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We extend to the $n$-dimensional ellipsoid contained in $\R^{n+1},$ the Darboux theory of integrability for polynomial vector fields in the $n$-dimensional sphere (Llibre et al., 2018). New results on the maximum number of invariant…
The eigenvalues of the potentials $V_{1}(r)=\frac{A_{1}}{r}+\frac{A_{2}}{r^{2}}+\frac{A_{3}}{r^{3}}+\frac{A_{4 }}{r^{4}}$ and $V_{2}(r)=B_{1}r^{2}+\frac{B_{2}}{r^{2}}+\frac{B_{3}}{r^{4}}+\frac{B_{4}}{r^ {6}}$, and of the special cases of…
We show that solutions of nonlinear nonlocal Fokker--Planck equations in a bounded domain with no-flux boundary conditions can be approximated by Cauchy problems with increasingly strong confining potentials defined in the whole space. Two…
Given two spherically symmetric and short range potentials $V_0$ and V_1 for which the radial Schrodinger equation can be solved explicitely at zero energy, we show how to construct a new potential $V$ for which the radial equation can…
The one dimensional Dirac equation with a rational potential is reducible to an ordinary differential equation with a Riccati-like coefficient. Its integrability can be studied with the help of differential Galois theory, although the…
We solve the eigenvalue spectra for two quasi exactly solvable (QES) Schr\"odinger problems defined by the potentials $V(x;\gamma,\eta) = 4\gamma^{2}\cosh^{4}(x) + V_{1}(\gamma,\eta) \cosh^{2}(x) + \eta \left( \eta-1 \right)\tanh^{2}(x)$…
New solvable one-dimensional quantum mechanical scattering problems are presented. They are obtained from known solvable potentials by multiple Darboux transformations in terms of virtual and pseudo virtual wavefunctions. The same method…
We consider equations of the form $\Delta u +\lambda^2 V(x)e^{\,u}=\rho$ in various two dimensional settings. We assume that $V>0$ is a given function, $\lambda>0$ is a small parameter and $\rho=\mathcal O(1)$ or $\rho\to +\infty$ as…
We consider Blanchet, Habegger, Masbaum and Vogel's universal construction of topological theories in dimension two, using it to produce interesting theories that do not satisfy the usual two-dimensional TQFT axioms. Kronecker's…
The formalism of exact 1D quantization is reviewed in detail and applied to the spectral study of three concrete Schr\"odinger Hamiltonians $[-\d^2/\d q^2 + V(q)]^\pm$ on the half-line $\{q>0\}$, with a Dirichlet (-) or Neumann (+)…
We characterize the virtually nilpotent finitely generated groups (or, equivalently by Gromov's theorem, groups of polynomial growth) for which the Domino Problem is decidable: These are the virtually free groups, i.e. finite groups, and…
We study the three-dimensional Dirac and Klein-Gordon equations with scalar and vector potentials of equal magnitudes as an attempt to give a proper physical interpretation of this class of problems which has recently been accumulating…
Domain wall, wormhole, particlelike, and cosmic string general relativistic solutions supported by two interacting phantom or ordinary scalar fields with 4th-, 6th-, and 8th-order potentials are studied. Numerical calculations indicate that…
We consider Darboux transformations for the derivative nonlinear Schr\"odinger equation. A new theorem for Darboux transformations of operators with no derivative term are presented and proved. The solution is expressed in quasideterminant…
For one-dimensional Schroedinger operators with complex-valued potentials, we construct pseudomodes corresponding to large pseudoeigenvalues. Our (non-semi-classical) approach results in substantial progress in achieving optimal conditions…
A relation between the deformed Hulth\'en potential and the Eckart one is used to write the bound-state wavefunctions of the former in terms of Jacobi polynomials and to calculate their normalization coefficients. The shape invariance…
Motivated by recent advances in inverse semigroup theory, we investigate the growth of and identities satisfied by free left and free two-sided adequate monoids. We explicitly compute the growth of the monogenic free left adequate monoid…
We employ separation of variables to prove weighted resolvent estimates for the semiclassical Schr\"odinger operator $-h^2 \Delta + V(|x|) - E$ in dimension $n \ge 2$, where $h, \, E > 0$, and $V: [0, \infty) \to \mathbb{R}$ is $L^\infty$…
In the present work, the isotropic and homogenous solutions with spatial curvature $k=0$ of four dimensional Gauss-Bonnet models are characterized. The main assumption is that the scalar field $\phi$ which is coupled to the Gauss-Bonnet…
We develop a holomorphic functional calculus for first-order operators $DB$ to solve boundary value problems for Schr\"{o}dinger equations $-\mathrm{div}\, A \nabla u + a V u = 0$ in the upper half-space $\mathbb{R}^{n+1}_+$ with…