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Mikhail Shubin

Publications and source records attributed to Mikhail Shubin.

14 recordsLinked to original sources
arXiv2013

The Geometry of Morphogenesis and the Morphogenetic Field Concept

The process of morphogenesis, which can be defined as an evolution of the form of an organism, is one of the most intriguing mysteries in the life sciences. It is clear, that gene expression patterns cannot explain the development of the precise geometry of an organism and its parts in space. Here, we suggest a set of postulates and possible approaches for discovering the correspondence between molecular biological information and its realization in a given geometry of an organism in space-time. First, we suggest that the geometry of the organism and its parts is coded by a molecular code located on the cell surfaces in such a way that, with each cell, there can be associated a corresponding matrix, containing this code. As a particular model, we propose coding by several types of oligosaccharide residues of glycoconjugates. Second, we provide a notion of cell event, and suggest a description of development as a tree of cell events, where under cell event we understand the changing of cell state, e.g. the processes of cell division, cell growth/death, shift or differentiation. Next we suggest describing these cell motion laws using the notion of a "morphogenetic field", meaning by the field an object in an "event space" over a "cell space", which govern the transformation of the coded biological information into an instructive signal for a cell event for a given cell, depending on the position of a cell in the developing embryo. The matrix on a cell surface will be changed after each cell event according to the rule(s) dictated by the morphogenetic field of an organism. Finally, we provide some ideas on the connection between morphogenetic code on a cell surface, cell motion law(s) and the geometry of an embryo.

q-bio.OT↗
arXiv2008

Semiclassical Asymptotics on Manifolds with Boundary

We discuss semiclassical asymptotics for the eigenvalues of the Witten Laplacian for compact manifolds with boundary in the presence of a general Riemannian metric. To this end, we modify and use the variational method suggested by Kordyukov, Mathai and Shubin (2005), with a more extended use of quadratic forms instead of the operators. We also utilize some important ideas and technical elements from Helffer and Nier (2006), who were the first to supply a complete proof of the full semi-classical asymptotic expansions for the eigenvalues with fixed numbers.

math.SP↗
arXiv2006

Geometric Theory of Lattice Vibrations and Specific Heat

We discuss, from a geometric standpoint, the specific heat of a solid. This is a classical subject in solid state physics which dates back to a pioneering work by Einstein (1907) and its refinement by Debye (1912). Using a special quantization of crystal lattices and calculating the asymptotic of the integrated density of states at the bottom of the spectrum, we obtain a rigorous derivation of the classical Debye $T^3$ law on the specific heat at low temperatures. The idea and method are taken from discrete geometric analysis which has been recently developed for the spectral geometry of crystal lattices.

math-ph↗
arXiv2006

Gauge optimization and spectral properties of magnetic Schrödinger operators

We establish new necessary and sufficient conditions for the discreteness of spectrum and strict positivity of magnetic Schrödinger operators with a positive scalar potential. They extend earlier results by Maz'ya and Shubin (2005), which were obtained in case when there is no magnetic field. We also derive two-sided estimates for the bottoms of spectrum and essential spectrum, extending results by Maz'ya and Otelbaev (1977). The main idea is to optimize the gauge of the magnetic field, thus reducing the quadratic form to one without magnetic field (but with an appropriately adjusted scalar potential).

math.SP↗
arXiv2005

Discreteness of spectrum and positivity criteria for Schrödinger operators

We provide a class of necessary and sufficient conditions for the discreteness of spectrum of Schrödinger operators with scalar potentials which are semibounded below. The classical discreteness of spectrum criterion by A.M.Molchanov (1953) uses a notion of negligible set in a cube as a set whose Wiener's capacity is less than a small constant times the capacity of the cube. We prove that this constant can be taken arbitrarily between 0 and 1. This solves a problem formulated by I.M.Gelfand in 1953. Moreover, we extend the notion of negligibility by allowing the constant to depend on the size of the cube. We give a complete description of all negligibility conditions of this kind. The a priori equivalence of our conditions involving different negligibility classes is a non-trivial property of the capacity. We also establish similar strict positivity criteria for the Schrödinger operators with non-negative potentials.

math.SP↗
arXiv2005

Can one see the fundamental frequency of a drum?

