# Mathematics Phd Dissertation Topics

By:

For information about ongoing research at the department, please see the webpages of the research groups and the personal homepages of our researchers. The department of mathematics has three research group in pure mathematics: Algebra, Geometry and Combinatorics, Analysis and Logic.

Current and/or potential PhD advisors are

Algebra, Geometry and Combinatorics: Gregory Arone, Jörgen Backelin, Alexander Berglund, Jonas Bergström, Rikard Bøgvad, Wushi Goldring, Samuel Lundqvist, Dan Petersen, Boris Shapiro.

Analysis: Pavel Kurasov, Annemarie Luger, Salvador Rodríguez-López, Jonathan Rohleder, Alan Sola.

Logic: Erik Palmgren, Peter LeFanu Lumsdaine.

A few suggestions for PhD topics are presented below.

### Calculus of functors and applications

Main supervisor:Gregory Arone

The goal of the project is to use calculus of functors, operads, moduli spaces of graphs, and other techniques from algebraic topology, to study spaces of smooth embeddings, and other important spaces. High-dimensional long knots constitute an important family of spaces that I am currently interested in. But it is by no means the only example.

Let $\mathbb{R}^m, \mathbb{R}^{m+i}$ be Euclidean spaces. An $m$-dimensional long knot in $\mathbb{R}^{m+i}$ is a smooth embedding $\mathbb{R}^m \hookrightarrow \mathbb{R}^{m+i}$ that agrees with the inclusion outside a compact set. Let $\text{Emb}_c(\mathbb{R}^m, \mathbb{R}^{m+i})$ be the space of all such knots.

The overarching goal of this project is to understand the dependence of the space $\text{Emb}_c(\mathbb{R}^m, \mathbb{R}^{m+i})$ on $m$ and $i$. The framework for doing this is provided by orthogonal calculus of functors, that was developed by Michael Weiss. The following are some of the specific objectives of this project.

• To analyse the structure to polynomial functors in orthogonal calculus.
• To show that it is possible to use orthogonal calculus to study the space of long knots $\text{Emb}_c(\mathbb{R}^m, \mathbb{R}^{m+i})$.
• To describe explicitly the derivatives (in the sense of orthogonal calculus) of functors such as $\text{Emb}_c(\mathbb{R}^m, \mathbb{R}^{m+i})$ in terms of moduli spaces of graphs similar to ones introduced by Culler and Vogtmann. The $n$-th derivative functor should be closely related to the moduli space of graphs that are homotopy equivalent to a wedge of $n$ circles.
• To show that rationally these derivatives are equivalent to hairy graph complexes that have been shown by Arone-Turchin and Turchin-Willwacher to calculate the rational homotopy of spaces of long knots.
• To compare two known $m+1$-fold deloopings of the space $\text{Emb}_c(\mathbb{R}^m, \mathbb{R}^{m+i})$. One of these deloopings, due to Dwyer-Hess, is homotopy-theoretic in nature and is given in terms of mapping spaces between operads. The other one is geometric in nature, and is given in terms of "topological Stiefel manifolds" $TOP(m+i)/TOP(i,m)$. I would like to show that the two deloopings are equivalent when $i\ge 3$, and also, by contrast, that they are not even rationally equivalent when $i=0$. The reason for this is that one delooping has the Pontryagin classes in its rational homotopy, while the other one does not.
• To clarify the connection between the delooping of $\text{Emb}_c(\mathbb{R}^m, \mathbb{R}^{m+i})$, and $G_i$ -- the group of self-homotopy equivalences of the sphere $S^{i-1}$. More precisely I want to show that $G_i$ is the limit of the delooping, as $m$ goes to infinity.
• To clarify the relationship of the first derivative to topological cyclic homology and to Waldhausen's algebraic K-theory.

### Algebraic models for spaces and manifolds

Main supervisor: Alexander Berglund

Algebraic topology studies continuous objects such as spaces or manifolds by attaching discrete invariants to them, e.g., the Euler characteristic, the fundamental group, cohomology groups, etc. As the invariants are refined by adding more algebraic structure, complete classification becomes possible in favorable situations. For example, for closed surfaces the fundamental group is a complete algebraic invariant, for simply connected manifolds the de Rham complex with its wedge product is a complete invariant of the real homotopy type, and for simply connected topological spaces the singular cochain complex with its E-infinity algebra structure is a complete invariant of the integral homotopy type.

