Recovery problem for a singularly perturbed differential equation with an initial jump

The article investigates the asymptotic behavior of the solution to reconstructing the boundary conditions and the right-hand side for second-order differential equations with a small parameter at the highest derivative, which have an initial jump. Asymptotic estimates of the solution of the reconstruction problem are obtained for singularly perturbed second-order equations with an initial jump. The rules for the restoration of boundary conditions and the right parts of the original and degenerate problems are established. The asymptotic estimates of the solution of the perturbed problem are determined as well as the difference between the solution of the degenerate problem and the solution of the perturbed problem. A theorem on the existence, uniqueness, and representation of a solution to the reconstruction problem from the position of singularly perturbed equations is proved. The results obtained open up possibilities for the further development of the theory of singularly perturbed boundary value problems for ordinary differential equations

Асимптотические разложения собственных значений периодической и антипериодической краевых задач для сингулярно возмущенных дифференциальных уравнений второго порядка с точками поворота

For a second order equation with a small factor at the highest derivative the asymptotic behavior of all eigenvalues of periodic and antiperiodic problems is studied. The main assumption is that the coefficient at the first derivative in the equation is the sign of the variable so that turning points exist an algorithm for computing all coefficients of asymptotic series for every considered eigenvalue is developed. It turns out that the values of these coefficients are defined by coefficient values of the original equation only in a neighborhood of turning points. Asymptotics for the length of Lyapunov zones of stability and instability was obtained. In particular, the problem of stability of solutions of second order equations with periodic coefficients and small parameter at the highest derivative was solvedИсследуется асимптотическое поведение всех собственных значений периодической и антипериодической краевых задач для уравнения второго порядка с малым множителем при старшей производной. Основное предположение состоит в том, что коэффициент при первой производной является знаком переменной, то есть имеются точки поворота. Разработан алгоритм вычисления всех коэффициентов асимптотических рядов для каждого из рассматриваемых собственных значений. Как оказалось, значения всех этих коэффициентов определяются по значениям коэффициентов исходного уравнения только в окрестности точек поворота. Получена асимптотика длин ляпуновских зон устойчивости и неустойчивости. В частности, решена задача об устойчивости решений уравнений второго порядка с периодическими коэффициентами и малым параметром при старшей производной

Contributions to Mathematics and Statistics : Essays in honor of Seppo Hassi

This Festschrift contains thirteen articles in honor of the sixtieth birthday of Professor Seppo Hassi (University of Vaasa). It centers on three topics: functional analysis and operator theory, boundary value problems, and statistics, stochastics, and the history of mathematics. The collection contains four papers on the topic of functional analysis and operator theory. More precisely, it includes a paper treating the transformation of operator-valued Nevanlinna functions and the congruence of their associated realizing operators, a paper treating Parseval frames in the setting of Krein spaces, a paper treating algebraic inclusions of relations as well as the generalized inverses of relations, and a paper treating Krein-von Neumann and Friedrichs extensions by means of energy spaces. Boundary value problems are considered in six of the contributions. In particular, singular perturbations of the Dirac operator are treated by means of the technique of boundary triplets, the connection between sectorial Schrödinger L-systems and certain classes of Weyl-Titchmarsh functions is considered, PT-symmetric Hamiltonians are treated from the perspective of couplings of dual pairs, the Riesz basis property of indefinite Sturm-Liouville problems is considered, the stability properties of spectral characteristics of boundary value problems are investigated, and the completeness and minimality of systems of eigenfunctions and associated functions of ordinary differential operators are treated. Finally, the collection also contains three contributions connected with the topics of statistics, stochastics, and the history of mathematics. More precisely, a new statistic is introduced for the testing of cumulative abnormal returns in the case of partially overlapping event windows, a new characterization of Brownian motion is established, and, finally, a history of (the department of) mathematics and statistics at the University of Vaasa is presented.fi=vertaisarvioimaton|en=nonPeerReviewed

