IWOTA 2019

International Workshop on

Operator Theory and its Applications

IWOTA 2019

International Workshop on

Operator Theory and its Applications

July 22-26, 2019

Instituto Superior Técnico

Lisbon, Portugal

Instituto Superior Técnico

Lisbon, Portugal

- Nuno António

CEAFEL, Instituto Superior Técnico, Universidade de Lisboa ## Separation of Variables in Integrable Systems

The method of separation of variables (SoV) is applied to the Neumann oscillator system. This integrable model describes the motion of a free particle, on the sphere, submitted to harmonic forces. Some recent applications of SoV to quantum integrable models are also discussed.

- Ioannis Arkoudis

National and Kapodistrian University of Athens, Department of Mathematics ## The method of fundamental solutions in electromagnetic scattering by a chiral obstacle

The scattering of a time-harmonic plane electromagnetic wave by a penetrable chiral obstacle in an achiral environment is considered. An extension of the standard method of fundamental solutions is applied in order to obtain numerically the solution of the problem. The electric fields are expressed in terms of the fundamental solutions of the corresponding equations in dyadic form. Completeness properties for appropriate systems of functions are proved which will be used in order to solve approximately the above scattering problem. Using the transmission conditions, the scattering problem is transformed into a linear algebraic system with coefficient matrix which consists of chiral and achiral blocks. Moreover, introducing the Beltrami fields, the chiral block matrix is further separated into a left circularly polarized part and a right circularly polarized part. The electric far-field pattern of this scattering problem is obtained in order to study the asymptotic behavior of the scattered field at infinity as well as to solve the inverse scattering problem. Additionally, the extinction cross-section is evaluated in order to calculate the energy that the scatterer removes from the incident wave. When the measure of chirality vanishes, our results cover the case of an achiral dielectric scatterer. Numerical results for scatterers of specific geometric form are presented.

Joint work with E. S. Athanasiadou.

- Harm Bart

Erasmus University Rotterdam ## How small can a sum of idempotents be?

The issue that will be discussed is: how small can a sum of idempotents be? Here smallness is understood in terms of nilpotency or quasinilpotency. Thus the question is: given idempotents $p_1,\ldots,p_n$ in a complex algebra or Banach algebra, is it possible that their sum $p_1+\cdots+p_n$ is quasinilpotent or (even) nilpotent (of a certain order)? The motivation for considering this problem comes from earlier work by the authors on the generalization of the logarithmic residue theorem from complex function theory to higher (possibly infinite) dimensions.

The talk is a report on work authored jointly with Torsten Ehrhardt (Santa Cruz, California) and Bernd Silbermann (Chemnitz, Germany).

- James Bell Cooper

Johannes Kepler Universität, Linz ## Analysis in the space of observables

Despite the fact that the observables are the basic quantities in quantum field theory, there is no adequate theory to carry over the basic methods of real and complex analysis to this context. We are particularly interested in the notions of holomorphic functions between spaces of observables and of operator-valued distributions. It is our intention in this talk to present such a treatment. It is based on a natural representation of the space of observables as a generalised spectrum of an algebra of functions on the real line, resp., the complex plane which is used to give it the structure of a polish space (when the underlying Hilbert space is separable). We use this fact to define various spaces of distributions with values in the space of observables and of holomorphic functions into or between them, whereby we employ methods developed for the classical case by J. Sebastião e Silva, G. Köthe and A. Grothendieck on the relations between duality and tensor products of spaces of test functions or analytic functions, resp., their vector-valued versions. In order to emphasise the naturalness of our approach we shall emphasise its axiomatic and functorial properties, rather than the technical details of the constructions.

- Maria Fragoulopoulou

Department of Mathematics, University of Athens ## Bounded and unbounded generalizations of Gelfand-Naimark type theorems

We shall present some

*generalizations*of $C^\ast$-algebras that occur on the one hand, in the context of Banach *-algebras and on the other hand, in the context of locally convex *-algebras. In the bounded case, the symmetric (equivalently hermitian) Banach *-algebras belong (D.A. Raikov, 1946); in the unbounded case, the GB*-algebras (abbreviation of generalized B*-algebras; G.R. Allan, 1967) and locally convex quasi $C^\ast$-algebras pertain. The latter were introduced in a joint work with F. Bagarello, A. Inoue and C. Trapani, in 2008.We shall exhibit examples of the aforementioned

*generalized*$C^\ast$*-algebras*and discuss Gelfand-Naimark type theorems for them. Furthermore, questions related with uniqueness of their topology and conditions under which some of them are reduced to classical $C^\ast$-algebras will be considered.- Rita Catarina Correia Guerra

CIDMA, Department of Mathematics, University of Aveiro ## New convolutions weighted by multi-dimensional Hermite functions and their applications

We introduce eight new convolutions weighted by multi-dimensional Hermite functions and prove two Young-type inequalities. As an application of those convolutions, we study the solvability of a general class of integral equations whose kernel depends on four different functions. Necessary and sufficient conditions for the unique solvability of such integral equations are here obtained. This is based on a joint work with L. P. Castro and N. M. Tuan.