My research revolves around algebraic models for spaces and their applications. Here are some topics for possible PhD projects within this area:

• #### Automorphisms of manifolds

The cohomology ring of the automorphism group of a manifold M is the ring of characteristic classes for fiber bundles with fiber M, which is an important tool for classification. Tractable differential graded Lie algebra models can be constructed for certain of these automorphism groups. A possible PhD project here is to further develop these algebraic models, and in particular to further investigate a newfound connection to Kontsevich graph complexes. This will involve a wide variety of tools from algebraic and differential topology as well as representation theory and homological algebra.

• #### String topology and free loop spaces

The space of strings in a manifold carries important information, e.g., about geodesics. Its homology carries interesting algebraic structure such as the Chas-Sullivan loop product. Tools such as Koszul duality theory, A-infinity algebras and Hochschild cohomology can be used to construct tractable algebraic models for free loop spaces. A possible PhD project is to further develop these models, in particular to endow them with more algebraic structure, and use them to make new computations.

### Moduli spaces, varieties over finite fields and Galois representations

Main supervisor:Jonas Bergström

Moduli spaces are spaces that parametrize some set of geometric objects. These spaces have become central objects of study in modern algebraic geometry. One way of getting a better understanding of a space is to find information about its cohomology.

In my research I have tried to extend the knowledge about the cohomology of moduli spaces when the objects parametrized are curves or abelian varieties. The main tool has been the so called Lefschetz fixed point theorem which connects the cohomology to counts over finite fields. That is, counting isomorphism classes of, say, curves defined over finite fields gives information about the cohomology (by comparison theorems also in characteristic zero) of the corresponding moduli space. I have often used concrete counts over small finite fields using the computer to find such information.

The cohomology of an algebraic variety (that is defined over the integers) comes with an action of the absolute Galois group of the rational numbers. Such Galois representations are in themselves very interesting objects. A count over finite fields also gives aritmethic information about the Galois representations that appear. In the case of Shimura varieties (at least according to a general conjecture which is part of the so called Langlands program) one has a good idea of which Galois representations that should appear, namely ones coming from the corresponding modular (and more generally, automorphic) forms. If one is not considering a Shimura variety, as for example the moduli space of curves with genus greater than one, it is much less clear what Galois representations to expect even though they are still believed to come from automorphic forms.

Key words: algebraic geometry, moduli spaces, curves, abelian varieties, finite fields, modular forms

### Asymptotic properties of zero-sets of polynomials in higher dimensions, currents and holonomic systems of differential equations

Consider the polynomials $p_n = x^n-1$. As the parameter $n$ increases, the n complex zeroes cluster in a very regular manner on a curve, the unit circle. This kind of behaviour is common in many other examples of sequences of polynomials, that, as here, are solutions to parameter dependent differential equations. The sequences occur in different areas, such as combinatorics, or special functions in Lie theory and algebraic geometry, and it is useful and interesting to understand the asymptotic properties of the polynomials through their zeroes.  A large amount of work has been done on this, in particular to determine what kind of curves in the complex plane that arise as asymptotic zero-sets.

The main  idea in these papers is often to consider the zero set as a measure and then use harmonic analysis, related to an algebraic curve, the so-called characteristic curve of the equation. There are as yet few papers that consider the corresponding problem in higher dimensions, and this is the suggested topic, and one that I have just started with. It is then natural to use the differential-geometric concept of currents, instead of measures, and connected complex algebraic geometry. Instead of having just one parameter dependent differential equation, one would consider holonomic systems of differential equations, such as GKZ-systems, that are important in some parts of algebraic geometry and algebraic topology. Holonomic systems come from the algebraic study of systems of differential equations, so-called D-module theory, and is a nice mixture of commutative algebra and analysis. In particular I am interested in understanding the relation to the characteristic variety better, since I expect this to also give a better understanding of the one-variable case.

Key words: complex algebraic geometry, D-module theory, varieties, hyper-geometric functions, harmonic analysis

### The quest for algebraicity: The Langlands Program and beyond

Main supervisor: Wushi Goldring

In 1967, R. P. Langlands wrote a letter to A. Weil. It would revolutionize mathematics. It launched the now-famous Langlands Program. For almost half-a-century, this program has been a driving force in several areas of mathematics, particularly harmonic analysis, representation theory, algebraic geometry, number theory and mathematical physics. It has seen spectacular progress and varied applications, such as Wiles' proof of Fermat's Last Theorem and Ngô's proof of the Fundamental Lemma. At the same time, most instances of Langlands' conjectures remain unsolved.

