Boris Mityagin

UNIFORM EMBEDDINGS OF METRIC SPACES AND OF BANACH SPACES INTO HILBERT SPACES

It is proved using positive definite functions that a normed spaceX is unifomly homeomorphic to a subset of a Hilbert space, if and only ifX is (linearly) isomorphic to a subspace of aL0(μ) space (=the space of the measurable functions on a probability space with convergence in probability). As a result we get thatlp (respectivelyLp(0, 1)), 2<p<∞, is not uniformly embedded in a bounded subset of itself. This answers negatively the question whether every infinite dimensional Banach space is uniformly homeomorphic to a bounded subset of itself. Positive definite functions are also used to characterize geometrical properties of Banach spaces.

Eigensystem of an L 2 -perturbed harmonic oscillator is an unconditional basis

For any complex valued Lp-function b(x), 2 ≤ p < ∞, or L∞-function with the norm ‖b↾L∞‖ < 1, the spectrum of a perturbed harmonic oscillator operator L = −d2/dx2 + x2 + b(x) in L2(ℝ1) is discrete and eventually simple. Its SEAF (system of eigen- and associated functions) is an unconditional basis in L2(ℝ).

Riesz bases consisting of root functions of 1D Dirac operators

with periodic matrix potentials v such that P,Q ∈ L([0, π],C), subject to periodic (Per) or antiperiodic (Per) boundary conditions (bc): (1.2) Per : y(π) = y(0); Per : y(π) = −y(0). Our goal is to give necessary and sufficient conditions on potentials v which guarantee that the system of periodic (or antiperiodic) root functions of LPer±(v) contains Riesz bases. The free operators LPer± = LPer±(0) have discrete spectrum: Sp(LPer±) = Γ , where Γ = {

Fourier Method for One-dimensional Schrödinger Operators with Singular Periodic Potentials

By using quasi-derivatives, we develop a Fourier method for studying the spectral properties of one-dimensional Schrodinger operators with periodic singular potentials.

Spectral gap asymptotics of one-dimensional Schrödinger operators with singular periodic potentials

By using quasi-derivatives we develop a Fourier method for studying the spectral properties of one-dimensional Schrödinger operators with periodic singular potentials. Our results reveal the close relationship between the smoothness of the potential and spectral gap asymptotics under the a priori assumption This extends and strengthens similar results proved in the classical case

The Spectrum of a Harmonic Oscillator Operator Perturbed by Point Interactions

We consider the operator Ly=−(d/dx)2y+x2y+w(x)y,yinL2(ℝ),

Differential operators admitting various rates of spectral projection growth

Ly = - (d/dx)^{2}y + x^{2} y + w(x) y, \quad y \text { in} L^{2}(\mathbb {R}),

Trace formula and Spectral Riemann Surfaces for a class of tri-diagonal matrices

where w(x)=sδ(x−b)+tδ(x+b),b≠0real,s,t∈ℂ

Smoothness of solutions of a nonlinear ode

w(x) = s \delta (x - b) + t \delta (x + b) , \quad b \neq 0 \, \, \text {real}, \quad s, t \in \mathbb {C}

Coupling constant behavior of eigenvalues of Zakharov-Shabat systems

. This operator has a discrete spectrum: eventually the eigenvalues are simple. Their asymptotic is given. In particular, if s=−t, λn=(2n+1)+s2κ(n)n+ρ(n)