Subdivisions of the spectra for D(r,0,s,0,t) operator on certain sequence spaces

In this paper we have examined the approximate point spectrum, defect spectrum and compression spectrum of the operator D(r,0,s,0,t) on the sequence spaces c0, c, and $bv_p (1<p<\infty)$.

CR embeddability of quotients of the Rossi sphere via spectral theory

We look at the action of finite subgroups of SU(2) on [Formula: see text], viewed as a CR manifold, both with the standard CR structure as the unit sphere in [Formula: see text] and with a perturbed CR structure known as the Rossi sphere. We show that quotient manifolds from these actions are indeed CR manifolds, and relate the order of the subgroup of SU(2) to the asymptotic distribution of the Kohn Laplacian’s eigenvalues on the quotient. We show that the order of the subgroup determines whether the quotient of the Rossi sphere by the action of that subgroup is CR embeddable. Finally, in the unperturbed case, we prove that we can determine the size of the subgroup by using the point spectrum.

Stationary multi-kinks in the discrete sine-Gordon equation

We consider the existence and spectral stability of static multi-kink structures in the discrete sine-Gordon equation, as a representative example of the family of discrete Klein–Gordon models. The multi-kinks are constructed using Lin’s method from an alternating sequence of well-separated kink and antikink solutions. We then locate the point spectrum associated with these multi-kink solutions by reducing the spectral problem to a matrix equation. For an m-structure multi-kink, there will be m eigenvalues in the point spectrum near each eigenvalue of the primary kink, and, as long as the spectrum of the primary kink is imaginary, the spectrum of the multi-kink will be as well. We obtain analytic expressions for the eigenvalues of a multi-kink in terms of the eigenvalues and corresponding eigenfunctions of the primary kink, and these are in very good agreement with numerical results. We also perform numerical time-stepping experiments on perturbations of multi-kinks, and the outcomes of these simulations are interpreted using the spectral results.

Absolutely continuous and pure point spectra of discrete operators with sparse potentials

We consider the discrete Schr\”odinger operator $H=-\Delta+V$ with a sparse potential $V$ and find conditions guaranteeing either existence of wave operators for the pair $H$ and $H_0=-\Delta$, or presence of dense purely point spectrum of the operator $H$ on some interval $[\lambda_0,0]$ with $\lambda_0<0$.

Balanced Gain-and-Loss Optical Waveguides: Exact Solutions for Guided Modes in Susy-QM

The construction of exactly solvable refractive indices allowing guided TE modes in optical waveguides is investigated within the formalism of Darboux–Crum transformations. We apply the finite-difference algorithm for higher-order supersymmetric quantum mechanics to obtain complex-valued refractive indices admitting all-real eigenvalues in their point spectrum. The new refractive indices are such that their imaginary part gives zero if it is integrated over the entire domain of definition. This property, called condition of zero total area, ensures the conservation of optical power so the refractive index shows balanced gain and loss. Consequently, the complex-valued refractive indices reported in this work include but are not limited to the parity-time invariant case.

Spectral analysis of dispersive shocks for quantum hydrodynamics with nonlinear viscosity

In this paper, we investigate spectral stability of traveling wave solutions to 1D quantum hydrodynamics system with nonlinear viscosity in the [Formula: see text], that is, density and velocity, variables. We derive a sufficient condition for the stability of the essential spectrum and we estimate the maximum modulus of eigenvalues with non-negative real part. In addition, we present numerical computations of the Evans function in sufficiently large domain of the unstable half-plane and show numerically that its winding number is (approximately) zero, thus giving a numerical evidence of point spectrum stability.