Fifty years later, all agree that the Langlands Program is indispensable for the unification of abstract mathematics. But many -- including perhaps Langlands himself -- grapple with the ultimate raison dêtre of the program. So what is really at the heart of the Langlands Program?

The prevailing common view has been that large swaths of the Langlands Program are inherently analysis-bound, that an algebraic understanding of them is impossible. Under this view, the Langlands Program is seen as injecting analytic methods to solve classical problems in number theory and algebraic geometry.

My research is focused on inverting the common view: My working algebraicity thesis is that, on the contrary, the Langlands Program is deeply algebraic and unveiling its algebraic nature leads to new results, both within it and in the myriad of areas it impinges upon.

In pursuit of my algebraicity theme, building on joint work with Jean-Stefan Koskivirta and other collaborators, I have begun a program to make simultaneous progress in the following four seemingly unrelated areas, by developing the connections between them:

(A) Algebraicity of automorphic representations
(B) The Deligne-Serre interchange of characteristic'' approach to
algebraicity, concerning both (A) but also a variety of other
questions
(C) The geometry of stacks of G-Zips, the Ekedahl-Oort (EO)
stratification of Shimura varieties and their Hasse invariants.
(D)  Algebraicity of Griffiths-Schmid manifolds.

### Discrete and continuous quantum graphs

Main supervisor:Pavel Kurasov

Credit: Erlend Davidson (Thomas Young Centre)

Quantum graphs - differential operators on metric graphs - is a rapidly growing branch of mathematical physics lying on the border between differential equations, spectral geometry and operator theory. The goal of the project is to compare dynamics given by discrete equations associated with (discrete) graphs with the evolution governed by quantum graphs. Discrete models can be successfully used to describe complex systems where the geometry of the connections between the nodes can be neglected. It is more realistic to use instead metric graphs with edges having lengths. The corresponding (continuous)
dynamics is described by differential equations coupled at the vertices. Such models are used for example in modern physics of nano-structures and microwave cavities.

Understanding the relation between discrete and continuous quantum graphs is a challenging task leaving a lot of freedom, since this area has not been studied systematically yet. In special cases such relations are straightforward, sometimes methods originally developed for discrete graphs can be generalized, but often studies lead to new unexpected results.

To find explicit connections between the geometry and topology of such graphs on one side and spectral properties of corresponding differential equations on the other is one of the most exciting directions in this research area. As an example one may mention an explicit formula connecting the asymptotics of eigenvalues to the number of cycles in the graph, or the estimate for the spectral gap (the difference between the two lowest eigenvalues) proved using a classical Euler theorem dated to 1736!

Possible directions of the research project are

•  Study spectral properties of continuous quantum graphs in relation to their connectivity and complexity (this question is well-understood for discrete graphs);
•  Investigate relations between quantum graphs and quasicrystals;
•  Develop new models combining features of discrete and continuous graphs and  study their properties;
•  Transport properties of networks and their complexity.

It is expected that the candidate will take an active part in the work of the Analysis
group at Stockholm University, Cooperation group "Continuous Models in the Theory of Networks" (ZIF, Bielefeld, http://www.uni-bielefeld.de/ZIF/KG/2012Models/) and Research and Training Network "QGRAPH" joining 15 research teams from all over the world.

### Herglotz-Nevanlinna functions

Main supervisor:Annemarie Luger

When complex analysis and functional analysis meet, many interesting things can happen. One such topic are questions involving so-called Herglotz-Nevanlinna functions, these are functions mapping the complex upper half plane analytically onto itself. They appear in surprisingly many situations, both in pure mathematics as well as in applications, for example, in connection with both ordinary and partial differential operators, in perturbation theory and extension theory, as transfer functions for passive systems, or as Fouriertransform of certain distributions, just to name a few occasions.

One typical example of how to utilize such connections is e.g. that eigenvalues of a differential operator are given as the zeros of a related Herglotz-Nevanlinna function.

