Related papers: Reflectionless Potentials for Difference Schr\"odi…
In solving $q$-difference equations, and in the definition of $q$-special functions, we encounter formal power series in which the $n$th coefficient is of size $q^{-\binom{n}{2}}$ with $q\in(0,1)$ fixed. To make sense of these formal…
The question of whether it is possible to compute scattering resonances of Schr\"odinger operators - independently of the particular potential - is addressed. A positive answer is given, and it is shown that the only information required to…
Spectral singularities are spectral points that spoil the completeness of the eigenfunctions of certain non-Hermitian Hamiltonian operators. We identify spectral singularities of complex scattering potentials with the real energies at which…
Consider the one-dimensional discrete Schr\"odinger operator $H_{\theta}$: $$(H_{\theta} q)_n=-(q_{n+1}+q_{n-1})+ V(\theta+n\omega) q_n \ , \quad n\in Z \ ,$$ with $\omega\in R^d$ Diophantine, and $V$ a real-analytic function on $ T^d=(…
We introduce an exactly integrable singular potential for which the solution of the one-dimensional stationary Schr\"odinger equation is written through irreducible linear combinations of the Gauss hypergeometric functions. The potential,…
A universal minimal spinor set of linear differential equations describing anyons and ordinary integer and half-integer spin fields is constructed with the help of deformed Heisenberg algebra with reflection. The construction is generalized…
For an open subset $\Omega$ of $\mathbb R^d$, symmetric with respect to a hyperplane and with positive part $\Omega_+$, we consider the Neumann/Dirichlet Laplacians $-\Delta_{N/D,\Omega}$ and $-\Delta_{N/D,\Omega_+}$. Given a Borel function…
We outline a recently developed theory of impedance-matching, or reflectionless excitation of arbitrary finite photonic structures in any dimension. It describes the necessary and sufficient conditions for perfectly reflectionless…
Quantum scattering in the presence of a potential valley followed by a barrier is examined for the case of a Morse potential, for which exact analytic solutions to the Schr\UNICODE{0xf6}dinger equation are known in terms of confluent…
We observe that in nonlinear quantum mechanics, unlike in the linear theory, there exists, in general, a difference between the energy functional defined within the Lagrangian formulation as an appropriate conserved component of the…
We present a new six-parameter family of potentials whose solutions are expressed in terms of the hypergeometric functions 3F2, 2F2 and 1F2. Both the scattering data and the bound states of these potentials are explicitly computed and the…
We develop direct scattering theory for one-dimensional Schr\"odinger operators with steplike potentials, which are asymptotically close to different Bohr almost periodic infinite-gap potentials on different half-axes.
We derive a generalized zero-range pseudopotential applicable to all partial wave solutions to the Schroedinger equation based on a delta-shell potential in the limit that the shell radius approaches zero. This properly models all higher…
A new approach to find exact solutions to one--dimensional quantum mechanical systems is devised. The scheme is based on the introduction of a potential function for the wavefunction, and the equation it satisfies. We recover known…
We consider the non-selfadjoint, semiclassical Schr\"odinger operator $\mathscr{L}(h) := -h^2\partial_x^2+e^{i\alpha}V$, where $\alpha \in (-\pi,\pi)$ and $V: \mathbb{R}\to \mathbb{R}_+$ is even and vanishes at exactly two (symmetric)…
A condition, at which the one-dimensional inverse power potential becomes reflectionless during propagation through it of a plane wave, is obtained on the basis of SUSY QM methods. A scattering of a particle on spherically symmetric inverse…
I consider general reflection coefficients for arbitrary one-dimensional whole line differential or difference operators of order $2$. These reflection coefficients are semicontinuous functions of the operator: their absolute value can only…
A generalization of associated Legendre functions is proposed and used to describe the scattering states of the Rosen-Morse potential. The functions are then given explicit formulas in terms of the hypergeometric function, their asymptotic…
All known additive shape invariant superpotentials in nonrelativistic quantum mechanics belong to one of two categories: superpotentials that do not explicitly depend on $\hbar$, and their $\hbar$-dependent extensions. The former group…
We wish to report here on a recent approach to the non-commutative calculus on $q$-Minkowski space which is based on the reflection equations with no spectral parameter. These are considered as the expression of the invariance (under the…