We establish two-sided estimates for the fundamental frequency (the lowest eigenvalue) of the Laplacian in an open subset G of R^n with the Dirichlet boundary condition. This is done in terms of the interior capacitary radius of G which is defined as the maximal possible radius of a ball B which has a negligible intersection with the complement of G. Here negligibility of a subset F in B means that the Wiener capacity of F does not exceed gamma times the capacity of B, where gamma is an arbitrarily fixed constant between 0 and 1. We provide explicit values of constants in the two-sided estimates.

math.SP↗
arXiv2005

The Miura Map on the Line

The Miura map (introduced by Miura) is a nonlinear map between function spaces which transforms smooth solutions of the modified Korteweg - de Vries equation (mKdV) to solutions of the Korteweg - de Vries equation (KdV). In this paper we study relations between the Miura map and Schroedinger operators with real-valued distributional potentials (possibly not decaying at infinity) from various spaces. We also investigate mapping properties of the Miura map in these spaces. As an application we prove existence of solutions of the Korteweg - de Vries equation in the negative Sobolev space H^{-1} for the initial data in the range of the Miura map.

math.SP↗
arXiv2004

Discreteness of spectrum and strict positivity criteria for magnetic Schrödinger operators

We establish necessary and sufficient conditions for the discreteness of spectrum and strict positivity of magnetic Schrödinger operators with a positive scalar potential. They are expressed in terms of Wiener's capacity and the local energy of the magnetic field. The conditions for the discreteness of spectrum depend on two functional parameters. One of them is a decreasing function of one variable whose argument is the normalized local energy of the magnetic field. This function enters the negligibility condition of sets for the scalar potential. We give a description for the range of admissible functional parameters which is precise in a certain sense. In case when there is no magnetic field, our results extend the discreteness of spectrum and positivity criteria by A.Molchanov (1953) and V.Maz'ya (1973).

math.SP↗
arXiv2002

Essential self-adjointness of Schroedinger type operators on manifolds

We obtain several essential self-adjointness conditions for a Schroedinger type operator D*D+V acting in sections of a vector bundle over a manifold M. Here V is a locally square-integrable bundle map. Our conditions are expressed in terms of completeness of certain metrics on M; these metrics are naturally associated to the operator. We do not assume a priori that M is endowed with a complete Riemannian metric. This allows us to treat e.g. operators acting on bounded domains in the euclidean space. For the case when the principal symbol of the operator is scalar, we establish more precise results. The proofs are based on an extension of the Kato inequality which modifies and improves a result of Hess, Schrader and Uhlenbrock.

math.SP↗
arXiv2001

Essential self-adjointness for semi-bounded magnetic Schrödinger operators on non-compact manifolds

We prove essential self-adjointness for semi-bounded below magnetic Schrödinger operators on complete Riemannian manifolds with a given positive smooth measure which is fixed independently of the metric. Some singularities of the scalar potential are allowed. This is an extension of the Povzner--Wienholtz--Simader theorem. The proof uses the scheme of Wienholtz but requires a refined invariant integration by parts technique, as well as a use of a family of cut-off functions which are constructed by a non-trivial smoothing procedure due to Karcher.

math.SP↗
arXiv2001

Discreteness of spectrum for the magnetic Schrödinger operators. I

We provide sufficient conditions for the discreteness of spectrum for magnetic Schrödinger operators. They generalize the classical result by K.Friedrichs (1934) and earlier results by J.Avron, I.Herbst and B.Simon (1978), A.Dufresnoy (1983) and A.Iwatsuka (1990) which were obtained in the absence of electric field. Our conditions are formulated as conditions of growth at infinity for some effective potentials. More precise sufficient conditions are formulated by use of the Wiener capacity and extend earlier work by A.M.Molchanov (1953) and V.Kondratiev and M.Shubin (1999) who considered the case of the operator without magnetic field. These conditions become necessary and sufficient in case when there is no magnetic field and the electric potential is semi-bounded below.

math.SP↗
arXiv1996

De Rham theorem for extended L^2-cohomology

We prove an analogue of the de Rham theorem for the extended L^2-cohomology introduced by M. Farber. This is done by establishing that the de Rham complex over a compact closed manifold with coefficients in a flat Hilbert bundle E of A-modules over a finite von Neumann algebra A is chain-homotopy equivalent (with bounded morphisms and homotopy operators) to a combinatorial complex with the same coefficients. This is established by using the Witten deformation of the de Rham complex. We also prove that the de Rham complex is chain-homotopy equivalent to the spectrally truncated de Rham complex which is also finitely generated.

dg-ga↗
arXiv1996

Twisted L^2 invariants of non-simply connected manifolds

We develop the theory of twisted L^2-cohomology and twisted spectral invariants for flat Hilbertian bundles over compact manifolds. They can be viewed as functions on the first de Rham cohomology of M and they generalize the standard notions. A new feature of the twisted L^2-cohomology theory is that in addition to satisfying the standard L^2 Morse inequalities, they also satisfy certain asymptotic L^2 Morse inequalities. These reduce to the standard Morse inequalities in the finite dimensional case, and when the Morse 1-form is exact. We define the extended twisted L^2 de Rham cohomology and prove the asymptotic L^2 Morse-Farber inequalities, which give quantitative lower bounds for the Morse numbers of a Morse 1-form on M.

dg-ga↗