In this project the PhD-student will work with such functions and in particular their generalisations. It will also be possible to connect the research questions to concrete applications in electro-engineering.

### Mathematical Logic - constructive and category-theoretic foundations for mathematics

Main supervisor:  Erik Palmgren

Project description: Constructive or algorithmic methods are important in mathematics, and are often the nal result when a mathematical theory is to be applied. In this wide-ranging research project, general logical and categorytheoretic methods are developed and studied in order to ensure constructive content of mathematical theorems. This covers for instance the study of constructive type theories and set theories with the help of models or properties of formal proofs. It could also include case studies where a limited area of mathematics
is constructivized. Apart from the purely mathematical-logical questions there are also interesting philosophical aspects, and also applications in computer science, for example extraction of computer programs from mathematical proofs.

For further information contact Erik Palmgren.

### Geometry and topology of moduli spaces

Main supervisor: Dan Petersen

I am interested broadly in the interface between algebraic geometry and algebraic topology. For example, the cohomology of algebraic varieties carries all kinds of "extra" structures that has no counterpart in the cohomology of, say, a manifold or a cell complex. I have been particularly interested in questions concerning moduli spaces in algebraic geometry.

Some ideas for projects are:

• Partially compactified moduli spaces of genus zero curves, operads, multiple zeta values. In work with Johan Alm, we studied a partial compactification of the moduli space of marked genus zero curves from an operad-theoretic perspective. There are still several open questions from that project. Optimistically, these kinds of results could be useful for the study of period integrals over these spaces; after the work of Francis Brown, these period integrals are known to be linear combinations of multiple zeta values.
• Cohomology of moduli spaces of principally polarized abelian varieties. Let A(g) be the moduli space of principally polarized abelian varieties of genus g. Recent work in the theory of automorphic representations of Chenevier, Renard, Lannes, Taïbi and others has made it possible to calculate the intersection cohomology of the Satake compactification of A(g) with twisted coefficients in a large range. It would be interesting to try to leverage these calculations to understand the cohomology of A(g) itself with twisted coefficients, by trying to understand the "perverse" Leray spectral sequence associated to the inclusion of A(g) into its Satake compactification.
• Tautological classes with twisted coefficients. The tautological ring of the moduli space of curves is an interesting subring of its cohomology ring/Chow ring. In work with Qizheng Yin and Mehdi Tavakol, we defined a notion of tautological classes inside the cohomology with twisted coefficients of the uncompactified moduli space. It would be useful to have a similar theory also on the compactified moduli space, but it is not clear how to set everything up. This will require working with intersection cohomology and perverse sheaves. Optimistically, it should be possible to do this "motivically".

### Analysis of Rough Linear and Multilinear Pseudodifferential Operators

In the study of Partial Differential Equations and in Harmonic Analysis, an important role is played by the so-called pseudodifferential operators. For instance, for equations that describe electric potential and steady-state heat flow (elliptic equations) one can construct explicit solutions using pseudodifferential operators. Roughly speaking, these operators act on functions (or signals) by filtering (attenuating or amplifying) specific frequencies of those. For equations that describe wave propagation (hyperbolic equations), a similar role is played by Fourier integral operators. These tools allow us to obtain a priori estimates for the solutions, and study their behaviour and properties. Therefore, being able to estimate these operators in different function spaces is important for measuring the size and regularity of the solutions of PDEs in those spaces.

In controlling height and width of a solution, the most important example of such spaces are the Lebesgue spaces Lp. Due to their rearrangement-invariant nature, these spaces are blind to the description of where solutions are concentrated, and thus the consideration of Lebesgue spaces with weights appears naturally. An important role is played by the so-called Muckenhoupt Ap weights.

For nonlinear PDEs , the multilinear counterpart of pseudodifferential and Fourier integral operators play a crucial role.

My recent research interests have been dealing with questions regarding both linear and multilinear operators of those described above, and in particular with those of rough type.

To get involved in a project in these areas requires a strong background and interest in Harmonic Analysis and PDEs. Some examples of lines of research that one could pursue:

• To develop an Ap-weighted theory for some classes of rough and mildly regular pseudodifferential operators, and find up-to-end-point improvements of existing results in the literature.
• To investigate the validity of corresponding end-point estimates for such operators.
• To develop the theory of spectral properties of rough pseudodifferential operators.
• Study multilinear end-point results and results of minimal regularity assumptions, for paraproducts and their application to the study of boundedness properties of multilinear pseudodifferential and Fourier integral operators.