Topological systems with strong electron-electron interactions

Over the last few decades, topological phases of matter have become an omnipresent topic in modern solid state physics. While conventional phases of matter and the phase transitions between them---like, for example, the transition from water to ice---can be fully understood from local properties of a system, topological phases of matter are characterized by global invariants that can be defined and described within the mathematical framework of topology. An early milestone in the field was the discovery of a peculiar class of materials---later termed topological insulators (TIs)---that exhibit a fully insulating bulk while their surfaces are conducting. The so-called gapless surface states that are responsible for this effect allow for dissipationless transport of electrons along the surfaces of the system and exhibit a surprising robustness against perturbations. Indeed, it turns out that the existence of these surface states is guaranteed by topological---and therefore global---properties of the system, leaving them unaffected by any local imperfections of a particular sample. Soon after the initial ideas had spread, it was realized that not only insulating but also superconducting systems can, at the mean-field level, be described within the framework of topology. One of the most striking features of topological superconductors (TSCs) is the fact that they can host so-called Majorana bound states. These exotic quasiparticles are neither bosons nor fermions but so-called non-Abelian anyons. This means that, upon the spatial exchange of two Majorana bound states, the overall wave function of the system does not simply acquire a phase factor, but undergoes a more complicated rotation in a degenerate manifold of ground states. Apart from their fundamental interest, Majorana bound states---and non-Abelian anyons in general---are considered particularly interesting due to their potential use for quantum computation. Indeed, it was predicted that Majorana bound states could in principle be used as a means to encode and process quantum information in a non-local way. This, in turn, would provide an intrinsic protection against quantum errors, which necessarily occur in any quantum computing device but can be expected to act locally in physically realistic scenarios. Following the seminal works on topological insulators and superconductors, the field has been driven by the desire to access topological phases of matter with increasingly exotic properties. While the original theory of TIs and TSCs was built on single-particle band structure considerations, it has been found that the effects of strong electron-electron interactions can lead to even more exotic phases of matter, many properties of which remain elusive up to date. One of the most remarkable features of strongly interacting phases of matter is the fractionalization of quantum numbers: For example, when a two-dimensional electron gas is driven into the so-called fractional quantum Hall regime, quasiparticle excitations carrying only a fraction of the electronic charge $e$ exist. Another intriguing consequence of strong interactions is the possible emergence of exotic bound states such as parafermions. Indeed, to some extent, parafermions can be seen as the fractionalized cousins of Majorana bound states. With even richer non-Abelian exchange statistics than their conventional counterparts, parafermions are---at least theoretically---predicted to harbor significant potential as building blocks for future quantum computing devices. Motivated both by potential technical applications as well as by fundamental theoretical interest, this Thesis is dedicated to studies of novel topological phases of matter with a particular focus on the effects of strong electron-electron interactions. To begin with, we give an introduction to Majorana bound states and topological superconductors in Chapter 1. While focusing mainly on non-interacting systems, this Chapter introduces some of the basic theoretical concepts that will frequently reappear throughout this Thesis. Next, in Chapters 2 and 3, we move on to strongly interacting phases of matter and study the emergence of parafermions in so-called higher-order TSCs. In particular, in Chapter 2, we construct a theoretical model for a fractional second-order TSC with parafermion corner states at two opposite corners of a rectangular sample. To treat the strong electron-electron interactions analytically, we make use of a coupled-wires construction based on weakly coupled Rashba nanowires. In Chapter 3, we propose an alternative model that can host Majorana and parafermion corner states. Instead of coupled Rashba nanowires, this model is based on coupled quasi-one-dimensional channels arising in bilayer graphene due to electrostatic gating. While the models discussed in Chapters 2 and 3 explicitly break time-reversal symmetry, it turns out that a magnetic field is not a necessary ingredient to obtain a second-order TSC. In Chapter 4, we present a theoretical construction of a time-reversal invariant second-order TSC with Kramers pairs of Majorana corner states. Our model is based on a layered structure consisting of two tunnel-coupled TI layers that are `sandwiched' between two $s$-wave superconductors with a phase difference of $\pi$ between them. The competition between interlayer tunneling and proximity-induced superconductivity can then bring the system into the second-order phase. In this Chapter, we restrict our