Related papers: Reflectionless analytic difference operators II. R…

In a recent paper Birke and Bissantz (2008) considered the problem of nonparametric estimation in inverse regression models with convolution-type operators. For multivariate predictors nonparametric methods suffer from the curse of…

Reference [1] established an index theory for a class of linear selfadjoint operator equations covering both second order linear Hamiltonian systems and first order linear Hamiltonian systems as special cases. In this paper based upon this…

Scaling similarity solutions of three integrable PDEs, namely the Sawada-Kotera, fifth order KdV and Kaup-Kupershmidt equations, are considered. It is shown that the resulting ODEs may be written as non-autonomous Hamiltonian equations,…

We introduce a relativistic version of the non-self-adjoint operator obtained by a dilation analytic transformation of the quantum harmonic oscillator. While the spectrum is real and discrete, we show that the eigenfunctions do not form a…

An overview is given of basic models combining discreteness in their linear parts (i.e. the models are built as dynamical lattices) and nonlinearity acting at sites of the lattices or between the sites. The considered systems include the…

We study the time evolution of quantum systems with a time-dependent non-Hermitian Hamiltonian given by a linear combination of SU(1,1) and SU(2) generators.With a time-dependent metric, the pseudo-Hermitian invariant operator is…

We review recent developments of soliton theories and integrable systems on noncommutative spaces. The former part is a brief review of noncommutative gauge theories focusing on ADHM construction of noncommutative instantons. The latter…

The Jacobi matrices with bounded elements whose spectrum of multiplicity 2 is separated from its simple spectrum and contains an interval of absolutely continuous spectrum are considered. A new type of spectral data, which are analogous for…

We rigorously analyze the asymptotics of soliton gases to the short-pulse (SP) equation. The soliton gas is formulated in terms of a RH problem, which is derived from the RH problems of the $N$-soliton solutions with $N \to \infty$.…

We address the problem of possible deformations of exactly solvable potentials having finitely many discrete eigenvalues of arbitrary choice. As Kay and Moses showed in 1956, reflectionless potentials in one dimensional quantum mechanics…

We study the properties of reflectionless measures for an $s$-dimensional Calder\'on-Zygmund operator $T$ acting in $\mathbb{R}^d$, where $s\in (0,d)$. Roughly speaking, these are the measures $\mu$ for which $T(\mu)$ is constant on the…

In $L_2(\mathbb{R}^d;\mathbb{C}^n)$, we consider a selfadjoint matrix strongly elliptic second order differential operator $\mathcal{A}_\varepsilon$, $\varepsilon >0$. The coefficients of the operator $\mathcal{A}_\varepsilon$ are periodic…

We consider the problem of discrete-time signal denoising, focusing on a specific family of non-linear convolution-type estimators. Each such estimator is associated with a time-invariant filter which is obtained adaptively, by solving a…

We study spaces of reflectionless Jacobi matrices. The main theme is the following type of question: Given a reflectionless Jacobi matrix, is it possible to approximate it by other reflectionless and, typically, simpler Jacobi matrices of a…

In this paper, we consider a second order nonlinear ordinary differential equation of the form $\ddot{x}+k_1\frac{\dot{x}^2}{x}+(k_2+k_3x)\dot{x}+k_4x^3+k_5x^2+k_6x=0$, where $k_i$'s, $i=1,2,...,6,$ are arbitrary parameters. By using the…

A class of non-local non-linear Schrodinger equations(NLSE) is considered in an external potential with space-time modulated coefficient of the nonlinear interaction term as well as confining and/or loss-gain terms. This is a generalization…

We consider the Jacobi operator (T,D(T)) associated with an indeterminate Hamburger moment problem, i.e., the operator in $\ell^2$ defined as the closure of the Jacobi matrix acting on the subspace of complex sequences with only finitely…

We review some results on the logarithmic convexity for evolution equations, a well-known method in inverse and ill-posed problems. We start with the classical case of self-adjoint operators. Then, we analyze the case of analytic…

We turn back to the well known problem of interpretation of the Schrodinger operator with the pseudopotential being the first derivative of the Dirac function. We show that the problem in its conventional formulation contains hidden…

Let D be a self-adjoint operator on a Hilbert space H and x a bounded operator on H. We say that x is n-times weakly D-differentiable, if for any pair of vectors a, b from H the function < exp(itD)x exp(-itD) a, b> is n-times…