### Spectral properties of differential operators on domains and graphs

Main supervisor: Jonathan Rohleder

Eigenvalues and, more generally, spectra of differential operators appear naturally in numerous physical models, for instance as frequencies of vibrating strings and membranes, or as energies of quantum systems. Their investigation is an important and lively field in mathematical physics. Since the spectra of most models cannot be calculated explicitly, there is a strong need for qualitative and quantitative estimates. In my current research I focus on the spectral investigation of Laplacian and Schrödinger operators on domains in the Euclidean space and on metric graphs. Possible research areas for a PhD include:

• Eigenvalue inequalities for Schrödinger operators on domains with mixed boundary conditions and their dependence on the geometry of the boundary.
• Spectral estimates for Sturm-Liouville operators on metric graphs in relation to the vertex conditions and the geometry and topology of the graph.
• Estimation of non-real spectra for Schrödinger operators with non-self-adjoint boundary conditions or non-real potentials.
• Spectral properties of infinite quantum graphs.
• Connections between the spectra of domains and graphs.

### Operator theory and function theory in polydisks

Main supervisor: Alan Sola

Coordinate shifts acting on Banach spaces of analytic functions represent a concrete and compelling incarnation of operator theory. At first glance, considering the action of a such simple operators on specific function spaces may appear to lose much of the generality that makes operator theory a powerful and flexible tool in mathematics. One can show, however, that many Hilbert space contractions are unitarily equivalent to the shift acting on model subspaces of the Hardy space. The class of analytic 2-isometries has the Dirichlet shift as a natural realization, and there are many other such models. Thus, understanding the invariant subspaces and cyclic vectors of coordinate shifts is important, and leads to a better understanding of more general operators.

Moreover, by working with analytic functions, we are able to connect operator-theoretic questions to deep problems in complex function theory such as boundary behavior, vanishing properties, and so on. For example, answering the question of whether a specific analytic function is cyclic with respect to shifts acting on a function space frequently amounts to analyzing the size and properties of the zero set of the function.

These types of questions have been studied by many mathematicians in the one-variable setting of the unit disk, and many remarkable results have been obtained. However, the higher-dimensional analogs of coordinate shifts acting on function spaces in polydisks have received somewhat less attention, especially for function spaces beyond the Hardy spaces. In recent years, I have been particularly interested in weighted Dirichlet spaces, which can be defined in terms of area-integrability of partial derivatives of an analytic function. These spaces are challenging due to the relative smoothness of their elements, yet are rich enough to allow for an interesting subspace structure. In a recent series of papers, my coauthors and I have started making headway on the problem of identifying cyclic vectors in weighted Dirichlet spaces in the bidisk, and we have found techniques for checking membership in such spaces of functions having singularities on the boundary of the bidisk.

Possible directions for a PhD project might be to:

• Study the multiplier algebra of weighted Dirichlet spaces on polydisks.
• Develop a machinery to analyze integrability and regularity of rational functions by examining the geometry of their singularities.
• Develop an understanding of the structure of invariant subspaces for the coordinate shifts.
• Study zero sets and boundary zero sets for function spaces in polydisks.
Project 1: Spectral and scattering properties of Quantum Hamiltonians (Prof. M. Melgaard)

Quantum Operator Theory concerns the analytic properties of mathematical models of quantum systems. Its achievements are among the most profound and most fascinating in Quantum Theory, e.g., the calculation of the energy levels of atoms and molecules which lies at the core of Computational Quantum Chemistry.

Among the many problems one can study, we give a short list:

• The atomic Schrödinger operator (Kato's theorem and all that);
• The periodic Schrödinger operator (describing crystals);
• Scattering properties of Schrödinger operators (describing collisions etc);
• Spectral and scattering properties of mesoscopic systems (quantum wires, dots etc);
• Phase space bounds (say, upper bounds on the number of energy levels) with applications, e.g., the Stability of Matter or Turbulence.

Key words: differential operators, spectral theory, scattering theory.

Recommended modules: Functional Analysis, Measure and Integration theory, Partial Differential Equations.

