\in (0,1)$. That condition defines the Steklov’s class. If $\{\phi_n\}$ are the corresponding orthonormal polynomials, then the following upper bound is well-known and easy to prove:

$\|\phi_n\|_\infty=o(\sqrt n)$

In the paper, we prove Theorem. For every sequence $\{\beta_n\}: \beta_n\to 0$, there is an absolutely continuous probability measure $\sigma^*$ from the Steklov class such that

$\|\phi_{k_n}(z,\sigma^*)\|_\infty>\beta_{k_n}\sqrt{k_n}$

for some sequence $k_n$. The proof is constructive and can be used when the measure satisfies different lower bounds. There are other interesting questions that naturally follow from the paper, some of them do not seem to be so hard anymore. The project on getting the uniform bound on the polynomials is mainly finished. What is left wide open is the problem on controlling the $\sup_{n}|\phi_n|$ for a.e. $\theta$ provided the constructive information on measure is known. This is the so-called nonlinear Luzin’s conjecture and we know very little about it. ____________________________________________________________ November 6, 2014  Fix $\delta\in (0,1)$, large $n$ and $p$ such that $p. Consider the class of probability measures defined as $P_{p,\delta}=\{\sigma=\delta \mu/(2\pi)+\sum_{j=1}^p m_j\delta(\theta-\theta_j)\}$ where $d\mu=d\theta$$\{\theta_j\}$ are arbitrary points on $[-\pi,\pi]$, and $m_j\geq 0$. Let $M_{n,p}=\max_{\sigma\in P_{p,\delta}}\|\phi_n(z,\sigma)\|_{L^\infty(\mathbb{T})}$. In the recent paper, I proved the following estimate $M_{n,p}1.5$ It implies, for example, that the maximizer in the Steklov problem has $\sim n$ point masses. In the same paper, I also proved the following inequality improving the bound by Kos: if $\lambda_1<\ldots<\lambda_p$ and $T(x)=\sum_{j=1}^p x_j e^{i\lambda_j x}$, then $|T(0)|\leq p\pi/2\|T\|_{L^2[0,1]}$. That was done by using the duality between two extremal problems for one of which the bound is easy to obtain if the generalization of the Halasz’ result is used. ___________________________________________________________________________________________________________________ May 19, 2015 The solution to a problem by Steklov discussed above allows to study other variational problems. For example, $A_{n,\epsilon}=\sup_{|w-1|\leq \epsilon} \|\phi_n(z,w)\|_{L^\infty(\mathbb{T})}$ Then, the old idea by S.Bernstein and our method used for getting the lower bounds can be combined to get $\log A_{n,\epsilon}\sim \epsilon \log n$ when $\epsilon$ is fixed and $n\to \infty$. In the case when the deviation of the weight is large, we have two results: 1. For every large $T$, there is $\alpha(T)>0$ such that $\|\phi_n(z,w)\|_{L^\infty(\mathbb{T})}\lesssim n^{1/2-\alpha(T)}$ for every weight $w: \delta\leq w\leq T$. 2. For every small $\beta>0$, there is large $T(\beta)$ and a weight $w: \delta\leq w\leq T$ such that $\phi_n(1,w)>n^{1/2-\beta}$. One can see that the $\sqrt n$ growth can not be achieved by weights uniformly bounded in $L^\infty$.

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July 2015.  We significantly simplified several proofs in the paper “On a problem by Steklov” and submitted a new version to arXiv. We also fixed some minor mistakes and typos.

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September 2015. In his original paper on the Steklov’s conjecture, Rakhmanov used a formula for orthogonal polynomial obtained by adding mass points to the measure of orthogonality. Although its uniform norm can grow not faster than $\log n$ (where $n$ is a degree), it is interesting to deduce this polynomial by method used for getting the sharp lower bound in the problem by Steklov. This was carried out in my recent preprint. The interesting feature is a different kind of cancellation. It is less effective than the one giving the $\sqrt n$ bound but it still might be interesting.

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November 2015. The method used to solve the problem of Steklov proves the possibility of $\sqrt n$ growth of polynomials with weights $w$ that satisfy $w>\delta/2\pi$ and $\|w\|_{L^p[0,2\pi]} with arbitrary $p<\infty$. If $w, w^{-1}\in L^\infty[0,2\pi)$ then the upper bound gives an exponent smaller than $1/2$: $\|\phi_n\|_{L^\infty[0,2\pi)}. In a joint project with my student Keith Rush, we proved that $w,w^{-1}\in {\rm BMO}$ is sufficient to have $\|\phi_n\|_{L^\infty[0,2\pi)}. The proof gives more that that. Under these assumptions, we can show that the polynomial entropies are uniformly bounded. In terms of the regularity of weight, this results is rather sharp since the entropies are known to grow logarithmically for $w: w>\delta/(2\pi), \|w\|_{L^p[0,2\pi)} with any $p<\infty$.