One Hat Cyber Team

View File Name : plan_en.txt

A. The tasks performed by the Russian participants of the project (with participation of Ukrainian researchers).

1. Spectral analysis of Laplace and Schrödinger operators on graphs with different topological structures (including the case of graph contains cycles) in the case of self-adjoint and non-self-adjoint conditions in the vertices. Both in  the self-adjoint and non-self-adjoint cases we plan to apply  functional models based on generalized Weyl-Titchmarsh function (M-function). Investigation of the spectrum in this case is based on the local version of the generalized Cayley identity developed by us earlier, and in case of on-self-adjoint operators with almost Hermitian spectrum - based on almost spectral decomposition. This new approach is essentially based on the wide application of methods of complex analysis to be completed using (in the boundary space of a quantum graph) methods developed by the Ukrainian side for classification sets of projectors satisfying various algebraic relations, which we expect will be very productive. Particular attention will be paid to the problems of restoring of the topology of quantum graph from its spectral characteristics, as well as the inverse spectral problem in the classical formulation. The first phase of the project (in 2011) will work out the proposed methodology in the case of the Laplace operator on the graph; on the second stage - extention of the results to the case of the Schrödinger operator.

2. The study of inverse spectral problem for (self-adjoint) quantum graphs we plan to develop the appropriate generalization of de Branges approach, based on the analysis of structural properties of Hilbert spaces of entire functions that arise as a result of spectral transformation of the canonical system corresponding, to the case of the matrix canonical systems. In the first phase (2011) we expect to get the matrix analogue of the classical Hermite-Biler theorem which gives necessary and sufficient condition that all zeros of the polynomial belong to a half-plane in terms of zeros of its real and imaginary parts. We expect to find such an analogue in terms of spectral flow, which corresponds to the matrix argument of the corresponding polynomial. In the second phase we expect to consider a generalization of the lattice property of subspaces of the de Branges space to the case of vector de Branges spaces using the method recently developed, which is based on estimates of harmonic measures. The said lattice property in the scalar case is crucial for the uniqueness theorems in inverse spectral problem. In parallel, in the case of quantum graphs with various structures we will explore the relationship of the model space of vector analytic functions built using functional models based on matrix M-function (see above) with the corresponding vector space of de Branges type. We believe that the parallel development of these approaches will provide in the second phase of the project new results in  inverse spectral problem for quantum graphs.

3. We plan to investigate the spectral properties of Schrödinger operators on the graph with PT-symmetric potentials. The study of PT-symmetric Schrödinger operators is motivated by the development of modern quantum physics. These operators are J-self-adjoint and therefore their study requires the use of non-self-adjoint methods and approaches, for example, based on functional models. We plan to study stability of (real) spectra of PT-symmetric operators on a graph in the case of small coupling constants. Also we propose the development of functional models for PT-symmetric operators. In the first phase (2011) we plan to consider PT-symmetric quantum graphs with "self-adjoint" conditions in the vertices. In the second stage (2012) we will study the possibility to realize  PT-symmetric Schrödinger operator on a graph as a functional model of some non-dissipative operator.

B. The tasks performed by the Ukrainian project participants (with the participation of Russian researchers).

1. We plan to continue studying systems of subspaces of a Hilbert space satisfying Temperley-Lieb type relations, as well as more complex form of algebraic relations, which contains several parameters. Description of the set of parameter values for which there exist relevant irreducible finite system, and a description of the possible dimensions of such systems. In the first phase we will consider systems associated with the tree graph (2011), then - with more complex graphs: unicyclic and containing more than one cycle (2012).

2. Description up to an isomorphism of algebras generated by two generators satisfying a quadratic relations, separation of these algebras in terms of difficulty of describing their representations in a finite-dimensional linear space and representations by operators close to self-adjoint ones in a Hilbert space, into wild and tame (2011). Study important for applications representations of such algebras (2012) using the methods of functional models that are developed by the Russian team.

3. A study of the properties of rearrangement invariant spaces, including Orlich-Lorentz spaces and investigating specific operators in these spaces (Hardy, Hilbert operators etc.). In the first phase (2011) we expect to investigate the behavior of Cesaro means built by absolute contractions in Orlich-Lorentz spaces of measurable functions on the half-axis (boundedness, convergence almost everywhere, by measure, ordinal). For Cesare means generated by measure preserving transformations of measurable space with infinite measure we plan (on the second stage) to find sufficient conditions for convergence in the strong operator topology. Also we propose a joint study with the Russian side of applicability of the developed technique for the development of functional calculus, developed recently by Russian project participants for non-self-adjoint operators with almost Hermitian spectrum.

6. Expected scientific results.

We expect to obtain the following results according to the research program:

1. Develop equally applicable to the analysis of both self-adjoint and non-self-adjoint operators a new approach to spectral analysis of quantum graphs, based on extensive use of complex analysis, functional models, and by using (in the boundary space of a quantum graph) developed by the Ukrainian team methods of classification of sets of projectors satisfying various algebraic relations. Getting in the way of new results in inverse problems of recovering the topology of the quantum graph from its spectral properties.

2. Obtain new results by a generalization of de Branges approach to inverse spectral problem to the case of the matrix canonical systems; apply the results to the analysis of inverse spectral problem for quantum graphs. Establish internal relations between de Branges method of construction of spectral representation of the investigated operator in the corresponding Hilbert space of analytic functions by spectral transformation of the corresponding canonical system on the one hand, and the method of constructing of such a representation based on generalized Weyl-Titchmarsh function, and approach based on functional model for a suitable dissipative operator on the other hand.

3. Obtain new results on the stability of the real spectrum of Schrödinger operators on graph with PT-symmetric potential. Develop new methods of spectral analysis of PT-symmetric Schrödinger operators on the graph based on functional model.

4. Get a description of certain classes of finite systems of subspaces satisfying the Temperley-Lieb type conditions and associated with graphs, cyclomatic number of which is greater than or equal to two. Obtain new results in classification problem for the system of subspace associated with Coxeter graphs with different types of relations. Apply the methods developed to study quantum graphs.

5. Get a description, up to an isomorphism, of the algebras generated by two generators connected with a quadratic relations. Separation of these algebras in terms of difficulty of describing their representations in a finite-dimensional linear spaces up to similarity, and in a Hilbert space up to a unitary equivalence, into the wild and tame. Study of various representations of these algebras and their applications to spectral theory.

6. Obtain new results about the behavior of Cesare means built by absolute contractions in symmetric spaces of measurable functions on the half-axis (boundedness, convergence almost everywhere, by measure, ordinal), and in some cases finding sufficient conditions for convergence in the strong operator topology. Application of the developed methods and results to the development of functional calculus of non-self-adjoint operators with almost Hermitian spectrum.