attention to the non-interacting case for simplicity and brevity. In Chapter 5, we move on to second-order phases in three dimensions and construct a coupled-wires model for a time-reversal invariant second-order topological insulator with helical hinge states. For suitably chosen interwire hoppings, we demonstrate that the system has a fully gapped bulk as well as fully gapped surfaces, but hosts two Kramers pairs of gapless helical hinge states that propagate along a path of hinges determined by the hierarchy of interwire hoppings and the boundary termination of the system. Furthermore, we show that sufficiently strong electron-electron interactions can drive the system into a fractional second-order TI phase with hinge states carrying only a fraction of the electronic charge $e$. Via the coupled-wires approach, all our studies of strongly interacting phases of matter heavily relied on the one-dimensional bosonization formalism. However, many intricate details concerning technical aspects of the bosonization formalism are traditionally glossed over in such studies. For example, in bosonized language, Majorana and parafermion zero modes are usually derived from a semi-classical picture in the limit of infinitely strongly pinned bosonic fields in the bulk of the system, leaving the true spatial profile of the bound states unknown. This is why, in Chapter 6, we take one step back and study the bosonized formulation of the simplest possible toy model for a TSC---the Kitaev chain---in an abundance of technical detail. Next, in Chapters 7 and 8 of this Thesis, we turn our attention to signatures of topological phases of matter, i.e., characteristic features that could be detected in experiments. In Chapter 7, we study an observable that we refer to as the fractional boundary charge. As suggested by the name, boundary charges are excess charges located at the boundary of a system with respect to some average background charge of the bulk. We use a coupled-wires construction to describe the fractional quantum Hall effect (FQHE) at odd filling factors and calculate the fractional boundary charge arising in a Corbino disk geometry. If the hole of the disk is threaded by an external flux, we find that the fractional boundary charge depends linearly on the flux with a quantized slope that is determined by the filling factor. Furthermore, different branches of the FBC directly correspond to different degenerate ground states of the system. Subsequently, in Chapter 8, we shift our attention back to topological superconducting systems and study the effects of dilute classical magnetic impurities a two-dimensional time-reversal invariant TSC with helical Majorana edge states. First, we demonstrate that the spin of a single magnetic impurity close to the edge of the TSC tends to align along the edge. We then compute the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between two magnetic impurities placed close to the edge of the TSC. We find that, in the limit of large interimpurity distances, the RKKY interaction between the two impurities is mainly mediated by the Majorana edge states and leads to a ferromagnetic alignment of both spins along the edge. All of these effects are absent in trivial $s$-wave superconductors. As such, spectroscopy of dilute magnetic impurities could be a powerful tool to probe helical TSCs or topological materials with helical edge states in general. Last but not least, in Chapter 9, we turn our attention to systems that exhibit one or more completely dispersionless---or so-called flat---bands. While such a peculiar band structure is interesting already in its own right, flat band systems have attracted particular attention since they can realize a variety of strongly correlated phases of matter. Indeed, since the kinetic energy is completely quenched in the flat band, even arbitrarily weak interactions can drastically modify the properties of the system. The same is true for disorder as well as for `perturbations' due to, e.g., the presence of dilute impurities. This has motivated us to study the RKKY interaction between two classical magnetic impurities in two different one-dimensional lattice models that host flat bands. We start by obtaining exact results for the RKKY interaction in both models by numerical exact diagonalization and find that, in both cases, the RKKY interaction exhibits peculiar features that can directly be traced back to the presence of a flat band. Next, we compare our numerical data to results obtained via different analytical techniques. We discuss how the presence of a flat band can invalidate the conventional RKKY approximation based on non-degenerate second-order perturbation theory and highlight the need for degenerate perturbation theory or even non-perturbative approaches to accurately capture the effect of the flat band

Modelling and Adaptive Control; Proceedings of an IIASA Conference, Sopron, Hungary, July 1986

One of the main purposes of the workshop on Modelling and Adaptive Control at Sopron, Hungary, was to give an overview of both traditional and recent approaches to the twin theories of modelling and control which ultimately must incorporate some degree of uncertainty. The broad spectrum of processes for which solutions of some of these problems were proposed was itself a testament to the vitality of research on these fundamental issues. In particular, these proceedings contain new methods for the modelling and control of discrete event systems, linear systems, nonlinear dynamics and stochastic processes