arXiv subjects
Publications and source records attributed to Evgeny L. Korotyaev.
We consider time periodic Hamiltonians with complex potentials on the lattice and determine trace formulas. As a corollary we estimate eigenvalues of the quasienergy operator in terms of the norm of potentials.
We solve inverse scattering problem for Schrödinger operators with compactly supported potentials on the half line. We discretize S-matrix: we take the value of the S-matrix on some infinite sequence of positive real numbers. Using this sequence obtained from S-matrix we recover uniquely the potential by a new explicit formula, without the Gelfand-Levitan-Marchenko equation.
We consider Schrödinger operators on the line with potentials that are periodic with respect to the coordinate variable and real analytic with respect to the energy variable. We prove that if the imaginary part of the potential is bounded in the right half-plane, then the high energy spectrum is real, and the corresponding asymptotics are determined. Moreover, the Dirichlet and Neumann problems are considered. These results are used to analyze the good Boussinesq equation.
We consider an inverse problem for Laplacians on rotationally symmetric manifolds, which are finite for the transversal direction and periodic with respect to the axis of the manifold, i.e., Laplacians on tori. We construct an infinite dimensional analytic isomorphism between the space of profiles (the radius of the rotation) of the torus and the spectral data as well as the stability estimates: those for the spectral data in terms of the profile and conversely, for the profile in term of the spectral data.
We consider the discrete Laplacian $Δ$ on the cubic lattice $\mathbb Z^d$, and deduce estimates on the group $e^{itΔ}$ and the resolvent $(Δ-z)^{-1}$, weighted by $\ell^q(\mathbb Z^d)$-weights for suitable $q\geq 2$. We apply the obtained results to discrete Schrödinger operators in dimension $d\geq 3$ with potentials from $\ell^p(\mathbb Z^d)$ with suitable $p\geq 1$.
We consider resonances for fourth order differential operators on the half-line with compactly supported coefficients. We determine asymptotics of a counting function of resonances in complex discs at large radius, describe the forbidden domain for resonances and obtain trace formulas in terms of resonances. We apply these results to the Euler-Bernoulli operator on the half-line. The coefficients of this operator are positive and constants outside a finite interval. We show that this operator does not have any eigenvalues and resonances iff its coefficients are constants on the whole half-line.
We consider the Stark operator perturbed by a compactly supported potential (of a certain class) on the real line. We prove the following results: (a) upper and lower bounds on the number of resonances in complex discs with large radii, (b) the trace formula in terms of resonances only, (c) all resonances determine the potential uniquely.
We solve the inverse spectral problem for rotationally symmetric manifolds, which include the class of surfaces of revolution, by giving an analytic isomorphism from the space of spectral data onto the space of functions describing the radius of rotation. An analogue of the Minkowski problem is also solved.
We show the existence of a real analytic isomorphism between a space of impedance function $ρ$ of the Sturm-Liouville problem $- ρ^{-2}(ρ^2f')' + uf$ on $(0,1)$, where $u$ is a function of $ρ, ρ', ρ''$, and that of potential $p$ of the Schr{ö}dinger equation $- y'' + py$ on $(0,1)$, keeping their boundary conditions and spectral data. This mapping is associated with the classical Liouville transformation $f \to ρf$, and yields a global isomorphism between solutions to inverse problems for the Sturm-Liouville equations of the impedance form and those to the Schr{ö}dinger equations.
We consider the Schrödinger operator with a periodic potential $p$ on the real line. We assume that $p$ belongs to the Sobolev space $\mH_m$ on the circle for some $m\ge -1$, and we determine the asymptotics of the quasimomentum and the Titchmarsh-Weyl functions, the Bloch functions at high energy.
We consider the Schr{ö}dinger operator $-Δ+V(x)$ in $L^2({\bf R}^3)$ with a real short-range (integrable) potential $V$. Using the associated Fredholm determinant, we present new trace formulas, in particular, the ones in terms of resonances and eigenvalues only. We also derive expressions of the Dirichlet integral, and the scattering phase. The proof is based on the change of view points for the above mentioned problems from the operator theory to the complex analytic (entire) function theory.
We consider the Schrödinger operator $H$ on the half-line with a periodic potential $p$ plus a compactly supported potential $q$. For generic $p$, its essential spectrum has an infinite sequence of open gaps. We determine the asymptotics of the resonance counting function and show that, for sufficiently high energy, each non-degenerate gap contains exactly one eigenvalue or antibound state, giving asymptotics for their positions. Conversely, for any potential $q$ and for any sequences $(\s_n)_{1}^\iy, \s_n\in \{0,1\}$, and $(\vk_n)_1^\iy\in \ell^2, \vk_n\ge 0$, there exists a potential $p$ such that $\vk_n$ is the length of the $n$-th gap, $n\in\N$, and $H$ has exactly $\s_n$ eigenvalues and $1-\s_n$ antibound state in each high-energy gap. Moreover, we show that between any two eigenvalues in a gap, there is an odd number of antibound states, and hence deduce an asymptotic lower bound on the number of antibound states in an adiabatic limit.
We consider the Schrödinger operators on zigzag nanoribbons (quasi-1D tight-binding models) in external magnetic fields and an electric potential $V$. The magnetic field is perpendicular to the plane of the ribbon and the electric field is perpendicular to the axis of the nanoribbon and the magnetic field. If the magnetic and electric fields are absent, then the spectrum of the Schrödinger (Laplace) operator consists of two non-flat bands and one flat band (an eigenvalue with infinite multiplicity) between them. If we switch on the magnetic field, then the spectrum of the magnetic Schrödinger operator consists of some non-flat bands and one flat band between them. Thus the magnetic field changes the continuous spectrum but does not the flat band. If we switch on a weak electric potential $V\to 0$, then there are two cases: (1) the flat band splits into the small spectral band. We determine the asymptotics of the spectral bands for small fields. (2) the unperturbed flat band remains the flat band. We describe all potentials when the unperturbed flat band remains the flat band and when one splits into the small band of the continuous spectrum. Moreover, we solve inverse spectral problems for small potentials.
We consider the defocussing NLS equation with small periodic initial condition. A new approach to study the Hamiltonian as a function of action variables is demonstrated. The problems for the NLS equation is reformulated as the problem of conformal mapping theory corresponding to quasimomentum of the Zakharov-Shabat operator. The main tool is the Löwner type equation for the quasimomentum. In particular, we determine the asymptotics of the Hamiltonian for small action variables. Moreover, we determine the gradient of Hamiltonian with respect to action variables. This gives so called frequencies and determines how the angles variables depend on the time.
We consider the KdV equation on the Sobolev space of periodic distributions. We obtain estimates of the solution of the KdV in terms of action variables.