PhD in Pure and Applied Mathematics

In the lecture series the spectrum of waveguides will be analysed by means of the asymptotic and computational methods.

 Contents

1. Functional analytic foundations: spectrum of a self-adjoint operator

 2. The spectrum of self-adjoint elliptic operators in bounded domains

3. The spectrum of self-adjoint elliptic operators in unbounded domains 

4. Examples of waveguides in engineering applications

5. Bounded states in quantum waveguides;

6. Trapped modes, i.e. embedded eigenvalues in the continuous spectrum; 

7. Band gap structure of the spectrum for periodic waveguides

8. Computational methods for approximating the spectrum

 

In this course the propagation of time-harmonic waves in waveguides will be studied. The waveguide may have a periodic structure or is an unbounded domain with cylindrical outlets into infinity. The spectral properties both the Dirichlet-Laplacian and Neumann-Laplacian will be studied by means of asymptotic analysis. The main theoretical tools are min-max principle, matched asymptotic expansions, scattering theory and FEM.

 

In chapters 1-3 the basics of spectral theory for self-adjoint elliptic oper- ators will be recalled and necessary mathematical foundations will be pre- sented.

 In chapter 4 some practical examples of waveguides in engineering appli- cations will be discussed.

 In chapter 5 the bound states of thin quantum waveguides with finite number of cylindrical outlets to infinity will be considered. Using the min- max principle it will be shown that the discrete spectrum is non-empty pro- vided the thickness of the waveguide is small enough. Several examples of waveguides will be presented.

 In chapter 6 we consider trapped modes in , e.g. embedded eigenvalues in the continuous spectrum. The main result is the criteria for the existence of trapped modes. The criteria is closely related to the properties of the generalization of the scattering matrix, either an augmented or a fictitious scattering matrix.

 

The band gap structure of the Laplacian with Neumann or Dirichlet boundary conditions on periodic waveguides will be studied in chapter 7. Also we present analogous results for the Steklov spectral problem in the linearized theory of water waves. Finally, in the final chapter we address the problem, how the spectrum will be approximated and how to compute the numerical approximation of the scattering matrices (augmented and fictitious).

 The course will be lectured in two months.