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Title:  Behaviour of Eigenfunction Subsequences for DeltaPerturbed 2D Quantum Systems 
Authors:  Newman, Adam 
Keywords:  Delta potential Laplacian operator Quantum chaos Rankone perturbations Semiclassical 
Issue Date:  2016 
Publisher:  © Adam Newman 
Abstract:  We consider a quantum system whose unperturbed form consists of a selfadjoint $\Delta$ operator on a 2dimensional compact Riemannian manifold, which may or may not have a boundary. Then as a perturbation, we add a delta potential / point scatterer at some select point $p$. The perturbed selfadjoint operator is constructed rigorously by means of selfadjoint extension theory. We also consider a corresponding classical dynamical system on the cotangent/cosphere bundle, consisting of geodesic flow on the manifold, with specular reflection if there is a boundary.
Chapter 2 describes the mathematics of the unperturbed and perturbed quantum systems, as well as outlining the classical dynamical system. Included in the discussion on the deltaperturbed quantum system is consideration concerning the strength of the delta potential. It is reckoned that the delta potential effectively has negative infinitesimal strength.
Chapter 3 continues on with investigations from [KMW10], concerned with perturbed eigenfunctions that approximate to a linear combination of only two "surrounding" unperturbed eigenfunctions. In Thm. 4.4 of [KMW10], conditions are derived under which a sequence of perturbed eigenfunctions exhibits this behaviour in the limit. The approximating pair linear combinations belong to a class of quasimodes constructed within [KMW10]. The aim of Chapter 3 in this thesis is to improve on the result in [KMW10].
In Chapter 3, preliminary results are first derived constituting a broad consideration of the question of when a perturbed eigenfunction subsequence approaches linear combinations of only two surrounding unperturbed eigenfunctions. Afterwards, the central result of this Chapter, namely Thm. 3.4.1, is derived, which serves as an improved version of Thm. 4.4 in [KMW10]. The conditions of this theorem are shown to be weaker than those in [KMW10]. At the same time though, the conclusion does not require the approximating pair linear combinations to be quasimodes contained in the domain of the perturbed operator. Cor. 3.5.2 allows for a transparent comparison between the results of this Chapter and [KMW10].
Chapter 4 deals with the construction of nonsingular rankone perturbations for which the eigenvalues and eigenfunctions approximate those of the deltaperturbed operator. This is approached by means of direct analysis of the construction and formulae for the rankoneperturbed eigenvalues and eigenfunctions, by comparison that of the deltaperturbed eigenvalues and eigenfunctions. Successful results are derived to this end, the central result being Thm. 4.4.19. This provides conditions on a sequence of nonsingular rankone perturbations, under which all eigenvalues and eigenbasis members within an interval converge to those of the deltaperturbed operator.
Comparisons have also been drawn with previous literature such as [Zor80], [AK00] and [GN12]. These deal with rankone perturbations approaching the delta potential within the setting of a whole Euclidean space $\mathbb{R}^n$, for example by strong resolvent convergence, and by limiting behaviour of generalised eigenfunctions associated with energies at every $E\in(0,\infty)$. Furthermore in Chapter 4, the suggestion from Chapter 2 that the delta potential has negative infinitessimal strength is further supported, due to the coefficients of the approximating rankone perturbations being negative and tending to zero. This phenomenon is also in agreement with formulae from [Zor80], [AK00] and [GN12].
Chapter 5 first reviews the correspondence between certain classical dynamics and equidistribution in position space of almost all unperturbed quantum eigenfunctions, as demonstrated for example in [MR12]. Equidistribution in position space of almost all perturbed eigenfunctions, in the case of the 2D rectangular flat torus, is also reviewed. This result comes from [RU12], which is only stated in terms of the "new" perturbed eigenfunctions, which would only be a subset of the full perturbed eigenbasis. Nevertheless, in this Chapter it is explained how it follows that this position space equidistribution result also applies to a fulldensity subsequence of the full perturbed eigenbasis.
Finally three methods of approach are discussed for attempting to derive this position space equidistribution result in the case of a more general deltaperturbed system whose classical dynamics satisfies the particular key property.
[AK00] S. Albeverio and P. Kurasov: Singular Perturbations of Differential Operators. London Math. Soc. Lecture Note Ser. 271. Cambridge University Press (2000).
[GN12] P. G. Grinevich and R. G. Novikov: Faddeev eigenfunctions for point potentials in two dimensions. Phys. Lett. A 376, 1102 (2012).
[KMW10] J. P. Keating, J. Marklof and B. Winn: Localized eigenfunctions in \v{S}eba billiards. J. Math. Phys. 51, 062101 (2010).
[MR12] J. Marklof and Z. Rudnick: Almost all eigenfunctions of a rational polygon are uniformly distributed. J. Spectr. Theory 2, 107 (2012).
[RU12] Z. Rudnick and H. Uebersch\"{a}r: Statistics of Wave Functions for a Point Scatterer on the Torus. Commun. Math. Phys. 316, 763 (2012).
[Zor80] J. Zorbas: Perturbation of selfadjoint operators by Dirac distributions. J. Math. Phys. 21(4), 840 (1980). 
Description:  A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University. 
Sponsor:  Loughborough University Graduate School 
URI:  https://dspace.lboro.ac.uk/2134/21568 
Appears in Collections:  PhD Theses (Maths)

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