dc.date.accessioned 2020-10-29T11:54:33Z dc.date.available 2020-10-29T11:54:33Z dc.date.issued 2020-10-29 dc.identifier.uri http://hdl.handle.net/123456789/31852 dc.title Two results on the essential self-adjointness of the Schrödinger operator en ethesis.discipline none und ethesis.department none und ethesis.faculty Matemaattis-luonnontieteellinen tiedekunta fi ethesis.faculty Faculty of Science en ethesis.faculty Matematisk-naturvetenskapliga fakulteten sv ethesis.faculty.URI http://data.hulib.helsinki.fi/id/8d59209f-6614-4edd-9744-1ebdaf1d13ca ethesis.university.URI http://data.hulib.helsinki.fi/id/50ae46d8-7ba9-4821-877c-c994c78b0d97 ethesis.university Helsingin yliopisto fi ethesis.university University of Helsinki en ethesis.university Helsingfors universitet sv dct.creator Nyberg, Jonne dct.issued 2020 dct.language.ISO639-2 eng dct.abstract Spectral theory is a powerful tool when applied to differential equations. The fundamental result being the spectral theorem of Jon Von Neumann, which allows us to define the exponential of an unbounded operator, provided that the operator in question is self-adjoint. The problem we are considering in this thesis, is the self-adjointness of the Schr\"odinger operator $T = -\Delta + V$, a linear second-order partial differential operator that is fundamental to non-relativistic quantum mechanics. Here, $\Delta$ is the Laplacian and $V$ is some function that acts as a multiplication operator. en We will study $T$ as a map from the Hilbert space $H = L^2(\mathbb{R}^d)$ to itself. In the case of unbounded operators, we are forced to restrict them to some suitable subspace. This is a common limitation when dealing with differential operators such as $T$ and the choice of the domain will usually play an important role. Our aim is to prove two theorems on the essential self-adjointness of $T$, both originally proven by Tosio Kato. We will start with some necessary notation fixing and other preliminaries in chapter 2. In chapter 3 basic concepts and theorems on operators in Hilbert spaces are presented, most importantly we will introduce some characterisations of self-adjointness. In chapter 4 we construct the test function space $D(\Omega)$ and introduce distributions, which are continuous linear functionals on $D(\Omega).$ These are needed as the domain for the adjoint of a differential operator can often be expressed as a subspace of the space of distributions. In chapter 5 we will show that $T$ is essentially self-adjoint on compactly supported smooth functions when $d=3$ and $V$ is a sum consisting of an $L^2$ term and a bounded term. This result is an application of the Kato-Rellich theorem which pertains to operators of the form $A+B$, where $B$ is bounded by $A$ in a suitable way. Here we will also need some results from Fourier analysis that will be revised briefly. In chapter 6 we introduce some mollification methods and prove Kato's distributional inequality, which is important in the proof of the main theorem in the final chapter and other results of similar nature. The main result of this thesis, presented in chapter 7, is a theorem originally conjectured by Barry Simon which says that $T$ is essentially self-adjoint on $C^\infty_c(\mathbb{R}^d)$, when $V$ is a non-negative locally square integrable function and $d$ is an arbitrary positive integer. The proof is based around mollification methods and the distributional inequality proven in the previous chapter. This last result, although fairly unphysical, is somewhat striking in the sense that usually for $T$ to be (essentially) self-adjoint, the dimension $d$ restricts the integrability properties of $V$ significantly. dct.subject Hilbert spaces dct.subject Schrödinger operators dct.subject Self-adjointness dct.subject Distribution theory dct.subject Kato's inequality dct.language en ethesis.isPublicationLicenseAccepted true ethesis.language.URI http://data.hulib.helsinki.fi/id/languages/eng ethesis.language englanti fi ethesis.language English en ethesis.language engelska sv ethesis.thesistype pro gradu -tutkielmat fi ethesis.thesistype master's thesis en ethesis.thesistype pro gradu-avhandlingar sv ethesis.thesistype.URI http://data.hulib.helsinki.fi/id/thesistypes/mastersthesis dct.identifier.ethesis E-thesisID:2b78f918-cf01-45bb-b5ea-e5bb6d167e06 dct.identifier.urn URN:NBN:fi:hulib-202010294365 dc.type.dcmitype Text ethesis.facultystudyline Matematiikka fi ethesis.facultystudyline Mathematics en ethesis.facultystudyline Matematik sv ethesis.facultystudyline.URI http://data.hulib.helsinki.fi/id/SH50_050 ethesis.mastersdegreeprogram Matematiikan ja tilastotieteen maisteriohjelma fi ethesis.mastersdegreeprogram Master's Programme in Mathematics and Statistics en ethesis.mastersdegreeprogram Magisterprogrammet i matematik och statistik sv ethesis.mastersdegreeprogram.URI http://data.hulib.helsinki.fi/id/MH50_001