SPECANSO · Spectral Analysis of Non-selfadjoint and Selfadjoint Operators - New Methods and Applications

Spectral Analysis of Non-selfadjoint and Selfadjoint Operators - New Methods and Applications

Non-selfadjoint operators and associated spectral problems arise naturally in many physical application areas such as hydrodynamics, magnetohydrodynamics, lasers and nuclear scattering. The spectral behaviour of these operators is considerably more complex than that of selfadjoint operators which leads to various new phenomena. In particular, the spectrum is no longer necessarily confined to the real axis and tools such as the spectral theorem and variational principles are no longer applicable. Unable to make use of these standard techniques, it is necessary to develop new methods to obtain results on spectral properties of non-selfadjoint operators. This, and applying these new methods to specific problems, are what this research project focusses on. The results obtained, as well as being of interest and importance in themselves, will have an impact on the methodology of the study of selfadjoint and non-selfadjoint problems and will be interesting to anybody working in the areas of spectral theory or scattering theory, in the mathematics, physics and also the engineering community. The original proposal had five main topics: the functional model using boundary triples, PT-symmetric operators, operators with almost Hermitian spectrum, abstract inverse problems - information contained in the M-function and the extension theory and spectral analysis of highly singular dissipative perturbations. Due to developments over the course of the grant, we have hardly begun work on PT-symmetric operators. Instead, there has been work on several areas related to the topic of the grant not set out in the original proposal. Over the lifetime of the project, we have worked on the following research topics. 1. Functional model: We are pursuing a new approach to the functional model based on the boundary triples idea. We have developed the translational form of the functional model in the boundary triple framework and compared our results to Kudryashov's previous construction of the selfadjoint dilation of maximal dissipative operators. The spectral form of the self-adjoint dilation for maximal dissipative operators in the boundary triple framework has also been considered, as well as examples of first order differential operators with singular potentials. 2. Operators with almost Hermitian spectrum: The Cayley identity theory started in earlier works by A. Kiselev and Sergey Naboko has been generalized to include Hermitian operators with purely singular spectrum. The generalization of the spectral theorem for the selfadjoint operators to the case of operators with almost Hermitian spectrum, including the completely nonselfadjoint operators of that class has also been obtained. It gives new results in the functional calculus for our class of operators. 3. Abstract inverse problems: The detectable subspace has been analyzed in detail. The relation between the generalized Weyl-Titchmarsh function (Dirichlet-to-Neumann map or M-function) and the operator resolvent bordered from both sides by the orthogonal projectors onto the detectable subspace and its adjoint counterpart were studied as well as the relation between the analytic continuation of M-function over the continuous spectrum of the operator and the analytic continuation of the quadratic form of the operator resolvent. Finally, the coincidence of the ranks of the jumps of bordered resolvent and M-functions was obtained for almost every point of the real axis, assuming the essential spectrum is real. The results have been applied to concrete examples, in particular Hain-Luest type differential models and the Friedrichs model. The detectability of Friedrichs model operators in the vector-valued case was considered by Sergey Naboko, together with C.Camara. On the basis of the application of the theory of Toeplitz operators with matrix symbols a full solution to the problem in that special case was obtained. 4. Extension theory and spectral analysis of highly singular dissipative perturbations: An analysis of some ODE and PDE examples whose coefficients have local power-like singularities of sufficiently high order leads to new interesting phenomena. This is due to the strong interaction of the singular perturbation with the main part of the operator. Work in this area has been carried out in collaboration with our PhD student Christoph Fischbacher. 5. Jacobi operators: These are infinite tri-diagonal matrices acting on spaces of sequences. They are closely related to three term recurrences and orthogonal polynomials. If the matrices are symmetric and their entries periodic, then the spectrum exhibits a band-gap structure with a finite number of bands of absolutely continuous (a.c.) spectrum. With our second PhD student, Edmund Judge, we have considered the problem of explicitly constructing embedded eigenvalues in the a.c. spectrum of periodic Jacobi operators by adding a Wigner-von Neumann potential. Together with S.Kupin (Bordeaux) and L.Golinskii (Kharkov), Sergey Naboko investigated a class of Hermitian Jacobi matrices with matrix elements with power-like growth of order greater than one. Some estimates from above and below for the eigenvalues were obtained. Also asymptotics of the eigenvectors at infinity in the integer index case were proved. Additionally, in work with J.Janas (Krakow), Sergey Naboko has studied the spectra of periodic block Jacobi matrices. Some explicit results on the location of eigenvalues were obtained. Most attention was paid to the difference between the scalar and matrix entries cases. 6. Hermitian linear pencils with a complicated point spectrum: The problem of describing the eigenvalues for the linear pencil A+ cB was studied, assuming the generating property of the range of B. It was shown that the point spectrum of the pencil has Lebesgue measure 0 under the condition that B is a nonnegative operator. On the other hand for more general B, even in the case of compact B and invertible positive part of A, the point spectrum may cover the whole complex plane, as shown by a constructive example. This is joint work of Sergey Naboko's with F.Gesztesy and R.Nichols. 7. Schroedinger operators with operator-valued potentials: For Schroedinger operators on the half-line and the full-line with operator-valued potentials the role in the spectral analysis of a new form of m-function, the so-called Donoghue m-function was analyzed by Sergey Naboko in collaboration with F.Gesztesy, R.Weikard and M.Zinchenko. 8. Spectra of quantum graphs: The direction of the eigenvalue shift under addition or removal of an edge of a quantum graph was considered. Using a geometrical approach together with a detailed analysis of the corresponding matrix Weyl function behavior, exhaustive results were obtained for the problem by Sergey Naboko with P.Kurasov (Stockholm). 9. The finite section method: For self-adjoint Jacobi matrices and Schroedinger operators on the half-line, perturbed by dissipative potentials, the finite section method does not omit any points of the spectrum. As a by-product some interesting estimates showing the rate of approximating the essential spectrum by the eigenvalues of the truncated dissipative problem were obtained. Main results achieved: The main results of the project are the development of the functional model, as described above and the deeper understanding of the relation of the detectable subspace and the M-function, both in an abstract setting and in concrete examples. These results, as well as being of interest and importance in themselves, will have an impact on the future study of selfadjoint and non-selfadjoint problems. Moreover, the activities on this grant have allowed Sergey Naboko to establish new and strengthen existing research collaborations throughout the EU, as well as at the University of Kent. Work on the grant has already resulted in two publications with three further ones submitted and many more in preparation. Researcher training and transfer of knowledge activities: We have organized two meetings on Spectral Theory during Sergey Naboko's time in Kent. The Kent Spectral Theory Meeting took place at the University of Kent from 14th-17th April 2014. Around 40 participants from across Europe and the UK gathered in Canterbury. One particular focus was the interaction between classical selfadjoint operator theory and modern non-selfadjoint operator theory. Earlier in Sergey Naboko's stay, we had already organised a half-day workshop as a special session of the London Analysis seminar to announce his arrival in the UK. Sergey Naboko has been strongly involved in the supervision of PhD students and has supervised three final year BSc dissertations. All these students gained the chance of working intensively with a World renowned expert and have been able to develop a much deeper understanding in an area of analysis. Ian Wood and Sergey Naboko have delivered two lecture series via the LTCC, one on dissipative operators, the other on Jacobi operators and orthogonal polynomials. Sergey Naboko has taught a final year undergraduate course in both of his years at Kent. Additionally, he has spoken at and attended various conferences and given seminars in the UK and many other parts of the European Union.

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