We study the duality between the two dimensional black hole and the sine-Liouville conformal field theories via exact operator quantization of a classical scattering problem. The ideas are first illustrated in Liouville theory, which is dual to itself under the interchange of the Liouville parameter b by 1/b. In both cases, a classical scattering problem does not determine uniquely the quantum ...

We establish Liouville-type theorems for stable and finite Morse index weak solutions of −∆pu = f(x)F (u) in R . For a general non-linearity F ∈ C(R) and f(x) = |x|, we prove such theorems in dimensions N ≤ 4(p+α) p−1 +p, for bounded radial stable solutions. Then, we give some point-wise estimates for not necessarily bounded solutions. Also, similar theorems will be proved for both radial finit...

This paper is the continuation of [6], where Victor Guillemin and I proved the following result: the Taylor expansion of a potential V (x) (x ∈ R) at a non degenerate critical point x0 of V , satisfying V (x0) 6= 0, is determined by the semi-classical spectrum of the associated Schrödinger operator near the corresponding critical value V (x0). Here, I prove results which are stronger in some as...

An inverse nodal problem has first been studied for the Sturm-Liouville equation with one turning point. The asymptotic representation of the corresponding eigenfunctions of the eigenvalues has been investigated and an asymptotic of the nodal points is obtained. For this problem, we give a reconstruction formula for the potential function. Furthermore, numerical examples have been established a...

An adaptive block elimination method based on recursive projection techniques is proposed for solving large linear systems which arise when applying Newton methods to parameter dependent nonlinear equations. The method is suitable for non-linear systems that have a Jacobian with a symmetric and almost positive deenite main block. The implicitly deened large symmetric and positive deenite subsys...

We describe a parallel algorithm for the numerical computation of guaranteed bounds for solutions of elliptic boundary value equations of second order. We use C 2-Hermite elements and a parallel Newton multigrid method to produce approximations of high accuracy. Then, we compute upper bounds for the defect and enclosures for the eigenvalues of the linearization. In order to obtain veriied bound...

Solving an inverse Sturm-Liouville problem requires a mathematical process to determine unknown function in the Sturm-Liouville operator from given data in addition to the boundary values. In this paper, we identify a Sturm-Liouville potential function by using the data of one eigenfunction and its corresponding eigenvalue, and identify a spatial-dependent unknown function of a SturmLiouville d...