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.