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Register a new userCopolymer-homopolymer blends: global energy minimisation and global energy bounds
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We study a variational model for a diblock-copolymer/homopolymer blend. The energy functional is a sharp-interface limit of a generalisation of the Ohta-Kawasaki energy. In one dimension, on the real line and on the torus, we prove existence of minimisers of this functional and we describe in complete detail the structure and energy of stationary points. Furthermore we characterise the conditions under which the minimisers may be non-unique. In higher dimensions we construct lower and upper bounds on the energy of minimisers, and explicitly compute the energy of spherically symmetric configurations.
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We study the stability of layered structures in a variational model for diblock copolymer-homopolymer blends. The main step consists of calculating the first and second derivative of a sharp-interface Ohta-Kawasaki energy for straight mono- and bilayers. By developing the interface perturbations in a Fourier series we fully characterise the stability of the structures in terms of the energy parameters. In the course of our computations we also give the Greens function for the Laplacian on a periodic strip and explain the heuristic method by which we found it.
One manifestation of quantum resonances is a large sojourn time, or autocorrelation, for states which are initially localized. We elaborate on Lavines time-energy uncertainty principle and give an estimate on the sojourn time. For the case of perturbed embedded eigenstates the bound is explicit and involves Fermis Golden Rule. It is valid for a very general class of systems. We illustrate the theory by applications to resonances for time dependent systems including the AC Stark effect as well as multistate systems.
Utilizing frameworks developed by Delsarte, Yudin and Levenshtein, we deduce linear programming lower bounds (as $Nto infty$) for the Riesz energy of $N$-point configurations on the $d$-dimensional unit sphere in the so-called hypersingular case; i.e, for non-integrable Riesz kernels of the form $|x-y|^{-s}$ with $s>d.$ As a consequence, we immediately get (thanks to the Poppy-seed bagel theorem) lower estimates for the large $N$ limits of minimal hypersingular Riesz energy on compact $d$-rectifiable sets. Furthermore, for the Gaussian potential $exp(-alpha|x-y|^2)$ on $mathbb{R}^p,$ we obtain lower bounds for the energy of infinite configurations having a prescribed density.
We develop the Riemann-Hilbert problem approach to inverse scattering for the two-dimensional Schrodinger equation at fixed energy. We obtain global or gener
The long-time asymptotics is analyzed for all finite energy solutions to a model $mathbf{U}(1)$-invariant nonlinear Dirac equation in one dimension, coupled to a nonlinear oscillator: {it each finite energy solution} converges as $ttopminfty$ to the set of all `nonlinear eigenfunctions of the form $(psi_1(x)e^{-iomega_1 t},psi_2(x)e^{-iomega_2 t})$. The {it global attraction} is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersive radiation. We justify this mechanism by the strategy based on emph{inflation of spectrum by the nonlinearity}. We show that any {it omega-limit trajectory} has the time-spectrum in the spectral gap $[-m,m]$ and satisfies the original equation. This equation implies the key {it spectral inclusion} for spectrum of the nonlinear term. Then the application of the Titchmarsh convolution theorem reduces the spectrum of $j$-th component of the omega-limit trajectory to a single harmonic $omega_jin[-m,m]$, $j=1,2$.
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