Multidimensional L2 conjecture: a survey
In this survey, we give a short overview of the recent progress on the multidimensional L2 conjecture. It can also serve as an introduction to the subject.
math.AP↗
arXiv subjects
Sergey A. Denisov
Publications and source records attributed to Sergey A. Denisov.
Ito's diffusion in multidimensional scattering with sign-indefinite potentials
This paper blends two techniques recently developed in [2] and [3] to prove the presence of absolutely continuous spectrum for the multidimensional Schrodinger operator provided that the potential is summable over trajectory with positive probability. These trajectories are defined by Ito's equation and escape to infinity almost surely.
math.AP↗
An evolution equation as the WKB correction in long-time asymptotics of Schrodinger dynamics
We consider 3d Schrodinger operator with long-range potential that has short-range radial derivative. The long-time asymptotics of non-stationary problem is studied and existence of modified wave operators is proved. It turns out, the standard WKB correction should be replaced by the solution to certain evolution equation.
math.AP↗
Continuous analogs of polynomials orthogonal on the unit circle. Krein systems
This survey contains the introduction to the subject. Many new results are also included.
math.CA↗
Schrodinger operators and associated hyperbolic pencils
For a large class of Schrodinger operators, we introduce the hyperbolic quadratic pencils by making the coupling constant dependent on the energy in the very special way. For these pencils, many problems of scattering theory are easier to study. Then, we give some applications to the original Schrodinger operators.
math.SP↗
Infinite superlinear growth of the gradient for the two-dimensional Euler equation
For two-dimensional Euler equation on the torus, we prove that the uniform norm of the gradient can grow superlinearly for some infinitely smooth initial data. We also show the exponential growth of the gradient for the finite time.
math.AP↗
Wave equation with slowly decaying potential: asymptotics and wave operators
For the one-dimensional case, we establish the long-time asymptotics of solution to Cauchy problem and prove existence of modified wave operators. In particular, we show that the part of the wave travels ballistically if the potential is square summable and this result is sharp.
math.AP↗
Weak asymptotics for Schrodinger evolution
We apply technique developed in [2] to study the long-time behavior of Schrodinger evolution.
math.AP↗
The generic behavior of solutions to some evolution equations: asymptotics and Sobolev norms
We study generic behavior of solutions to a large class of evolution equations. The methods are applied to Schrodinger evolution on the circle.
math.AP↗
Wave propagation through sparse potential barriers
We prove that 3-dimensional Schrodinger operator with slowly decaying sparse potential has an a.c. spectrum that fills positive half-line. A new kind of WKB asymptotics for Green's function is found. The absense of positive eigenvalues is established as well.
math-ph↗
Spectral Properties of Schrödinger Operators with Decaying Potentials
We review recent advances in the spectral theory of Schrödinger operators with decaying potentials. The area has seen spectacular progress in the past few years, stimulated by several conjectures stated by Barry Simon starting at the 1994 International Congress on Mathematical Physics in Paris. The one-dimensional picture is now fairly complete, and provides many striking spectral examples. The multidimensional picture is still far from clear and may require deep original ideas for further progress. It might hold the keys for better understanding of a wide range of spectral and dynamical phenomena for Schrödinger operators in higher dimensions.
math.SP↗
On the preservation of absolutely continuous spectrum for Schrodinger operators
We present general principles for the preservation of a.c. spectrum under weak perturbations. The Schrodinger operators on the strip and on the Caley tree (Bethe lattice) are considered.
math-ph↗
Zeros of orthogonal polynomials on the real line
Let $p_n(x)$ be orthogonal polynomials associated to a measure $dμ$ of compact support in $R$. If $E\not\in supp(dμ)$, we show there is a $δ>0$ so that for all $n$, either $p_n$ or $p_{n+1}$ has no zeros in $(E-δ, E+δ)$. If $E$ is an isolated point of $supp(dμ)$, we show there is a $δ$ so that for all $n$, either $p_n$ or $p_{n+1}$ has at most one zero in $(E-δ, E+δ)$. We provide an example where the zeros of $p_n$ are dense in a gap of $supp(dμ)$.
math.CA↗