Posudek vedouc´ıho diplomové práce

Posudek vedoucı́ho diplomové práce
Název: Spektra kvantových grafů
Autor: Bc. Gabriela Malenová
Vedoucı́: Mgr. David Krejčiřı́k, Ph.D., DSc.
Studentka se ve své diplomové práci zabývá spektrálnı́mi vlastnostmi schrödingerovských
operátorů na metrických grafech. Hlavnı́ výsledky jsou dvojı́ho typu.
1. Primárnı́m úkolem bylo vyvinout efektivnı́ a uživatelsky přátelskou numerickou metodu pro výpočet mnoha vlastnı́ch hodnot při zadaných parametrech grafu. Studentka si
úspešně osvojila a aplikovala spektrálnı́ metodu založenou na uzlech Chebyshevových polynomů, a to iniciativně vytvořenı́m nové knihovny modernı́ nástavby chebfun prostředı́
Matlab. Vyvinutý kód lze účinně použı́t pro výpočet spekter kvantových grafů za
přı́tomnosti elektrického i magnetického pole a v diplomové práci je mimojiné použit
pro studium inversnı́ úlohy na grafu. Nutno podotknout, že nově vyvinutá knihovna
studentky si zı́skala pozornost britského projektu chebfun representovaného Nickem Trefethenem.
2. Numerická analysa spektrálnı́ch vlastnostı́ grafů přirozeně vedla k formulaci teoretických předpovědı́ a jejich rigorosnı́m důkazům. Dokázané věty lze považovat za grafové
analogie isoperimetrických nerovnostı́ pro spektrum Laplaceova operátoru na oblastech
či varietách. Studentka napřı́klad ukazuje, že spojenı́m dvou vrcholů grafu se zvyšuje
přechodová energie základnı́ho stavu a že tato energie se naopak snižuje dodánı́m nové
hrany do grafu apod. Tato část diplomové práce byla přijata k opublikovánı́ v časopise
Journal of Physics A: Mathematical and Theoretical se spoluautory Pavel Kurasov a Sergey Naboko, z nichž prvnı́ byl hlavnı́m vedoucı́m práce během ročnı́ho Erasmus pobytu
studentky ve Stockholmu.
Podle názoru českého vedoucı́ho práce dosažené výsledky přesahujı́ standardnı́ kritéria
kladená na diplomové práce. Na závěr bych si také dovolil poznamenat, že autorka
výsledky svého bádánı́ úspěšně presentovala na několika konferencı́ch a seminářı́ch doma
i v zahraničı́. Předchozı́ verse diplomové práce byla obhájena v listopadu 2012 na Stockholm University ve Švédsku. Celkově lze diplomovou práci studentky považovat jako
dobrý start do jejı́ budoucı́ vědecké kariéry.
Navrhuji ohodnocenı́ známkou A - výborně.
Řež, 13. května 2013
David Krejčiřı́k
Universität Ulm | D-89081 Ulm | Germany
Faculty of Mathematics
and Economics
Institute of Analysis
Jun.-Prof. Dr. Delio Mugnolo
Helmholtzstr. 18
D-89081 Ulm, Germany
Phone: +49(0)731-5023603
Fax: +49(0)731-5023648
[email protected]
May 21, 2013
Report on the Diploma thesis Spectra Of Quantum Graphs by Gabriela Malenová
In recent years, much attention has been devoted – both in analysis and mathematical physics – to so-called
quantum graphs: these consist of differential operators that do not – as usual – act on function defined on
domains, but rather on a certain class of singular one-dimensional manifolds that can be most naturally looked
at as graphs whose edges are endowed with a suitable metric structure. By extension, in recent years also the
graph Laplacian and some further difference operators (i.e., matrices) on finite or infinite (non-metric) graphs
have been re-discovered by analysts. This interesting diploma thesis is devoted to the study of several problems
in the spectral theory of such quantum or discrete graphs.
One of the most interesting features of quantum graphs is their dual nature – partly discrete, partly continuous
objects. Indeed, it is well-known that the spectrum of the Laplacian or, more generally, of a Schrödinger operator
will typically depend both on the connectivity of the associated graph – which in turn determines the spectrum
of said difference operators – and on the typical features of 1-dimensional Schrödinger operators. Then, the
influence on the spectrum of both the former (e.g., in terms of dependence from certain kinds of graph surgery)
– or the latter (say, under addition of magnetic and/or electric potentials, or modification of the edge lengths)
can be naturally investigated. This was the candidate’s task.
After the introductory Section 1, which contains some reasonably comprehensive overview of the literature in
this field, and Section 2, where the notation is fixed, Ms Malenová devotes in Section 3 her attention to the
determination of the spectrum of the Laplacian on a few easy classes of continuous graphs. This is certainly not
difficult, and may in principle be done for arbitrarily complicated graphs, but the straightforward formulae quickly
become unhandy as the connectivity of the graphs gets more and more intricate.
It is mostly for this reason that numerical analysis is a good aid – also because the (almost) 1-dimensional setting
may make numerical schemes particularly easy and efficient. Section 4 is thus devoted to the description of
a new class for a certain MATLAB-add-on that has been programmed by the candidate: It takes connectivity,
edge lengths, boundary conditions, and potentials as inputs, and yields in return the first few eigenvalues
and eigenvectors. I am not a numerical analyst, so that I can hardly tell how deep Ms Malenová’s numerical
contribution is; but I can certainly say that many of her results and observations about the spectrum of differential
and difference operators bear some interest for the analyst or mathematical physicist working on quantum
graphs. For instance, by means of this software it is observed in Section 5 that deleting an edge from a
continuous graph may typically produce a shift in the spectrum, as one expects; but also that this shift may in
general be either positive or negative, unless more is known about the the graph: this might indeed be surprising,
as deleting edges always leads to larger eigenvalues in the case of the Laplacian matrix on a discrete graph.
Motivated by this observation, in the last section Ms Malenová focuses her attention on comparison of eigenvalues, and in particular of the spectral gap, on different but related graphs. Sections 6.1–6.2 contain a systematic
study of the behaviour of the first non-zero eigenvalue of either the discrete or the continuous Laplacian under
certain more or less natural graph operations: edge/node deletion, edge addition, decoration, and so on. Section 6.3 is devoted to proving a Faber–Krahn-type result by an ingenious method based on turning the given
graph into a new Eulerian by suitably decorating it.
The present work elaborates on an article written by Ms Malenová jointly with P. Kurasov and S. Naboko,
substantially complemented by her own numerical investigations on the topic. I do not know the Czech academic
system well enough to competently assess this work in relation to its context, but I can say that in Germany
its original content would certainly make it an outstanding MSc thesis. It shows enthusiasm, ability to develop
interesting mathematical ideas, solid knowledge of the existing literature and unusual skills in relating and
comparing new and old results. In my opinion it deserves to be graded
A
Jun.-Prof. PD Dr. Delio Mugnolo
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