Three Solutions to the Nonlinear Schrödinger Equation for a Constant Potential

We introduce three sets of solutions to the nonlinear Schrödinger equation for the free particle case. A well-known solution is written in terms of Jacobi elliptic functions, which are the nonlinear versions of the trigonometric functions sin, cos, tan, cot, sec, and csc. The nonlinear versions of the other related functions like the real and complex exponential functions and the linear combinations of them is the subject of this chapter. We also illustrate the use of these functions in Quantum Mechanics as well as in nonlinear optics

Emergence of Classical Distributions from Quantum Distributions: The Continuous Energy Spectra Case

We explore the properties of quantum states and operators that are conjugate to the Hamiltonian eigenstates and operator when the Hamiltonian spectrum is continuous, i.e., we find time-like operators T^ such that [T^,H^]=iℏ. This is a property expected for a time operator. We explicitly unfold the momentum sign degeneracy of energy states. We consider the free-particle case, and we find, among other things, that the time states are also the solution of the quantized version of the classical motion of the particle

Matrices Which are Discrete Versions of Linear Operations

We introduce and study a matrix which has the exponential function as one of its eigenvectors. We realize that this matrix represents a set of finite differences derivation of vectors on a partition. This matrix leads to new expressions for finite differences derivatives which are exact for the exponential function. We find some properties of this matrix, the induced derivatives and of its inverse. We provide an expression for the derivative of a product, of a ratio, of the inverse of vectors, and we also find the equivalent of the summation by parts theorem of continuous functions. This matrix could be of interest to discrete quantum mechanics theory

Exact Finite Differences for Quantum Mechanics

We introduce a finite difference derivative, on a non-uniform partition, with the characteristic that the derivative of the exponential function is the exponential function itself, times a constant, which is similar to what happens in the continuous variable case. Aside from its application to perform numerical computations, this is particularly useful in defining a quantum mechanical discrete momentum operator

The Inverse of the Discrete Momentum Operator

In the search of a quantum momentum operator with discrete spectrum, we obtain some properties of the discrete momentum operator for nonequally spaced spectrum. We find the inverse operator. We use the matrix representation of these operators, and we find that there is one more eigenvalue and eigenfunction than the dimension of the matrix. We apply the results to obtain the discrete adjoint of the momentum operator. We conclude that we can have discrete operators which can be self-adjoint and that it is possible to define a self-adjoint extension of the corresponding Hilbert space. These results help us understand the quantum time operator

Comparative evaluation of therapeutic DNA vaccines against Trypanosoma cruzi in mice

Chagas disease, caused by the protozoan parasite Trypanosoma cruzi, is a major public health problem in most of Latin America. A key priority is the development of new treatments, due to the poor efficacy of current ones. We report here the comparative evaluation of therapeutic DNA vaccines encoding various T. cruzi antigens. ICR mice infected with 500 parasites intraperitoneally were treated at 5 and 12 days postinfection with 20 mu g of plasmid DNA encoding T. cruzi antigens TSA-1, TS, ASP-2-like, Tc52 or Tc24. Treatment with plasmid encoding TS and/or ASP-2-like antigens had no significant effect on parasitemia or survival. Treatment with Tc52 DNA significantly reduced parasitemia, as well as cardiac parasite burden, and improved survival, although myocarditis was not significantly affected. Finally, treatment with plasmids encoding Tc24 and TSA-1 induced the most complete control of disease as evidenced by significant reductions in parasitemia, mortality, myocarditis and heart parasite burden. These data demonstrate that therapeutic vaccine efficacy is dependent on the antigen and suggest that DNA vaccines encoding Tc24, TSA-1, and Tc52 represent the best candidates for further studies of a therapeutic vaccine against Chagas disease.Univ Autonoma Yucatan, Parasitol Lab, Ctr Invest Reg Dr Hideyo Noguchi, Merida, Yucatan, MexicoINSERM, IRD, UR 008, Ctr IRD Montpellier, Montpellier, FranceUniversidade Federal de São Paulo, CINTERGEN, Dept Microbiol Inmunol & Parasitol, São Paulo, BrazilUniversidade Federal de São Paulo, CINTERGEN, Dept Microbiol Inmunol & Parasitol, São Paulo, BrazilWeb of Scienc