Thesis (Ph.D.)--University of Nebraska-Lincoln, 2007. / Title from title screen (site viewed July 9, 2007). PDF text: 96 p. UMI publication number: AAT 3252832. Includes bibliographical references. Also available in microfilm and microfiche formats.

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We consider a formally self-adjoint elliptic differential operator in IRⁿ, denoted by τ. T₀ and T are operators given by τ with specific domains. We determine conditions under which T₀ is essentially self-adjoint, introducing the topic by means of a brief historical survey of some results in this field. In Part I, we consider an operator of order 4, and in Part II, we generalise the results obtained there to ones for an operator of order 2m. Thus, the two parts run parallel. In Chapter 1, we determine the domain of T₀*, denoted by D(T₀*), where T₀* denotes the adjoint of T₀, and introduce operators <u>T</u>₀ and <u>T</u> which are modifications of T₀ and T. In Chapter 2, we use a theorem of Schechter to give conditions under which <u>T</u>₀ is essentially self-adjoint. Working with the operator <u>T</u>, in Chapter 3 ve show that we can approximate functions u in D(T₀*) by a particular sequence of test-functions, which enables us to derive an identity involving u, Tu and the coefficient functions of the operator concerned. In Chapter 4, we determine an upper bound for the integral of a function involving a derivative of u in D(T₀*) whose order is half the order of the operator concerned, and we use the identity from the previous chapter to reformulate this upper bound. In Chapter 5, we give conditions which are sufficient for the essential self-adjointness of T₀. In the main theorem itself, the major step is the derivation of the integral of the function involving the particular derivative of u in D(T₀*) whose order is half the order of the operator concerned, referred to above, itself as a term of an upper bound of an integral we wish to estimate. Hence, we can employ the upper bound from Chapter 4. This "sandwiching" technique is basic to the approach we have adopted. We conclude with a brief discussion of the operators we considered, and restate the examples of operators which we showed to be essentially self-adjoint.

Thesis (Ph.D.)--Baylor University, 2009. / Subscript in abstract: n and n=0 in {Pn([alpha],[beta])(x)} [infinity] n=0, [mu] in (f,g)[mu], and R in [integral]Rfgd[mu]. Superscript in abstract: ([alpha],[beta]) and [infinity] in {Pn([alpha],[beta])(x)} [infinity] n=0. Includes bibliographical references (p. 115-119).

Let $\Omega\subset\mathbb{R}^n$ be an open and connected subset of $\mathbb{R}^n$. We say that a function $F\colon \Omega-\Omega\to\mathbb{C}$, where $\Omega-\Omega=\{x-y\colon x,y\in\Omega\}$, is positive definite if for any $x_1,\ldots,x_m\in\Omega$ and any $c_1,\ldots,c_m\in \mathbb{C}$ we have that $\sum_{j,k=1}^m F(x_j-x_k)c_j\overline{c_k}\geq 0$. Let $F\colon\Omega-\Omega\to\mathbb{C}$ be a continuous positive definite function. We give necessary and sufficient conditions for $F$ to have an extension to a continuous and positive definite function defined on the entire Euclidean space $\mathbb{R}^n$. The conditions are formulated in terms of strong commutativity of some certain selfadjoint operators defined on a Hilbert space associated to our positive definite function.

Thesis (Ph. D.)--University of Washington, 1997. / Vita. Includes bibliographical references (p. [77]-79).

L'instabilité du spectre des opérateurs non-autoadjoints constitue la thématique centrale de cette thèse. Notre premier objectif est de mettre en évidence ce phénomène dans le cas de certains modèles naturels tels que l'opérateur d'Airy, l'oscillateur harmonique ou l'oscillateur cubique complexes. Dans ce but, nous nous intéressons au comportement des projecteurs spectraux associés aux valeurs propres de ces opérateurs, poursuivant une démarche initiée par E. B. Davies. Le second objectif de notre travail consiste à montrer de quelle manière ces modèles peuvent contribuer à la compréhension de certains problèmes issus de domaines mathématiques et physiques aussi variés que la mécanique quantique, la supraconductivité ou la théorie du contrôle. Nos résultats sur l'instabilité spectrale de l'oscillateur cubique complexe viennent ainsi corroborer un travail de B. Krejcirik et P. Siegl, soulignant l'impossibilité de fournir une justification rigoureuse aux théories actuelles de la mécanique quantique non-hermitienne. Par ailleurs, nous nous appuyons sur les propriétés des modèles mentionnés ci-dessus pour obtenir des résultats sur le spectre et la résolvante d'opérateurs de Schrödinger à potentiels imaginaires purs dans des ouverts bornés. Ces résultats peuvent en particulier être appliqués à l'étude du système de Ginzburg-Landau dépendant du temps en supraconductivité. Enfin, nous présentons des résultats sur la contrôlabilité d'équations paraboliques dégénérées qui reposent sur une étude spectrale et pseudospectrale de l'opérateur d'Airy et de l'oscillateur harmonique complexes. Ce dernier travail est le fruit d'une collaboration avec K. Beauchard, B. Helffer et L. Robbiano. / Spectral instability of non-selfadjoint operators is the main subject of this thesis. Our first goal is to understand the pseudospectral behavior of natural models such as the complex Airy operator, harmonic oscillator and cubic oscillator. To this purpose, we analyze the asymptotic behavior of the spectral projections associated with the eigenvalues of these operators, following a work initiated by E.B. Davies. Our second goal is to illustrate how such models can be used in several problems arising in quantum mechanics, superconductivity or control theory. For instance, our results on the spectral instability of the complex cubic oscillator enable us to confirm that the current theory of non-hermitian quantum mechanics can not be rigorously justified, as recently pointed out by B. Krejcirik and P. Siegl. On the other hand, we obtain spectral information and resolvent estimates for semi-classical Schrödinger operators with purely imaginary potentials in a bounded domain, by using the properties of the models mentioned above. In particuler, these results entail some information on the time-dependent Ginzburg-Landau system in superconductivity. Finally, we reproduce a joint work with K. Beauchard, B. Helffer et L. Robbiano in which the controllability of some degenerate parabolic operators is investigated. An analysis of the spectrum and resolvent of the complex Airy operator and harmonic oscillator yields some controllability and non-controllability results for the equation under consideration.