相关论文: Explicit solution of the (quantum) elliptic Caloge…
We establish elliptic regularity for nonlinear inhomogeneous Cauchy-Riemann equations under minimal assumptions, and give a counterexample in a borderline case. In some cases where the inhomogeneous term has a separable factorization, the…
The quantum trigonometric Calogero-Sutherland models related to Lie algebras admit a parametrization in which the dynamical variables are the characters of the fundamental representations of the algebra. We develop here this approach for…
Jack superpolynomials are eigenfunctions of the supersymmetric extension of the quantum trigonometric Calogero-Moser-Sutherland. They are orthogonal with respect to the scalar product, dubbed physical, that is naturally induced by this…
The Hamiltonian of the trigonometric Calogero-Sutherland model coincides with some limit of the Hamiltonian of the elliptic Calogero-Moser model. In other words the elliptic Hamiltonian is a perturbed operator of the trigonometric one. In…
On compact Riemannian manifolds, we prove a decomposition theorem for arbitrarily bounded energy sequence of solutions of a singular elliptic equation.
We provide a list of explicit eigenfunctions of the trigonometric Calogero-Sutherland Hamiltonian associated to the root system of the exceptional Lie algebra E8. The quantum numbers of these solutions correspond to the first and second…
We prove local boundedness, Harnack's inequality and local regularity for weak solutions of quasilinear degenerate elliptic equations in divergence form with Rough coefficients. Degeneracy is encoded by a non-negative, symmetric, measurable…
The classical continuous finite element method with Lagrangian $Q^k$ basis reduces to a finite difference scheme when all the integrals are replaced by the $(k+1)\times (k+1)$ Gauss-Lobatto quadrature. We prove that this finite difference…
The U(1) Calogero Sutherland Model (CSM) with anti-periodic boundary condition is studied. The Hamiltonian is reduced to a convenient form by similarity transformation. The matrix representation of the Hamiltonian acting on a partially…
Various quasi-exact solvability conditions, involving the parameters of the periodic associated Lam{\'e} potential, are shown to emerge naturally in the quantum Hamilton-Jacobi approach. It is found that, the intrinsic nonlinearity of the…
We consider the eigenvalues of an elliptic operator for systems with bounded, measurable, and symmetric coefficients. We assume we have two non-empty, open, disjoint, and bounded sets and add a set of small measure to form the perturbed…
Shape invariance is an important ingredient of many exactly solvable quantum mechanics. Several examples of shape invariant ``discrete quantum mechanical systems" are introduced and discussed in some detail. They arise in the problem of…
Two sets of infinitely many exceptional orthogonal polynomials related to the Wilson and Askey-Wilson polynomials are presented. They are derived as the eigenfunctions of shape invariant and thus exactly solvable quantum mechanical…
The wave functions of the Calogero-Sutherland model are known to be expressible in terms of Jack polynomials. A formula which allows to obtain the wave functions of the excited states by acting with a string of creation operators on the…
We construct exactly solvable models for four particles moving on a real line or on a circle with translation invariant two- and four-particle interactions.
We show that Calogero-Sutherland models for interacting particles have a natural supersymmetric extension. For the construction, we use Jacobians which appear in certain superspaces. Some of the resulting Hamiltonians have a direct physics…
We consider a system of quasilinear elliptic equations, with indefinite super-linear nonlinearity, depending on two real parameters $\lambda,\mu$. By using the Nehari manifold and the notion of extremal parameter, we extend some results…
We develop a unified framework for a broad class of nonlocal elliptic problems, encompassing a wide spectrum of nonlocal terms, including the classical Kirchhoff and Carrier-type equations as particular cases, and nonlinearities having…
We extend the notion of Dirac oscillator in two dimensions, to construct a set of potentials. These potentials becomes exactly and quasi-exactly solvable potentials of non-relativistic quantum mechanics when they are transformed into a…
The degeneracy structure of the eigenspace of the N-particle Calogero-Sutherland model is studied from an algebraic point of view. Suitable operators satisfying SU(2) algebras and acting on the degenerate eigenspace are explicitly…