continuation for Schroedinger operators. Towards
an optimal result" (with L.Shartser), J.
Funct. Anal., 258 (2010), p. 2662-2681
We prove the property of unique continuation for solutions of differtial inequality |\Delta u| \leq |Vu| (extending a number of classical results), and apply it (using Kato's result) to the problem of absence of positive eigenvalues of self-adjoint Schroedinger operators with form-bounded potentials V.
|§2|| "Towards Oka-Cartan
theory for algebras of fibrewise bounded
holomorphic functions on coverings of Stein
manifolds I. Cartan theorems A and B" (with A.Brudnyi), Preprint, 36
 "Towards Oka-Cartan theory for algebras of fibrewise bounded holomorphic functions II. Elements of function theory" (with A.Brudnyi), Preprint, 53 p (2012) bru_kinz_2.pdf
 "Holomorphic almost periodic functions on coverings of complex manifolds" (with A.Brudnyi)
New York J. Math, 17a (2011), p. 267-300 nyjm.pdf
We establish the basic results of complex function theory within certain algebras of holomorphic functions on coverings of Stein manifolds, including
(1) Bohr's holomorphic almost periodic functions,
(2) algebras of all fibrewise bounded holomorphic functions arising, e.g., in the corona problem.
In particular, we obtain results on holomorphic extension from complex submanifolds, properties of divisors, corona type theorems, holomorphic analogues of the Peter-Weyl approximation theorem, Hartogs type theorems, characterizations of uniqueness sets, etc. Our proofs are based on analogues of Cartan theorems A and B for coherent type sheaves on maximal ideal spaces of these algebras.
semi-almost periodic functions" (with A.Brudnyi), Integr. Equ. Operat.
Theory, 66 (2010),
 "On algebras of holomorphic functons with semi-almost periodic boundary values" (with A.Brudnyi), C. R. Math. Rep. Acad. Sci. Canada, 32 (2010), p.1-12.
 "On uniform subalgebras of L\infty on the unit circle generated by almost periodic functions" (with A.Brudnyi), St. Petersburg Math. J., 19 (2008), p.495-518 BK - SPbJMath.pdf
We obtain a number of functin-theoretic results on the subalgebra of Hardy algebra of bounded holomorphic functions on the unit disk that have, in a sense, the weakest possible boundary discontinuities (corona-type theorem, approximation property etc). The techniques developed in this paper were used in the subsequent papers -.
|§4|| "A note on
Gagliardo-Nirenberg type inequalities on analytic
sets", C. R. Math. Rep. Acad. Sci. Canada, 30 (2009), p.97-105.
We study a new invariant of singularities of complex analytic sets that arises as a `correcting exponent' in a family of inequalities relating norms of functions on these sets.
Long yet productive :-) journey from the 2nd year of my B.Sc, through M.Sc, to the very begining of Ph.D:
|§5|| "Systems with
distributions and viability theorem", J. Math. Anal. Appl., 331 (2007), p.
 "Dynamical generalized functions and the multiplication problem" (with V.Derr), Russian Math., 51 (2007), p.32-43 0603351.pdf
We introduce two spaces of distributions endowed with continuous operation of multiplication by first-kind discontinuous functions, as needed for some classes of ordinary differential equations with distributions arising, e.g. in singular optimal control problems.
|§6|| "On linear
perturbations of Ricker model" (with E.Braverman), Math Biosci., 202, (2006), p.
We generalize a classical model of population dynamics.