- Klaus Gürlebeck

Bauhaus-Universität Weimar, Germany ## On some boundary value problems in the micropolar theory of elasticity

Micropolar elasticity is a refined version of the classical elasticity. Equations of micropolar elasticity are given by a coupled system of differential equations connecting fields of displacements and rotations. The construction of solution methods for boundary value problems of micropolar elasticity is still a challenging mathematical task, mostly due to the coupled nature of the resulting system of partial differential equations. Especially, only few results are available for spatial problems of micropolar elasticity. Therefore, in this contribution, we present a quaternionic operator calculus-based approach to construct general solutions to three-dimensional problems of micropolar elasticity. Moreover, we prove solvability of the boundary value problem of micropolar elasticity, as well as we provide an explicit estimate for the difference between the classical elasticity and the micropolar model.

- Sanne ter Horst

North West University ## Does equivalence after extension imply Schur coupling for Banach space operators?

The relations between two Banach space operators now known as Equivalence After Extension (EAE), Matricial Coupling (MC) and Schur Coupling (SC) originated from the study of holomorphic operator functions and integral operators and provide information about the relative spectral properties of the operators near the origin, e.g., Fredholm properties, level of compactness, etc. In the application of these relations it is essential that they coincide, i.e., the operators in question are EAE if and only if they are MC if and only if they are SC. It was observed in the 1980s and 1990s that, at the level of (bounded) Banach space operators, indeed EAE and MC coincide, and both are implied by SC. The remaining implication, i.e., that EAE (or MC) implies SC, remained open for several decades. Some affirmative partial results were obtained, mainly in the last decade, most notably for Hilbert space operators. In this talk we show that in general the answer is negative, EAE does not imply SC, but we will also discuss various classes of Banach space operators for which there is a affirmative answer.

The talk is based on joint work with M. Messerschmidt, A. C. M. Ran, M. Roelands and M. Wortel.

- Angeliki Kaiafa

University of Piraeus, School of Financial and Statistics, Department of Statistics and Insurance Science ## Mixed Impedance Problems in Elasticity: The Case of a Non-Penetrable Partially Coated Cavity

Elastic scattering problems for partially coated obstacles are mathematically modeled by interior and exterior mixed impedance boundary value problems for equations of steady-state elastic oscillations. Our scattering problem deals with a non-penetrable

*partially coated cavity*which is formulated due to Navier equation and we assume that the scattered field satisfies mixed Dirichlet-Robin boundary conditions on the Lipschitz boundary of the cavity. We consider the case where both point-sources (incident waves) and scattered waves (measurements) are located inside the object, which leads to an*interior scattering problem*. We employ an appropriate variational method together with a suitable Sobolev space setting, in order to establish uniqueness- existence results as well as stability ones. We also give the corresponding inverse scattering problem and analogous issues to the direct problem for well-posedness are discussed. Finally, useful remarks and conclusions concerning the direct scattering problem and its linchpin with the shape reconstruction and location of the partially coated cavity are given.- Anastasiia Legatiuk

Bauhaus University Weimar ## Application of the discrete potential theory to numerical modelling of the induction heating process

Discrete potential theory is a natural extension of the continuous theory to functions defined on lattices. The idea of the discrete potential theory is to transfer all important aspects of the continuous theory directly to the discrete level. Direct formulation on the discrete level is advantageous for applications in mathematical physics, since important physical quantities, such as for example conservation laws or asymptotic conditions, are modelled and satisfied on the discrete level and not only approximated as in conventional approaches. Moreover, rectangular lattices, i.e. lattices allowing two different stepsizes, are more beneficial in real-world applications. Thus, in this contribution, we present recent developments in the discrete potential theory on rectangular lattices and its application to numerical modelling of the induction heating process.

