Related papers: Simultaneous separation for the Neumann and Chaply…
For the elliptic Gaudin model (a degenerate case of XYZ integrable spin chain) a separation of variables is constructed in the classical case. The corresponding separated coordinates are obtained as the poles of a suitably normalized…
Two new methods for investigation of two-dimensional quantum systems, whose Hamiltonians are not amenable to separation of variables, are proposed. 1)The first one - $SUSY-$ separation of variables - is based on the intertwining relations…
Several explicit examples of multi-particle quasi exactly solvable `discrete' quantum mechanical Hamiltonians are derived by deforming the well-known exactly solvable multi-particle Hamiltonians, the Ruijsenaars-Schneider-van Diejen…
A relationship between two old mathematical subjects is observed: the theory of hypergeometric functions and the separability in classical mechanics. Separable potential perturbations of the integrable billiard systems and the Jacobi…
On the basis of loop group decompositions (Birkhoff decompositions), we give a discrete version of the nonlinear d'Alembert formula, a method of separation of variables of difference equations, for discrete constant negative Gauss curvature…
We introduce a new infinite class of superintegrable quantum systems in the plane. Their Hamiltonians involve reflection operators. The associated Schr\"odinger equations admit separation of variables in polar coordinates and are exactly…
This paper is concerned with quantitative homogenization of second-order parabolic systems with periodic coefficients varying rapidly in space and time, in different scales. We obtain large-scale interior and boundary Lipschitz estimates as…
In this paper, we determine the convergence speed of an upscaling of a pseudo-parabolic system containing drift terms with scale separation of size $\epsilon \ll 1$. Both the upscaling and convergence speed determination exploit a natural…
The examples are considered of integrable hyperbolic equations of third order with two independent variables. In particular, an equation is found which admits as evolutionary symmetries the Krichever--Novikov equation and the modified…
Motivated by the concept of shape invariance in supersymmetric quantum mechanics, we obtain potentials whose spectrum consists of two shifted sets of equally spaced energy levels. These potentials are similar to the Calogero-Sutherland…
In this paper, we establish two new types of invariant sets for the coupled nonlinear Schrodinger system on $\mathbb{R}^n$, and derive two sharp thresholds of blow-up and global existence for its solution. Some analogous results for the…
Cubic invariants for two-dimensional degenerate Hamiltonian systems are considered by using variables of separation of the associated St\"ackel problems with quadratic integrals of motion. For the superintegrable St\"ackel systems the cubic…
Translationally invariant symmetric polynomials as coordinates for $N$-body problems with identical particles are proposed. It is shown that in those coordinates the Calogero and Sutherland $N$-body Hamiltonians, after appropriate gauge…
We review the theory of orthogonal separation of variables on pseudo-Riemannian manifolds of constant non-zero curvature via concircular tensors and warped products. We then apply this theory simultaneously to both the three-dimensional…
We describe a method for constructing $n$-orthogonal coordinate systems in constant curvature spaces. The construction proposed is a modification of Krichever's method for producing orthogonal curvilinear coordinate systems in the…
We present in this paper new multiscale transforms on the sphere, namely the isotropic undecimated wavelet transform, the pyramidal wavelet transform, the ridgelet transform and the curvelet transform. All of these transforms can be…
We study existence and nonexistence of diagonal and separating coordinates for Riemannian symmetric spaces of rank 1. We generalize the results of Gauduchon and Moroianu, 2020, by showing that a symmetric space of rank 1 has diagonal…
The Neumann--Neumann method is a commonly employed domain decomposition method for linear elliptic equations. However, the method exhibits slow convergence when applied to semilinear equations and does not seem to converge at all for…
The entanglement produced by a bilinear Hamiltonian in continuous variables has been thoroughly studied and widely used. In contrast, the physics of entanglement resulting from nonlinear interaction described by partially degenerate…
A description of Lagrangian and Hamiltonian formalisms naturally arisen from the invariance structure of given nonlinear dynamical systems on the infinite--dimensional functional manifold is presented. The basic ideas used to formulate the…