References:

!$[1]$! M. Melgaard, G. Rozenblum, Schrödinger operators with singular potentials, in: Stationary partial differential equations Vol. II, 407--517, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2005.

!$[2]$! Reed, M., Simon, B., Methods of modern mathematical physics. Vol. I-IV. Academic Press, Inc., New York, 1975, 1978,1979,1980.

Project 2: Variational approach to Kohn-Sham models with magnetic fields (Prof. M. Melgaard)

Quantum Mechanics (QM) has its origin in an effort to understand the properties of atoms and molecules. Its first achievement was to establish the Schrödinger equation by explaining the stability of the hydrogen atom; but hydrogen is special because it is exactly solvable. When we proceed to a molecule, however, the QM problem cannot be solved in its full generality. In particular, we cannot determine the solution (i.e., the ground state) to !$HΨ=EΨ$!, where !$H$! denotes the Hamiltonian of the molecular system, !$Ψ$! is the wavefunction of the system, and !$E$! is the lowest possible energy. This problem corresponds to finding the minimum of the spectrum of !$H$! or, equivalently, !$$E= \inf \{ \, \mathcal{E}^{\rm QM}(Ψ) \, : \, Ψ \in \mathcal{H}, \:\: \| Ψ \|_{L^{2}} =1 \, \}, where \ \mathcal{E}^{\rm QM}(Ψ):= \langle Ψ, H Ψ \rangle_{L^{2}}$$! and !$\mathcal{H}$! is the variational (Hilbert) space. For systems involving a few (say today six or seven) electrons, a direct Galerkin discretization is possible, which is known as Full CI in Computational Chemistry. For larger systems, with !$N$! electrons, say, this direct approach is out of reach due to the excessive dimension of the space !$ℜ^{3N}$! on which the wavefunctions are defined and the problem has to be approximated. Quantum Chemistry (QC), as pioneered by Fermi, Hartree, Löwdin, Slater, and Thomas, emerged in an attempt to develop various ab initio approximations to the full QM problem. The approximations can be divided into wavefunction methods and density functional theory (DFT). For both, the fundamental questions include minimizing configuration, divided into Question I (i) necessary and sufficient conditions for existence of a ground state (=a minimizer), and Question I (ii) uniqueness of a minimizer, and Question II, necessary and sufficient conditions for multiple (nonminimal) solutions (i.e., excited states).

A magnetic field has two effects on a system of electrons: (i) it tends to align their spins, and (ii) it alters their translational motion. The first effect appears when one adds a term of the form !$-eħm^{-1} {s} \cdot \mathcal{B}$! to the Hamiltonian, while the second, diamagnetic effect arises from the usual kinetic energy !$(2m)^{-1} | {\mathbf p} |^{2}$! being replaced by !$(2m)^{-1} | \mathbf {p} -(e/c) \mathcal{A}|^{2}$!. Here !${\mathbf p}$! is the momentum operator, !$\mathcal{A}$! is the vector potential, !$\mathcal{B}$! is the magnetic field associated with !$\mathcal{A}$!, and !${s}$! is the angular momentum vector. Within the numerical practice, one approach is to apply a perturbation method to compute the variations of the characteristic parameters of, say, a molecule, with respect to the outside perturbation. It is interesting to go beyond and consider the full minimization problem of the perturbed energy. In Hartree-Fock theory, one only takes into account the effect (ii), whereas in nonrelativistic DFT it is common to include the spin-dependent term and to ignore (ii) and to study the minimization of the resulting nonlinear functional, which depends upon two densities, one for spin "up" electrons and the other for spin "down" electrons. Each density satisfies a normalisation constraint which can be interpreted as the total number of spin "up" or "down" electrons.

The proposed project concerns the above-mentioned problems within the context of DFT in the presence of an external magnetic field. More specifically, molecular Kohn-Sham (KS) models, which turned DFT into a useful tool for doing calculations, are studied for the following settings:

Recent results on rigorous QC are found in the references.

1. As a first step towards systems subject to a magnetic field, Question I(i) is addressed for the unrestricted KS model, which is suited for the study of open shell molecular systems (i.e., systems with a odd number of electrons such as radicals, and systems with an even number of electrons whose ground state is not a spin singlet state). The aim is to consider the (standard and extended) local density approximation (LDA) to DFT.
2. The spin-polarized KS models in the presence of an external magnetic field with constant direction are studied while taking into account a realistic local spin-density approximation, in short LSDA.

