Related papers: Intertwined isospectral potentials in an arbitrary…
A real band condition is shown to exist for one dimensional periodic complex non-hermitian potentials exhibiting PT-symmetry. We use an exactly solvable ultralocal periodic potential to obtain the band structure and discuss some spectral…
To gain insight into the mechanisms behind machine learning methods, it is crucial to establish connections among the features describing data points. However, these correlations often exhibit a high-dimensional and strongly nonlinear…
The technique of differential intertwining operators (or Darboux transformation operators) is systematically applied to the one-dimensional Dirac equation. The following aspects are investigated: factorization of a polynomial of Dirac…
A theory for constructing integrable couplings of soliton equations is developed by using various perturbations around solutions of perturbed soliton equations being analytic with respect to a small perturbation parameter. Multi-scale…
It is shown how nucleon-nucleon potentials can be defined in N dimensions, using dimensional regularization to continue amplitudes. This provides an easy way to separate out contact ($\delta$-function) terms arising from renormalization. An…
The supersymmetrical approach is used to analyse a class of two-dimensional quantum systems with periodic potentials. In particular, the method of SUSY-separation of variables allowed us to find a part of the energy spectra and the…
A general method to find an effective potential of interaction between far separated 2D and 3D solitons is elaborated, including the case of 2D vortex solitons. The method is based on explicit calculation of the overlapping term in the full…
We develop a systematic approach to construct novel completely solvable rational potentials. Second-order supersymmetric quantum mechanics dictates the latter to be isospectral to some well-studied quantum systems. $\cal PT$ symmetry may…
We discuss one of the many topics that illustrate the interaction of Blaine Lawson's deep geometric and analytic insights. The first author is extremely grateful to have had the pleasure of collaborating with Blaine over many enjoyable…
We review the current status of one dimensional periodic potentials and also present several new results. It is shown that using the formalism of supersymmetric quantum mechanics, one can considerably enlarge the limited class of…
The connection between the strictly isospectral construction in supersymmetric quantum mechanics and the general zero mode solutions of the Schroedinger equation is explained by introducing slightly generalized first-order intertwining…
Quantum versions of the hydrogen atom and the harmonic oscillator are studied on non Euclidean spaces of dimension N. 2N-1 integrals, of arbitrary order, are constructed via a multi-dimensional version of the factorization method, thus…
This paper employs layer potential techniques to investigate wave scattering in two-dimensional elastic media exhibiting high contrasts in both Lam\'{e} parameters and density. Our contributions are fourfold. First, we construct an…
Solitons are typically stable objects in 1D models, but their straightforward extensions to 2D and 3D settings tend to be unstable. In particular, the ubiquitous nonlinear Schroedinger (NLS) equation with the cubic self-focusing, creates…
General first- and higher-order intertwining relations between non-stationary one-dimensional Schr\"odinger operators are introduced. For the first-order case it is shown that the intertwining relations imply some hidden symmetry which in…
We obtain analytical expressions for an effective potential of interaction between two- and three-dimensional (2D and 3D) solitons (including the case of 2D vortex solitons) belonging to two different modes which are incoherently coupled by…
L-Infinity structures have been a subject of recent interest in physics, where they occur in closed string theory and in gauge theory. This paper provides a class of easily constructible examples of $L_n$ and $L_{\infty}$ structures on…
The conditions for superintegrable systems in two-dimensional Euclidean space admitting separation of variables in an orthogonal coordinate system and a functionally independent third-order integral are studied. It is shown that only…
Recent technological advances in integrated photonics have spurred on the study of topological phenomena in engineered bosonic systems. Indeed, the controllability of silicon ring-resonator arrays has opened up new perspectives for building…
Three-dimensional isospectral systems are constructed using the framework of supersymmetric quantum mechanics. In case the supercharge of first order in momentum is used, it is proved that the constructed systems reduce to a trivial…