Isospectral Mathieu-Hill Operators

In this paper we prove that the spectrum of the Mathieu-Hill Operators with potentials ae^{-i2{\pi}x}+be^{i2{\pi}x} and ce^{-i2{\pi}x}+de^{i2{\pi}x} are the same if and only if ab=cd, where a,b,c and d are complex numbers. This result implies some corollaries about the extension of Harrell-Avron-Simon formula. Moreover, we find explicit formulas for the eigenvalues and eigenfunctions of the t-periodic boundary value problem for the Hill operator with Gasymov's potential

On the structure of eigenfunctions corresponding to embedded eigenvalues of locally perturbed periodic graph operators

The article is devoted to the following question. Consider a periodic self-adjoint difference (differential) operator on a graph (quantum graph) G with a co-compact free action of the integer lattice Z^n. It is known that a local perturbation of the operator might embed an eigenvalue into the continuous spectrum (a feature uncommon for periodic elliptic operators of second order). In all known constructions of such examples, the corresponding eigenfunction is compactly supported. One wonders whether this must always be the case. The paper answers this question affirmatively. What is more surprising, one can estimate that the eigenmode must be localized not far away from the perturbation (in a neighborhood of the perturbation's support, the width of the neighborhood determined by the unperturbed operator only). The validity of this result requires the condition of irreducibility of the Fermi (Floquet) surface of the periodic operator, which is expected to be satisfied for instance for periodic Schroedinger operators.Comment: Submitted for publicatio

Bethe-Sommerfeld conjecture for periodic operators with strong perturbations

We consider a periodic self-adjoint pseudo-differential operator $H=(-\Delta)^m+B$, $m>0$, in $\R^d$ which satisfies the following conditions: (i) the symbol of $B$ is smooth in \bx, and (ii) the perturbation $B$ has order less than $2m$. Under these assumptions, we prove that the spectrum of $H$ contains a half-line. This, in particular implies the Bethe-Sommerfeld Conjecture for the Schr\"odinger operator with a periodic magnetic potential in all dimensions.Comment: 61 page

Relative Oscillation Theory, Weighted Zeros of the Wronskian, and the Spectral Shift Function

We develop an analog of classical oscillation theory for Sturm-Liouville operators which, rather than measuring the spectrum of one single operator, measures the difference between the spectra of two different operators. This is done by replacing zeros of solutions of one operator by weighted zeros of Wronskians of solutions of two different operators. In particular, we show that a Sturm-type comparison theorem still holds in this situation and demonstrate how this can be used to investigate the finiteness of eigenvalues in essential spectral gaps. Furthermore, the connection with Krein's spectral shift function is established.Comment: 26 page

Bounds for the positive eigenvalues of the p-Laplacian with decaying potential

An upper bound is obtained for the positive eigenvalues of the p-Laplacian with decaying potential on [0,∞). The bound is expressed in terms of the potential and is shown to be the best possible of its kind

Spectral concentration and perturbed discrete spectra

AbstractWe examine spectral concentration for a class of Sturm-Liouville problems on [0, ∞), a typical example being y″(x) + (λ − x + cx2)y(x) = 0. The discrete spectrum for c = 0 leads to spectral concentration in the continuous spectrum for c > 0, and we use a new formula for the spectral function to make a detailed computational investigation of the way in which spectral concentration occurs. In particular, we find that, as c decreases, spectral concentration arises first from the lowest unperturbed eigenvalue and then in turn from these eigenvalues in increasing order of size

Chinese target market research for virtual reality technology in education field

This thesis studies three Chinese target markets for virtual learning environment technology. The emphasis is to contribute the valuable market information to Commissioner Applebones Ltd. The research releaser problems are: 1) What is the situation of the Chinese Educational Market? a) What is the customer behaviour of the target markets? b) What are the basic policies and regulations in the relevant business field? 2) Who are the possible business contacts or what is the focus group? A qualitative method was chosen for this research and the data was gathered by desktop research and in-depth interviews. The scope of this research was defined as junior high schools and the criteria were defined by the commissioner as technological-oriented, foreign language oriented, experimental type of schools. The basic theoretical framework for this study was obtained from pertinent literature and complemented by updated information gained from online source. The findings and results of this study are of concern to the general introduction of the Chinese education market. This also includes the relevant regulation and policies, customer behaviour in terms of school principals’ role in decision making. The lists of the schools that are fulfilling the given criteria in Shanghai and Hangzhou region and the current information communication technology (ICT) situation and development of Hong Kong Fung Kai Innovative School are presented, as well as the possibility of pilot testing for Virtual learning environment technology. There are several schools recommended as the possible focus group. Although the Hong Kong key contact was not positive with the immediate business cooperation due to Fung Kai Innovative School’s development schedule and technologically limitation, the commissioner seeks long-term relationship and future possibility. It is suggested that further study could be made with narrow scale and emphasis can lay on the focus group. The market segmentation, the case study for competitors locally or internationally as well as the benching marking of successful entries to Chinese market is proposed topics

Conditions for the spectrum associated with an asymptotically straight leaky wire to comprise the interval (??, ?)

We consider a quantum (or leaky) wire in the plane, and the wire supports a singular attraction which becomes large at distant points on the wire. An analogous regular potential arises from the motion of a hydrogen atom in an electric field. We prove that, as in the regular case, the spectrum is the whole of (??, ?)