1
$\begingroup$

We start with the following recurrence relation for complex coefficients $c_{n,m}$: $$i\dot{c}_{n,m}(t) = \sqrt{(n+1)(n+2)(m-1)m}c_{n-2,m+2} + \sqrt{n(n-1)(m+1)(m+2)}c_{n+2,m-2}$$ where $\dot{c}_{n,m}$ denotes time derivative and $n,m = 0,1,2,...$.

In order to solve this equation and find the time evolution of $c_{n,m}$ (we assume that at $t=0$ we know values of $c_{n,m}$) we define a generation function: $$G(x,y,t) = \sum\limits_{n,m=0}^{\infty}x^{n}y^{m}\alpha_{n,m}c_{n,m}(t)$$ with some unknown coefficient $\alpha_{n,m}$ that can be adjusted manually. Recurrence relation can be rewritten in terms of differential equation for $G$: $$i\partial_{t}G(x,y,t) = (y^2\partial_{x}^2 + x^2\partial_{y}^2)G(x,y,t)$$ if we set $$\alpha_{n,m} = \alpha_{n+2,m-2}\sqrt{\frac{(n+1)(n+2)}{m(m-1)}}$$ We can assume that $G(x,y,0)$ is known and converges in the domain $[0,1] \times [0,1]$ for any $t$.

Is there a way to find function $G(x,y,t)$ that is not represented by infinite series? This is a special kind of Schrödinger equation.

$\endgroup$
4
  • 3
    $\begingroup$ Minor comment: it is not a diffusion equation. It is a Schrodinger equation with variable coefficients. $\endgroup$ Feb 12, 2016 at 14:22
  • $\begingroup$ Are you sure your recursion relation is stated correctly? I think that, in the first term on the right hand side, you should have a $c_{n-2,m+2}$, not $c_{n-2,m-2}$. Note that the problem completely uncouples into a sequence of finite recursion problems because for a fixed sum $s = m+n$ and (even-odd) parity of $m-n$, the set of such $c_{n,m}$ (and this is a finite set of such, since you are assuming that $n,m\ge0$) only interacts with itself. Thus, you really should be thinking in terms of these finite recursion problems. It's really a collection of finite dimensional eigenvalue problems. $\endgroup$ Feb 12, 2016 at 15:42
  • $\begingroup$ @RobertBryant Indeed there was a mistake in recursion relation. This is a physics problem and as you noticed coefficient $c_{n,m}$ is linked with others such that $s=n+m$ is fixed. We can of course find eigenvalues for small $s$ and find time evolution explicitly. However this doesn't solve my problem cause imagine I want to know what happens when $s$ is large and theoretically tends to infinity. $\endgroup$
    – WoofDoggy
    Feb 14, 2016 at 14:56
  • $\begingroup$ @Nex_Friedrich: I understand that you want to know 'what happens' for large values of $s$, but, depending on what you mean by 'what happens', I am skeptical that you will learn more by any technique than by figuring out the asymptotics of the distribution of the eigenvalues in terms of $s$. At least you know that, for each $s$, the eigenvalues of $L=y^2\partial_x^2+x^2\partial_y^2$ on $H_s$, the homogeneous polynomials in $x$ and $y$ of total degree $s$, are real, symmetrical about 0, and have multiplicity $2$ when $s$ is odd and multiplicity $1$ when $s$ is even. Do you know any asymptotics? $\endgroup$ Feb 14, 2016 at 16:14

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.