Asymptotic Analysis

We propose an approach to calculate asymptotic series for low lying eigenvalues of Schrödinger operator based on normal forms in the formal graded Weyl–Heisenberg algebra. The difference from a traditional scheme is that we don’t use any symbol map (Weyl, p x , x p , etc.). We show that our method may be useful for different reasons. Firstly, it enables to estimate the growth of the eigenvalues expansion coefficients, and secondly it may be efficient for practical calculations, e.g. for treating inverse problems. In particular, we prove that under some restrictions in the one-dimensional case the knowledge of asymptotic series for any pair of low lying eigenvalues is enough to recover the potential.

We consider the Laplace operator on the finite-dimensional linear space of algebraic polynomials in N variables such that each variable occurs at most with the power n. The space of the polynomials is equipped with the Laguerre norm. We establish safe lower and upper bounds for the norm of the Laplace operator on this space, and we derive asymptotic lower and upper bounds for this norm as n goes to infinity. The asymptotic bounds are better than the safe bounds.

We study the asymptotic integration problem for a general class of n-th order nonlinear delay differential equations of the form L n x ( t ) = f ( t , x ( g ( t ) ) ) , where L n x = x ( n ) + a n − 1 ( t ) x ( n − 1 ) + ⋯ + a 0 ( t ) x . It is shown that if { u 1 , u 2 , … , u n } is a set of principal solutions for L n x = 0 , then the n-th order nonlinear delay differential equation above has solutions x k with the property that x k ( t ) = b 1 u 1 ( t ) + ⋯ + b k u k ( t ) + o ( u k ( t ) ) , t → ∞ for k = 1 , 2 , … , n .

For the Kac equation and homogeneous Boltzmann equation of Maxwellian without Grad’s angular cut-off, we prove an exponential convergence towards the equilibrium as t → ∞ in a weak norm which is equivalent to the weak convergence of measures, extending results of Gabetta, Toscani and Wennberg (J. Stat. Phys. 81 (1995), 901–934) and Carlen, Gabetta and Toscani (Commun. Math. Phys. 199 (1999), 521–546) from the cut-off case to the non-cut-off case. We give quantitative estimates of the convergence rate, which are governed by the spectral gap of the linearized collision operator. We then prove a uniform bound in time on Sobolev norms of the solutions. The results are then combined with some interpolation inequalities, to obtain the rate of the exponential convergence in the strong L 1 norm, as well as various Sobolev norms.