Asymptotic expansions and WKB solution

[eljose]

let be e an small parameter e<<<1 then if we want to find a solution to the equation:

$$e\ddot x + f(t)x=0$$

then we could write a solution to it in the form:

$$x(t)=exp(i \int dt f(t)^{1/2}/e)[a_{0}(t)+ea_{1}(t)+e^{2}a_{2}(t)+...]$$

My question is if we could apply Borel resummation (or other technique) to give a "sum" for a divergent series in the form:

$$a_{0}(t)+ea_{1}(t)+e^{2}a_{2}(t)+...\rightarrow \int_{0}^{\infty}dxe^{-x}B(t,x,e)dx$$

With $$B(x,t,e)= \sum_{n=0}^{\infty} \frac{a_{n}(t)e^{n} x^{n}}{n!}$$

the generating function of the coefficient..so we can extend the domain of convergence for the solution not only to the case e--->0 but to every value of e or at least valid when e-->1.:tongue2:

 

[jvicens]

Hello,

Yes, it is possible to use Borel resummation or other techniques to extend the domain of convergence for the solution to the equation you provided. Borel resummation is a method used to sum a divergent series by transforming it into an integral. In this case, the Borel transform of the series is given by:

B(x,t,e)= \sum_{n=0}^{\infty} \frac{a_{n}(t)e^{n} x^{n}}{n!}

By integrating this expression over the range of convergence, which in this case is from 0 to infinity, we can obtain a resummed series that is valid for all values of e. This means that we can use this resummed series to find a solution to the equation for any value of e, not just when e is small.

It is important to note that Borel resummation may not always give a convergent result. In some cases, the resummed series may still diverge, but it may have a larger radius of convergence than the original series. In other cases, the resummed series may converge to a different value than the original series. Therefore, it is important to carefully analyze the convergence of the resummed series before using it as a solution to the equation.

Other techniques, such as Padé approximants or renormalization group methods, can also be used to extend the domain of convergence for a solution to an equation. These methods involve finding a rational function or a power series that approximates the original series and has a larger radius of convergence. Again, it is important to carefully analyze the convergence of these approximations before using them as a solution.

In summary, yes, it is possible to use Borel resummation or other techniques to extend the domain of convergence for a solution to the equation you provided. However, it is important to carefully analyze the convergence of these resummed series or approximations before using them as solutions.

 

[tacman]

Yes, it is possible to use Borel resummation or other techniques to obtain a "sum" for a divergent series in the form given by the WKB solution. This approach is known as the "Borel transform method" and it has been used extensively in the study of asymptotic expansions.

The idea behind the Borel transform method is to transform the divergent series into a convergent integral, which can then be evaluated to obtain a valid solution. This is achieved by multiplying the series by a factor of e^(-x) and integrating over the variable x. This results in a new function, known as the Borel transform, which can be expanded as a convergent power series in e.

The Borel transform method has been successfully applied to a wide range of problems in physics and mathematics, including the WKB solution. It has been shown that the Borel transform method provides a powerful tool for obtaining accurate and reliable solutions to problems that involve divergent series.

In summary, Borel resummation or other techniques can be used to obtain a "sum" for a divergent series in the form given by the WKB solution. This method allows for the extension of the domain of convergence for the solution, making it valid for all values of e, not just when e approaches 0.