Iterative method for finding the eigenfunctions of a system of two Schrödinger equations with combined nonlinearity

An iterative method is proposed to determine the eigenfunctions of a system of two nonlinear Schrödinger equations governing the interaction of two femtosecond pulses in a medium with quadratic and cubic nonlinearity. The method produces soliton solutions of a new form for a wide range of nonlinearity coefficients corresponding to the first and second eigenvalues. A specially chosen initial approximation is required to determine the third and higher eigenfunctions.     

Witten-Helffer-Sjöstrand theory for -equivariant cohomology

Given an -invariant Morse function and an -invariant Riemannian metric , a family of finite dimensional subcomplexes , , of the Witten deformation of the -equivariant de Rham complex is constructed, by studying the asymptotic behavior of the spectrum of the corresponding Laplacian as . In fact the spectrum of can be separated into the small eigenvalues, finite eigenvalues and the large eigenvalues. Then one obtains as the complex of eigenforms corresponding to the small eigenvalues of . This permits us to verify the -equivariant Morse inequalities. Moreover suppose is self-indexing and satisfies the Morse-Smale condition, then it is shown that this family of subcomplexes converges as to a geometric complex which is induced by and calculates the -equivariant cohomology of .

     

Log-dimensional spectral properties of one-dimensional quasicrystals

We consider discrete one-dimensional Schrödinger operators on the whole line and establish a criterion for continuity of spectral measures with respect to log-Hausdorff measures. We apply this result to operators with Sturmian potentials and thereby prove logarithmic quantum dynamical lower bounds for all coupling constants and almost all rotation numbers, uniformly in the phase.

     

Energy transfer in scattering by rotating potentials

Quantum mechanical scattering theory is studied for time-dependent Schrödinger operators, in particular for particles in a rotating potential. Under various assumptions about the decay rate at infinity we show uniform boundedness in time for the kinetic energy of scattering states, existence and completeness of wave operators, and existence of a conserved quantity under scattering. In a simple model we determine the energy transferred to a particle by collision with a rotating blade.     

Absolutely continuous spectrum for one-dimensional Schrödinger operators with slowly decaying potentials: Some optimal results

The absolutely continuous spectrum of one-dimensional Schrödinger operators is proved to be stable under perturbation by potentials satisfying mild decay conditions. In particular, the absolutely continuous spectra of free and periodic Schrödinger operators are preserved under all perturbations satisfying This result is optimal in the power scale. Slightly more general classes of perturbing potentials are also treated. A general criterion for stability of the absolutely continuous spectrum of one-dimensional Schrödinger operators is established. In all cases analyzed, the main term of the asymptotic behavior of the generalized eigenfunctions is shown to have WKB form for almost all energies. The proofs rely on maximal function and norm estimates, and on almost everywhere convergence results for certain multilinear integral operators.

     

三个方程耦合的Schrödinger方程组的基态

研究由三个方程耦合的非线性Schr?dinger方程组,它们源于非线性光学和Bose-Einstein凝聚.考虑了两种类型:含有周期位势的方程组和含有势阱位势的方程组.借助于广义的Nehari流形以及精细的能量估计,证明了当相互作用位势适当小时,这两类非线性Schr?dinger方程组存在正的基态.     

BLOW-UP OF THE SOLUTIONS FOR THE INITIAL-BOUNDARY PROBLEMS OF THE NONLINEAR SCHR?DINGER EQUATIONS

Ｔｈｅｒｅａｒｅｍａｎｙｄｏｃｕｍｅｎｔｓｔｈａｔｄｉｓｃｕｓｓｂｌｏｗ＿ｕｐｏｆｔｈｅｓｏｌｕｔｉｏｎｓｆｏｒｔｈｅｉｎｉｔｉａｌ＿ｂｏｕｎｄａｒｙｖｌｕｅｐｒｏｂｌｅｍｓｏｆｔｈｅｎｏｎｌｉｎｅａｒＳｃｈｒ ｄｉｎｇｅｒｅｑｕａｔｉｏｎｓ,ｓｕｃｈａｓｐａｐｅｒ [1 ] .Ｂｕｔｔｈｅｐａｐｅｒ [1 ]ｈａｓｏｎｅｗｅａｋｐｏｉｎｔ,ｔｈａｔｉｓ,ｉｔｃａｎｎｏｔｕｎｉｔｅｄｉｓｃｕｓｓｉｏｎｆｏｒｂｌｏｗ＿ｕｐｏｆｔｈｅｓｏｌｕｔｉｏｎｓｆｏｒｔｈｅｉｎｉｔｉａｌ＿ｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌ…     

Interpretation for Probability Wave and Quantum Measured Problem

By using Hamilton-Jacobi-Bellman equation with complex time, we investigate quantum theory in timelike curve. State vectors of a physical system in the two-dimensional timelike curve not only obey Schrödinger equation in the observed timespace but also involve random motion in the traversal timespace. The random motion with hidden variables is successfully to explain why the wave function is a probability wave. Quantum measurement are discussed in present work. The results are in agreement with the conventional interpretation of quantum theory.     

Two optimized symmetric eight-step implicit methods for initial-value problems with oscillating solutions

In this paper, we present two optimized eight-step symmetric implicit methods with phase-lag order ten and infinite (phase-fitted). The methods are constructed to solve numerically the radial time-independent Schr?dinger equation with the use of the Woods–Saxon potential. They can also be used to integrate related IVPs with oscillating solutions such as orbital problems. We compare the two new methods to some recently constructed optimized methods from the literature. We measure the efficiency of the methods and conclude that the new method with infinite order of phase-lag is the most efficient of all the compared methods and for all the problems solved. T. E. Simos—Highly Cited Researcher, Active Member of the European Academy of Sciences and Arts.     

One-dimensional Schrödinger operator on the half-line: The differential equation for eigenfunctions with respect to the spectral parameter and an analog of the Freud equation

It is shown that the eigenfunctions of the Schrödinger operator on the half-line satisfy an explicitly constructed differential equation with respect to the spectral parameter. Such an equation was earlier obtained for orthogonal polynomials. An analog of the Freud equation is found.