Borel summation of adiabatic invariants

Borel summation of adiabatic invariants

Borel summation techniques are developed to obtain exact invariants from formal adiabatic invariants (given as divergent series in a small parameter) for a class of differential equations, under assumptions of analyticity of the coefficients; the method relies on the study of associated partial differential equations in the complex plane. The type and location of the singularities of these asso...

A Modified Borel Summation Technique

We compare and contrast three different perturbative expansions for the quartic anharmonic oscillator wavefunction and apply a modified Borel summation technique to determine the energy eigenvalues. In the first two expansions this provides the energy eigenvalues directly however in the third method we tune the wavefunctions to achieve the correct large x behaviour as first illustrated in [1]. ...

Gamow Vectors and Borel Summation

We analyze the detailed time dependence of the wave function ψ(x, t) for one dimensional Hamiltonians H = −∂ x + V (x) where V (for example modeling barriers or wells) and ψ(x, 0) are compactly supported. We show that the dispersive part of ψ(x, t), its asymptotic series in powers of t−1/2, is Borel summable. The remainder, the difference between ψ and the Borel sum, is a convergent expansion o...

Theory of Adiabatic Invariants

We review the basic theory on adiabatic invariants in classical and quantum mechanics. PACS numbers: 45.30.+s,05.45.-a,03.65.-w, 03.65.Sq

Notes on Adiabatic Invariants