Related papers: Bound States for Nano-Tubes with a Dislocation
We consider the twisted waveguide $\Omega_\theta$, i.e. the domain obtained by the rotation of the bounded cross section $\omega \subset {\mathbb R}^{2}$ of the straight tube $\Omega : = \omega \times {\mathbb R}$ at angle $\theta$ which…
We deepen the study of Dirichlet eigenvalues in bounded domains where a thin tube is attached to the boundary. As its section shrinks to a point, the problem is spectrally stable and we quantitatively investigate the rate of convergence of…
We consider the existence of localized modes corresponding to eigenvalues of the periodic Schr\"{o}dinger operator $-\partial_x^2+ V(x)$ with an interface. The interface is modeled by a jump either in the value or the derivative of $V(x)$…
In this paper, two-dimensional periodic capillary-gravity waves travelling under the effect of a vertical electric field are considered. The full system is a nonlinear, two-layered and free boundary problem. The interface dynamics arises…
Synthetic nanoscale complexes capable of mechanical movement are often studied theoretically using discrete-state models that involve instantaneous transitions between metastable states. A number of general results have been derived within…
In this paper, we consider the Schr\"odinger equation, \begin{equation*} Hu=-u^{\prime\prime}+(V(x)+V_0(x))u=Eu, \end{equation*} where $V_0(x)$ is 1-periodic and $V (x)$ is a decaying perturbation. By Floquet theory, the spectrum of…
Understanding the interface dynamics in non-equilibrium quantum systems remains a challenge. We study the interface dynamics of strongly coupled immiscible binary superfluids by using holographic duality. The full nonlinear evolution of the…
We consider a self-adjoint operator $T$ on a separable Hilbert space, with pure-point and simple spectrum with accumulations at finite points. Explicit conditions are stated on the eigenvalues of $T$ and on the bounded perturbation $V$…
A class of one-dimensional lattice models with incommensurate complex potential $V(\theta)=2[\lambda_r cos(\theta)+i \lambda_i sin(\theta)]$ is found to exhibit localization transition at $|\lambda_r|+|\lambda_i|=1$. This transition from…
A nonlocal Cahn-Hilliard model with a nonsmooth potential of double-well obstacle type that promotes sharp interfaces in the solution is presented. To capture long-range interactions between particles, a nonlocal Ginzburg-Landau energy…
Space-time varying media enable unprecedented control over electromagnetic waves, yet most existing studies assume idealized, nondispersive materials and thus fail to capture the intrinsic frequency dispersion of realistic platforms. Here,…
We consider the Hamiltonian of a system of three quantum mechanical particles on the three-dimensional lattice $\Z^3$ interacting via short-range pair potentials. We prove for the two-particle energy operator $h(k),$ $k\in \T^3$ the…
This paper is concerned with nonlinear stability of viscous contact discontinuity to a free boundary problem for the one-dimensional full compressible Navier-Stokes equations in half space $[0,\infty)$. For the case when the local stability…
We complexify a 1-d potential which exhibits bound, reflecting and free states to study various properties of a non-Hermitian system. This potential turns out a PT-symmetric non-Hermitian potential when one of the parameters becomes…
We extend the application of the techniques developed within the framework of the pseudo-Hermitian quantum mechanics to study a unitary quantum system described by an imaginary PT-symmetric potential v(x) having a continuous real spectrum.…
We point out that PT-symmetric potentials $V_{PT}(x)$ having imaginary asymptotic saturation: $V_{PT}(x=\pm \infty) =\pm i V_1, V_1 \in \Re$ are devoid of scattering states and spectral singularity. We show the existence of real (positive…
Two-dimensional (2D) superconductors provide a powerful building block for engineering emergent quantum states shaped by reduced dimensionality, enhanced quantum fluctuations, and interfacial symmetry breaking. In van der Waals…
We prove that any limit-interface corresponding to a locally uniformly bounded, locally energy-bounded sequence of stable critical points of the van der Waals--Cahn--Hilliard energy functionals with perturbation parameter tending to 0 is…
The aim of this paper is to show that a two-dimensional Schr\"odinger operator with the potential in the form of a `ditch' of a fixed profile can have a geometrically induced discrete spectrum; this happens if such a potential channel has a…
For the the Schr\"odinger operator $H=-\Delta+ V(x)\cdot$, acting in the space L_2(\R^d)\,(d\ge 3), with V(x)\ge 0 and V(\cdot)\in L_{1,loc}(\R^d), we obtain some constructive conditions for discreteness of its spectrum. Basing on the…