Key words: differential operators, spectral theory, scattering theory.

Recommended modules: Functional Analysis, Measure and Integration theory, Partial Differential Equations.

References:

!$[1]$! M. Melgaard, G. Rozenblum, Schrödinger operators with singular potentials, in: Stationary partial differential equations Vol. II, 407--517, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2005.

!$[2]$! Reed, M., Simon, B., Methods of modern mathematical physics. Vol. I-IV. Academic Press, Inc., New York, 1975, 1978,1979,1980.

Project 3: Resonances for Schrödinger and Dirac operators (Prof. M. Melgaard)

Resonances play an important role in Chemistry and Molecular Physics. They appear in many dynamical processes, e.g. in reactive scattering, state-to-state transition probabilities and photo-dissociation, and give rise to long-lived states well above scattering thresholds. The aim of the project is carry out a rigorous mathematical study on the use of Complex Absorbing Potentials (CAP) to compute resonances in Quantum Dynamics.

In a typical quantum scattering scenario particles with positive energy arrive from infinity, interact with a localized potential !$V(x)$! whereafter they leave to infinity. The absolutely continuous spectrum of the the corresponding Schrödinger operator !$T(\hbar)=-\hbar^{2}D+V(x)$! coincides with the positive semi-axis. Nevertheless, the Green function !$G(x,x'; z)= \langle x | (T(\hbar)-z)^{-1}| x \rangle$! admits a meromorphic continuation from the upper half-plane !$\{ \, {\rm Im}\, z >0 \,\}$! to (some part of) the lower half-plane !$\{ \, {\rm Im}\, z < 0 \,\}$!. Generally, this continuation has poles !$z_{k} =E_{k}-i Γ_{k}/2$!, !$Γ_{k}>0$!, which are called resonances of the scattering system. The probability density of the corresponding "eigenfunction" !$Ψ_{k}(x)$! decays in time like !$e^{-t Γ_{k}/ \hbar}$!, thus physically !$Ψ_{k}$! represents a metastable state with a decay rate !$Γ_{k}/ \hbar$! or, re-phrased, a lifetime !$\tau_{k}=\hbar / Γ_{k}$!. In the semi-classical limit !$\hbar \to 0$!, resonances !$z_{k}$! satisfying !$Γ_{k}=\mathcal{O}(\hbar)$! (equivalently, with lifetimes bounded away from zero) are called "long-lived".

Physically, the eigenfunction !$Ψ_{k}(x)$! only make sense near the interaction region, whereas its behaviour away from that region is evidently nonphysical (Outgoing waves of exponential growth). As a consequence, a much used approach to compute resonances approximately is to perturb the operator !$T(\hbar)$! by a smooth absorbing potential !$-iW(x)$! which is supposed to vanish in the interaction region and to be positive outside. The resulting Hamiltonian !$T_{W}(\hbar):=T(\hbar)-iW(x)$! is a non-selfadjoint operator and the effect of the potential !$W(x)$! is to absorb outgoing waves; on the contrary, a real-valued positive potential would reflect the waves back into the interaction region. In some neighborhood of the positive axis, the spectrum of !$T_{W}(\hbar)$! consists of discrete eigenvalues !$\tilde{z}_{k}$! corresponding to !$L^{2}$!-eigenfunctions !$\widetilde{Ψ}_{k}$!.

As mentioned above, the CAP method has been widely used in Quantum Chemistry and numerical results obtained by CAP are very good. The drawback with the use of CAP is that there are no proof that the correct resonances are obtained. (This is in stark contrast to the mathematically rigorous method of complex scaling). In applications it is assumed implicitly that the eigenvalues !$\tilde{z}_{k}$! near to the real axis are small perturbations of the resonances !$z_{k}$! and, likewise, the associated eigenfunctions !$\widetilde{Ψ}_{k}$!, !$Ψ_{k}(x)$! are close to each other in the interaction region. Stefanov (2005) proved that very close to the real axis (namely, for !$| {\rm Im}\, \tilde{z}_{k}| =\mathcal{O}(\hbar^{n})$! provided !$n$! is large enough), this is in fact true. Stefanov's proof relies on a series of ingenious developments by several people, most notably Helffer (1986), Sjöstrand (1986, 1991, 1997, 2001, 2002), and Zworski (1991, 2001).

