p Dissipative operator

In this paper the authors prove that the generalized positive p selfadjoint (GPpS) operators in Banach space satisfy the generalized Schwarz inequality, solve the maximal dissipative extension representation of p dissipative operators in Banach space by using the inequality and introducing the generalized indefinite inner product (GIIP) space, and apply the result to a certain type of Schrodinger operator.

N=2 supersymmetric QCD and elliptic potentials

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Positive Solution for a Class of Degenerate Quasilinear Elliptic Equations in R-N

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

A Bethe ansatz solvable model for superpositions of Cooper pairs and condensed molecular bosons

We introduce a general Hamiltonian describing coherent superpositions of Cooper pairs and condensed molecular bosons. For particular choices of the coupling parameters, the model is integrable. One integrable manifold, as well as the Bethe ansatz solution, was found by Dukelsky et al. [J. Dukelsky, G.G. Dussel, C. Esebbag, S. Pittel, Phys. Rev. Lett. 93 (2004) 050403]. Here we show that there is a second integrable manifold, established using the boundary quantum inverse scattering method. In this manner we obtain the exact solution by means of the algebraic Bethe ansatz. In the case where the Cooper pair energies are degenerate we examine the relationship between the spectrum of these integrable Hamiltonians and the quasi-exactly solvable spectrum of particular Schrodinger operators. For the solution we derive here the potential of the Schrodinger operator is given in terms of hyperbolic functions. For the solution derived by Dukelsky et al., loc. cit. the potential is sextic and the wavefunctions obey PT-symmetric boundary conditions. This latter case provides a novel example of an integrable Hermitian Hamiltonian acting on a Fock space whose states map into a Hilbert space of PE-symmetric wavefunctions defined on a contour in the complex plane. (c) 2006 Elsevier B.V. All rights reserved.

UNIQUENESS OF SOLUTIONS TO SCHRODINGER EQUATIONS ON H-TYPE GROUPS

This paper deals with the Schrodinger equation i partial derivative(s)u(z, t; s) - Lu(z, t; s) = 0; where L is the sub-Laplacian on the Heisenberg group. Assume that the initial data f satisfies vertical bar f(z, t)vertical bar less than or similar to q(alpha)(z, t), where q(s) is the heat kernel associated to L. If in addition vertical bar u(z, t; s(0))vertical bar less than or similar to q(beta)(z, t), for some s(0) is an element of R \textbackslash {0}, then we prove that u(z, t; s) = 0 for all s is an element of R whenever alpha beta < s(0)(2). This result holds true in the more general context of H-type groups. We also prove an analogous result for the Grushin operator on Rn+1.

MULTIPLICITY OF POSITIVE SOLUTIONS FOR A CLASS OF NONLINEAR SCHRODINGER EQUATIONS

This paper proves the multiplicity of positive solutions for the following class of quasilinear problems: {-epsilon(p)Delta(p)u+(lambda A(x) + 1)vertical bar u vertical bar(p-2)u = f(u), R(N) u(x)>0 in R(N), where Delta(p) is the p-Laplacian operator, N > p >= 2, lambda and epsilon are positive parameters, A is a nonnegative continuous function and f is a continuous function with subcritical growth. Here, we use variational methods to get multiplicity of positive solutions involving the Lusternick-Schnirelman category of intA(-1)(0) for all sufficiently large lambda and small epsilon.

On the Schrodinger equation involving a critical Sobolev exponent and magnetic field

We consider the semilinear Schrodinger equation -Delta(A)u + V(x)u = Q(x)vertical bar u vertical bar(2* -2) u. Assuming that V changes sign, we establish the existence of a solution u not equal 0 in the Sobolev space H-A,V(1) + (R-N). The solution is obtained by a min-max type argument based on a topological linking. We also establish certain regularity properties of solutions for a rather general class of equations involving the operator -Delta(A).