Related papers: Explicit solution of the 1D Schr\"odinger equation
We establish quantitative upper and lower bounds for Schr\"odinger operators with complex potentials that satisfy some weak form of sparsity. Our first result is a quantitative version of an example, due to S.\ Boegli (Comm. Math. Phys.,…
We show that the spectrum of a Schr\"odinger operator on $\mathbb{R}^n$, $n\ge 3$, with a periodic smooth Riemannian metric, whose conformal multiple has a product structure with one Euclidean direction, and with a periodic electric…
The Schr\"{o}dinger equation, in hyperspherical coordinates, is solved in closed form for a system of three particles on a line, interacting via pair delta functions. This is for the case of equal masses and potential strengths. The…
The radial Schrodinger equation for a spherically symmetric potential can be regarded as a one dimensional classical harmonic oscillator with a time-dependent spring constant. For solving classical dynamics problems, symplectic integrators…
The accuracy of different transfer matrix approaches, widely used to solve the stationary effective mass Schr\"{o}dinger equation for arbitrary one-dimensional potentials, is investigated analytically and numerically. Both the case of a…
We study the uniform resolvent estimates for Schr\"odinger operator with a Hardy-type singular potential. Let $\mathcal{L}_V=-\Delta+V(x)$ where $\Delta$ is the usual Laplacian on $\mathbb{R}^n$ and $V(x)=V_0(\theta) r^{-2}$ where $r=|x|,…
In this article, we prove that the equation of the Schr\"odinger maps from ${\bf R}^2$ to the hyperbolic 2-space ${\bf H}^2$ is SU(1,1)-gauge equivalent to the following 1+2 dimensional nonlinear Schr\"odinger-type system of unknown three…
For the first time, Schr\"odinger equations with cubic and more complex nonlinearities containing the unknown function with constant delay are analyzed. The physical considerations that can lead to the appearance of a delay in such…
In this paper, we consider matrix Schr\"odinger equation, dynamical Schr\"odinger equation and matrix KdV. We construct their explicit solutions using our GBDT version of B\"acklund--Darboux transformation and square roots of the…
We perform the complete group classification in the class of cubic Schr\"odinger equations of the form $i\psi_t+\psi_{xx}+\psi^2\psi^*+V(t,x)\psi=0$ where $V$ is an arbitrary complex-valued potential depending on $t$ and $x$. We construct…
An exact approach for the factorization of the relativistic linear singular oscillator is proposed. This model is expressed by the finite-difference Schr\"odinger-like equation. We have found finite-difference raising and lowering…
The Schwinger oscillator operator representation of SU(3) is analysed with particular reference to the problem of multiplicity of irreducible representations. It is shown that with the use of an $Sp(2,R)$ unitary representation commuting…
We extend the well-known trace formula for Hill's equation to general one-dimensional Schr\"odinger operators. The new function $\xi$, which we introduce, is used to study absolutely continuous spectrum and inverse problems.
We obtain new results about the high-energy distribution of resonances for the one-dimensional Schr\"odinger operator. Our primary result is an upper bound on the density of resonances above any logarithmic curve in terms of the singular…
We describe a certain "self-similar" family of solutions to the free Schroedinger equation in all dimensions, and derive some consequences of such solutions for two specific problems.
We carry out the complete group classification of the class of (1+1)-dimensional linear Schr\"odinger equations with complex-valued potentials. After introducing the notion of uniformly semi-normalized classes of differential equations, we…
We find an exact solution to strongly-coupled matrix models with a single-trace monomial potential. Our solution yields closed form expressions for the partition function as well as averages of Schur functions. The results are fully…
In this note, we derive explicit formulas for the Schroedinger wave operators in R^2 under the assumption that 0-energy is neither an eigenvalue nor a resonance. These formulas justify the use of a recently introduced topological approach…
It is proven that the absolutely continuous spectrum of matrix Schr\"{o}dinger operators coincides (with the multiplicity taken into account) with the spectrum of the unperturbed operator if the (matrix) potential is square integrable. The…
By constructing the commutative operators chain, we derive the integrable conditions for solving the eigenfunctions of Dirac equation and Schr\"odinger equation. These commutative relations correspond to the intrinsic symmetry of the…