The first part of the project would be to understand in details Stefanov's work [2] and, subsequently, several open problems await.

Key words: operator and spectral theory, semiclassical analysis, micro local analysis.

Recommended modules:Functional Analysis, Measure and Integration theory, Partial Differential Equations.

References:

!$[1]$! J. Kungsman, M. Melgaard, Complex absorbing potential method for Dirac operators. Clusters of resonances, J. Ope. Th., to appear.

!$[2]$! P. Stefanov, Approximating resonances with the complex absorbing potential method, Comm. Part. Diff. Eq. 30 (2005), 1843--1862.

Project 4: Critical point approach to solutions of nonlinear, nonlocal elliptic equations arising in Astrophysics (Prof. M. Melgaard)

The Choquard equation in three dimensions reads:

!$$$$\tag*{(0.1)} -Δ u - \left( \int_{ℜ^{3}} u^{2}(y) W(x-y) \, dy \right) u(x) = -l u ,$$$$! where !$W$! is a positive function. It comes from the functional:

!$$\mathcal{E}^{\rm NR}(u) = \int_{ℜ^{3}} | \nabla u |^{2} \, dx -\int \int | u(x) |^{2} W(x-y) |u(y)|^{2} \, dx dy,$$!

which, in turn, arises from an approximation to the Hartree-Fock theory of a one-component plasma when !$W(y) =1/ | y |$! (Coulomb case). Lieb (1977) proved that there exists a unique minimizer to the constrained problem !$E^{\rm NR}(\nu) = \inf \{ \, \mathcal{E}(u) \, : \, u \in \mathcal{H}^{1}(ℜ^{3}), \| u \|_{L^{2}} \leq \nu \, \}$!.

The mathematical difficulty of the functional is caused by the minus sign in !$\mathcal{E}^{\rm NR}$!, which makes it impossible to apply standard arguments for convex functionals. Lieb overcame the lack of convexity by using the theory of symmetric decreasing functions. Later Lions (1980) proved that the unconstrained problem (0.1) possesses infinitely many solutions. For the constrained problem, seeking radially symmetric, normalized functions !$\| u \|_{L^{2}} =+1$!, or more generally, seeking solutions belonging to:

!$$\mathcal{C}_{N}= \{ \, φ \in \mathcal{H}_{\rm r}^{1} (ℜ^{3}) \, : \, \| φ \|_{L^{2}} =N \, \} ,$$! the situation is much more complicated and conditions on !$W$! are necessary. In the Coulomb case, Lions proves that there exists a sequence !$(l_{j}, u_{j})$!, with !$l_{j} > 0$!, and !$u_{j}$! satisfies !${(0.1)}$! (with !$l=l_{j}$!) and belongs to !$\mathcal{C}_{1}$!

We may replace the negative Laplace operator by the so-called quasi-relativistic operator, i.e., the pseudodifferential operator !$\sqrt{ -δ +m^{2} } -m$!; this is the kinetic energy operator of a relativistic particle of mass !$m \geq 0$!. It is defined via multiplication in the Fourier space with the symbol !$\sqrt{k^{2} +m^{2}} -m$!, which is frequently used in relativistic quantum physics models as a suitable replacement of the full (matrix valued) Dirac operator. The associated time-dependent equation arises as an effective dynamical description for an !$N$!-body quantum system of relativistic bosons with two-body interaction given by Newtonian gravity, as recently shown by Elgart and Schlein (2007). This system models a Boson star.

Several questions arise for the quasi-relativistic Choquard equation (existence, uniqueness, positive solutions etc) and the first part of the project would be to get acquainted with recent (related) results, e.g., [1] and [2].

Key words: operator and spectral theory, semiclassical analysis, micro local analysis.

Recommended modules:Functional Analysis, Measure and Integration theory, Partial Differential Equations.

References:

!$[1]$! S. Cingolani, M. Clapp, S. Secchi, Multiple solutions to a magnetic nonlinear Choquard equation, Zeitschrift fr Angewandte Mathematik und Physik (ZAMP) , vol. 63 (2012), 233-248.

!$[2]$! M. Melgaard, F. D. Zongo, Multiple solutions of the quasi relativistic Choquard equation, J. Math. Phys. !${53}$!(2012), 033709 (12 pp).