Research Interests of Matthias Langer

My main research interests lie in Functional Analysis, Spectral Theory and Differential Equations and their applications in science and engineering. In particular, I am interested in the following topics.

Differential operators.
I work on elliptic differential operators with various boundary conditions, which may be non-local or depend on the spectral parameter. They may also include potentials of δ-type supported on hyper-surfaces. Using the concept of quasi boundary triples, introduced by Jussi Behrndt and myself, I study spectral problems of such differential operators. Moreover, I am work in differential operators with singular coefficients.
Spectral problems that depend nonlinearly on the eigenvalue parameter.
Such problems arise, e.g. after separation of variables or in the study of systems of equations. The dependence may be polynomially or rationally, but can also be more general. These problems can be studied with the help of operator functions, i.e. functions that are defined on some complex domain or real interval and whose values are closed operators in a Banach or Hilbert space. One particular topic I am interested in is the use of variational principles to obtain estimates for eigenvalues or other parts of the spectrum.
Block operator matrices.
These are matrices whose entries are operators in Banach or Hilbert spaces; the matrix acts in a direct sum of spaces. Such block operator matrices can be used, e.g. to study systems of differential equations of different order or type. The aim is to study properties of a block operator matrix using properties of its entries. Questions of interest are, e.g. spectral inclusions or basis properties of components of eigenvectors. One useful tool is the Schur complement, which is an operator function acting only in one component of the direct sum of spaces.
Inverse problems.
Here one tries to reconstruct the model properties, e.g. the coefficients in a differential equation, from measurement data. Applications appear, e.g. in non-destructive testing with the help of ultrasound. I am particularly interested in inverse problems for differential equations with interior or boundary singularities.
Operator semigroups.
To solve various evolution equations, like the heat equation or the time-dependent Schrödinger equation, operator semigroups are a very powerful tool. I am interested, e.g. in the application of operator semigroups to coagulation–fragmentation equations, which are integro-differential equations with singular kernels.
Spaces with indefinite inner products.
These are spaces with an inner product that is not positive definite. Often problems become only symmetric when an indefinite inner product is used; examples include the Klein–Gordon equation, Sturm–Liouville equations with an indefinite weight or operators connected with operator polynomials.