Indefinite Hamiltonian systems whose Titchmarsh–Weyl coefficients have no finite generalized poles of non-positive type

The two-dimensional Hamiltonian system (*)  y'(x)=zJH(x)y(x),  x∈(a,b), where the Hamiltonian H takes non-negative 2x2-matrices as values, and $J:= \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$, has attracted a lot of interest over the past decades.  Special emphasis has been put on operator models and direct and inverse spectral theorems.  Weyl theory plays a prominent role in the spectral theory of the equation, relating the class of all equations (*) to the class N0 of all Nevanlinna functions via the construction of Titchmarsh–Weyl coefficients. In connection with the study of singular potentials, an indefinite (Pontryagin space) analogue of equation (*) was proposed, where the 'general Hamiltonian' is allowed to have a finite number of inner singularities. Direct and inverse spectral theorems, relating the class of all general Hamiltonians to the class <N∞ of all generalized Nevanlinna functions, were established. In the present paper, we investigate the spectral theory of general Hamiltonians having a particular form, namely, such which have only one singularity and the interval to the left of this singularity is a so-called indivisible interval.  Our results can comprehensively be formulated as follows. — We prove direct and inverse spectral theorems for this class, i.e. we establish an intrinsic characterization of the totality of all Titchmarsh–Weyl coefficients corresponding to general Hamiltonians of the considered form. —  We determine the asymptotic growth of the fundamental solution when approaching the singularity. —  We show that each solution of the equation has 'polynomially regularized' boundary values at the singularity. Besides the intrinsic interest and depth of the presented results, our motivation is drawn from forthcoming applications: the present theorems form the core for our study of Sturm–Liouville equations with two singular endpoints and our further study of the structure theory of general Hamiltonians (both to be presented elsewhere)

Direct and inverse spectral theorems for a class of canonical systems with two singular endpoints

Part I of this paper deals with two-dimensional canonical systems $y'(x)=yJH(x)y(x)$, $x\in(a,b)$, whose Hamiltonian $H$ is non-negative and locally integrable, and where Weyl's limit point case takes place at both endpoints $a$ and $b$. We investigate a class of such systems defined by growth restrictions on H towards a. For example, Hamiltonians on $(0,\infty)$ of the form $H(x):=\begin{pmatrix}x^{-\alpha}&0\\ 0&1\end{pmatrix}$ where $\alpha<2$ are included in this class. We develop a direct and inverse spectral theory parallel to the theory of Weyl and de Branges for systems in the limit circle case at $a$. Our approach proceeds via - and is bound to - Pontryagin space theory. It relies on spectral theory and operator models in such spaces, and on the theory of de Branges Pontryagin spaces. The main results concerning the direct problem are: (1) showing existence of regularized boundary values at $a$; (2) construction of a singular Weyl coefficient and a scalar spectral measure; (3) construction of a Fourier transform and computation of its action and the action of its inverse as integral transforms. The main results for the inverse problem are: (4) characterization of the class of measures occurring above (positive Borel measures with power growth at $\pm\infty$); (5) a global uniqueness theorem (if Weyl functions or spectral measures coincide, Hamiltonians essentially coincide); (6) a local uniqueness theorem. In Part II of the paper the results of Part I are applied to Sturm--Liouville equations with singular coefficients. We investigate classes of equations without potential (in particular, equations in impedance form) and Schr\"odinger equations, where coefficients are assumed to be singular but subject to growth restrictions. We obtain corresponding direct and inverse spectral theorems

A function space model for canonical systems

Recently, a generalization to the Pontryagin space setting of the notion of canonical (Hamiltonian) systems which involves a finite number of inner singularities has been given. The spectral theory of indefinite canonical systems was investigated with help of an operator model. This model consists of a Pontryagin space boundary triple and was constructed in an abstract way. Moreover, the construction of this operator model involves a procedure of splitting-and-pasting which is technical but at the present stage of development in general inevitable. In this paper we provide an isomorphic form of this operator model which acts in a finite-dimensional extension of a function space naturally associated with the given indefinite canonical system. We give explicit formulae for the model operator and the boundary relation. Moreover, we show that under certain asymptotic hypotheses the procedure of splitting-and-pasting can be avoided by employing a limiting process. We restrict attention to the case of one singularity. This is the core of the theory, and by making this restriction we can significantly reduce the technical effort without losing sight of the essential ideas

Energy selective neutron imaging for the characterization of polycrystalline materials

This multipart dissertation focuses on the development and evaluation of advanced methods for material testing and characterization using neutron diffraction and imaging techniques. A major focus is on exploiting diffraction contrast in energy selective neutron imaging (often referred to as Bragg edge imaging) for strain and phase mapping of crystalline materials. The dissertation also evaluates the use of neutron diffraction to study the effect of multi-axial loading, in particular the role of applying directly shear strains from the application of torsion. A portable tension-torsion-tomography loading system has been developed for in-situ measurements and integrated at major user facilities around the world. Promising applications for the Bragg edge technique are implemented at the neutron imaging facility CONRAD at the reactor source BER-II as well as at neutron time of flight instruments. Strain mapping is successfully demonstrated for all cases to yield quantifiable results, but is limited in practicality due to limitations in choice of the scattering vector (direction of probed strain tensor component) and the gauge volume selection. The use of Bragg edge imaging for crystalline phase mapping was explored and appears to be a very promising technique. The extension to three-dimensionally resolved tomography is presented for samples exhibiting the TRansfomation Induced Plasticity (TRIP) effect, while challenges with characterizing textured samples are discussed. Individual crystallites within a polycrystalline material exhibit elastic anisotropy which is significant as that can lead to stress concentrations and inhomogeneities during plastic deformation. Characterization of elastic anisotropy is important to understand the effects of texture on the macroscopic mechanical properties. Diffraction methods can do this, by probing the response of individual lattice planes to externally applied mechanical stress. Past experimental data using diffraction based methods have largely been limited to uni‑axial tensile and/or compressive loading conditions, even though shear dominates most common failure mechanisms for structural materials. Within this dissertation, experimental techniques have been established for the measurement of lattice strains under applied torsion (pure shear) and lattice specific shear moduli are reported. This is accomplished using a (traditional) neutron diffractometer instrument, in conjunction with special alignment procedures and the specifically designed axial-torsional loading system

Wetting Behavior of Polymer Melts with Refractory Coatings at High Temperature

Within the scope of this thesis, an experimental system has been designed, developed and manufactured for the determination of the wetting behavior of liquids and polymer melts with solid surfaces (coated and uncoated) at high temperatures (\u3e 200 ºC). The measurement system incorporates a modified Wilhelmy plate technique, using a precision weighing module, a vertical linear stage, custom developed application software using LabView with suitable hardware and a high temperature furnace with thermocouple feedback control. Experiments have been performed and are reported to evaluate the performance of the testing system, using liquids of known wetting properties. A suitable testing procedure based on dynamic Wilhelmy plate theory is proposed, involving investigation of advancing and receding liquid-probe interactive forces and hysteresis loops. Interfacial wetting and wicking behavior of polystyrene melt with clay based refractory coatings, as used in the lost-foam casting (LFC) process, are presented a function of temperature using this measurement system. Experiments of particular interest were performed for two different types of refractory coating and for polymer melts at processing temperatures between 220°C and 300°C, where they show pronounced viscoelastic behavior. Different variables, obtained from the hysteresis loops, were utilized as quantitative indicators for comparison, including the area under the loop from contact onwards, the slope of advancing and receding lines in the force-displacement domain, the force hysteresis at zero displacement and Fast Fourier Transform (FFT) analysis of the hysteresis loop