- Anatoli Merzon

Universidad Michoacana de San Nicolás de Hidalgo ## Time-dependent approach to the uniqueness of the Sommerfeld solution of the diffraction problem by a half-plane

We consider the Sommerfeld problem of diffraction by an opaque half-plane interpreting it as the limiting case as $t \to \infty$ of the corresponding nonstationary problem. We prove that the Sommerfeld formula for the solution is the limiting amplitude of the solution of this nonstationary problem which belongs to a certain functional class and is unique in it. For the proof of uniqueness of solution to the nonstarionary problem we reduce this problem, after the Fourier-Laplace transform in $t$ , to a stationary diffraction problem with a complex wave number. This permits us to use the proof of the uniqueness in the Sobolev space $H^1$ as in [1]. Thus we avoid imposing the radiation condition from the beginning and instead obtain it in a natural way.

This is a joint work with Prof A. Komech and Prof. P. Zhevandrov.

### References

- Castro L. P., Kapanadze D. Wave diraction by wedges having arbitrary aperture angle. J Math Anal Appl. 2015;421(2):1295-1314. https://doi.org/10.1016/j.jmaa.2014.07.080
- A. Komech, A. Merzon, and J.E. Dela Paz Mendez, Time-dependent scattering of generalized plane waves by wedges, Mathematical Methods in the Applied Sciences. 2015; 38(18) : 4774-4785.
- A. Komech , A. Merzon, A. Esquivel Navarrete, J. E. De La Paz Méndez, T. J. Villalba Vega, Sommerfeld's solution as the limiting amplitude and asymptotics for narrow wedges, Mathematical Methods in the Applied Sciences (2018). 10.1002/mma.5075.
- A. Merzon, P. Zhevandrov, J. E. De La Paz Méndez, T. J. Villalba Vega, Time-dependent approach to the uniqueness of the Sommerfeld solution of the diraction problem by a half-plane. (In preparation.)

- David Natroshvili

Department of Mathematics, Georgian Technical University ## Investigation of interfacial crack problems of the thermo-electro-magneto-elasticity theory by the potential method

We investigate regularity properties of solutions to mixed boundary value problems for the system of partial differential equations of dynamics associated with the thermo-electro-magneto elasticity theory [1]. We consider piecewise homogeneous anisotropic elastic solid structures with interior and interfacial cracks. Using the potential method and theory of pseudodifferential equations, we prove the existence and uniqueness of solutions. The singularities and asymptotic behaviour of the thermo-mechanical and electro-magnetic fields are analyzed near the crack edges and near the curves where different types of boundary conditions collide. In particular, for some important classes of anisotropic media we derive explicit expressions for the corresponding stress singularity exponents and demonstrate their dependence on the material parameters. The questions related to the so-called oscillating singularities are analyzed in detail as well.

This is a joint work with O. Chkadua and T. Buchukuri.

This work was supported by Shota Rustaveli National Science Foundation of Georgia (SRNSF) (Grant number FS-18-126).

### References

- T. Buchukuri, O. Chkadua, D. Natroshvili,
*Mathematical Problems of Generalized Thermo-Electro-Magneto-Elasticity Theory*, Memoirs on Differential Equations and Mathematical Physics**68**(2016), pp. 1-165.

- T. Buchukuri, O. Chkadua, D. Natroshvili,
- Andre Ran

Vrije Universiteit ## Inner-outer factorization of wide rational matrix valued functions

In this lecture formulas for the full rank inner outer factorization of a wide matrix valued rational function $G$ with $H^\infty$ entries, that is, functions $G$ with more columns than rows will be given. State space formulas are derived for the inner and outer factor of $G$.

- Maria Manuela Fernandes Rodrigues

CIDMA & University of Aveiro ## Time-fractional telegraph equation and its first and second fundamental solutions

In this work we obtain the first and second fundamental solutions (FS) of the multidimensional time-fractional equation with Laplace operator, where the two time-fractional derivatives of orders $\alpha \in {]0,1]}$ and $\beta \in {]1,2]}$ are in the Caputo sense. We obtain representations of the FS in terms of Hankel transform, double Mellin-Barnes integrals, and H-functions of two variables. As an application, the FS are used to solve Cauchy problems of Laplace type. Some considerations about its applications in future work will be presented.

- Anabela de Sousa e Silva

CIDMA - Center for Research and Development in Mathematics and Applications, University of Aveiro ## New convolutions for Fourier sine and cosine transforms involving Hermite weights, and their applicability

We will introduce new convolutions, for Lebesgue integrable functions on the positive half-line, and study some of their properties. Namely, factorization identities for those convolutions will be derived, upon the use of Fourier sine and cosine transforms and Hermite polynomials. Moreover, such convolutions allow us to consider classes of systems of convolution type equations on the half-line. Conditions for the solvability of such systems are identified and, under such conditions, their solutions are obtained.

- Alberto Manuel Tavares Simões

Centre of Mathematics and Applications-University of Beira Interior (CMA-UBI) ## Hyers-Ulam-Rassias stabilities for classes of integral equations with oscillatory kernels

We will analyse the stability of classes of integral equations with kernels depending on sine and cosine functions. Conditions will be identified to ensure Hyers-Ulam stability, Hyers-Ulam-Rassias stability and some other intermediate stabilities for those integral equations. This will be done by taking profit of fixed point arguments in the framework of spaces of continuous functions endowed with generalizations of the Bielecki metric.

Joint work with L. P. Castro.

- Wolfgang Wendland

Universität Stuttgart, IANS, Mathematics ## Potential methods and solution spaces of Stokes and Brinkman problems

Potential methods have a long history going back to George Green and Carl F. Gauss and were first used for solving elliptic boundary value problems of the Laplacian and the Helmholtz equations as well as strongly elliptic second order equations with constant coefficients.

Then during the last century, the potential methods were extended to elasticity and flow problems with constant material properties and finally used for treating nonlinear problems as the Navier-Stokes equations.

In this lecture we consider the Stokes and Brinkman systems (and some extensions) and present the corresponding layer distribution spaces in the domains and on the boundaries beginning with $L^2$-based Sobolev-Slobodeckij spaces in $\mathbb{R}^n$ and also in Lipschitz domains on Riemannian manifolds and finally interpolation scales of Besov space layers.

Joint work with M. Kohr (Babes-Bolyai Univ. Cluj-Napoca) S. Mikhailov (Brunel Univ. West-London), Lanza di Cristoforis (Univ. Padova).

- Lorenzo Zanelli

Università degli studi di Padova ## Periodic coherent states decomposition and quantum dynamics on the flat torus

We provide a result on the periodic coherent states decomposition for functions $L^2 (\Bbb T^n)$ on the flat torus $\Bbb T^n := (\Bbb R / 2\pi \Bbb Z)^n$. Suddenly, we study such a decomposition with respect to the quantum dynamics related to semiclassical elliptic Pseudodifferential operators on $\Bbb T^n$, and we prove a related invariance result under time propagation.

- Petr Zhevandrov

Universidad Michoacana de San Nicolás de Hidalgo ## Rayleigh-Bloch waves trapped by a periodic perturbation and resonances

We consider the two-dimensional Helmholtz equation with constant wave speed perturbed by a periodic (say, in the $x$-direction) series of bumps of small amplitude. This perturbation generates a so-called Rayleigh-Bloch (RB) wave which is quasiperiodic in $x$, decays exponentially in the orthogonal direction and is a solution to the Helmholtz equation; the frequency of this wave lies outside the continuous spectrum of the problem. Moreover, the first embedded threshold of the continuous spectrum of the unperturbed problem also generates an RB mode but only when a certain geometric (orthogonality) condition is satisfied by the perturbation. When this condition is violated, the trapped mode becomes a complex resonance with small imaginary part and the reflection and transmission coefficients for the corresponding scattering problem present drastic changes in a neighborhood of the resonance. We obtain explicit formulas for these objects in the form of series in powers of the small parameter characterizing the magnitude of the perturbation by means of reducing the initial problem to an infinite system of integral equations, give a description of the Breit-Wigner and Fano resonances, and indicate the conditions when the perturbation induces total transmission and reflection.

This is joint work with A. Merzon and M. I. Romero Rodríguez.

- Stefania Zoi

National and Kapodistrian University of Athens, Department of Mathematics ## A thermoelastic penetrable scatterer in an elastic environment

In this work the scattering of time-harmonic plane elastic waves by a penetrable thermoelastic body is considered. The formulation of the direct scattering problem in a unified four-dimensional form is presented. The uniqueness of solution of the problem using radiation conditions and a Rellich's type Lemma is proved. Using single- and double-layer elastic and thermoelastic potentials the problem is transformed into a system of integral equations and via Riesz-Fredholm theory existence of solution is secured. Moreover, an integral representation of the solution, in which the physical parameters of the thermoelastic scatterer have been incorporated, is presented. The asymptotic form of the integral representation when the observation point tends to infinity gives the far-field patterns which describe the behavior of the scattered field at infinity. A far-field operator is presented and some of its properties are also proved. Finally, scattering theorems and reciprocity relations are derived and their linchpin with inverse scattering problems is discussed.

Joint work with E. S. Athanasiadou, V. Sevroglou.