20051222, 11:16  #1 
Dec 2003
Hopefully Near M48
2·3·293 Posts 
Asymptotic Behavior of a Differential Equation
Here's a question that I've been trying to answer for a while, with no real progress:
Suppose that is a complexvalued function that satisfies: where is the Laplacian operator (in spherical coordinates) and is a realvalued function that is always positive. Prove that , or give a counterexample (exclude the trivial case where at all points in space). Last fiddled with by jinydu on 20051222 at 11:19 
20051222, 14:07  #2 
Dec 2003
Hopefully Near M48
2·3·293 Posts 
If it helps, I can split the differential equation up into components to derive relations.
In Cartesian coordinates: In Polar coordinates: (which in fact tells us nothing, since is an arbitrary positive function; and the absolute values are nonnegative) If we use the polar coordinate relation (presumably because it is the simplest), the problem becomes to show that: Last fiddled with by akruppa on 20051222 at 16:29 Reason: fixed tex 
20051222, 14:23  #3 
Dec 2003
Hopefully Near M48
2×3×293 Posts 
Argh! I ran out of time editing my last message. The last line should read:
Last fiddled with by jinydu on 20051222 at 14:23 
20051222, 20:16  #4 
∂^{2}ω=0
Sep 2002
República de California
2^{2}·3^{2}·17·19 Posts 
Solutions of such elliptic PDEs are wellknown to be entirely dependent on their boundary conditions, so specifying the equations sans BCs is not meaningful. In the case of an infinite domain ("free boundary conditions"), I believe it is possible to show that so so long as the forcing function f({coordinates}) is L^{2}integrable (which implies that it must decay at infinity), the solutions of the corresponding elliptic must also be L^{2}integrable, i.e. must also decay at infinity  in fact I believe in the complex case the solutions must be entire if f is. I'll see if I can find a reference...

20051223, 00:36  #5 
Dec 2003
Hopefully Near M48
2·3·293 Posts 
This problem comes from my trying to prove a theorem in quantum mechanics: That there are no normalizable solutions to the timeindependent Schrodinger equation where the total energy is less than the potential energy at every point in space. I've managed to show, after some algebra, that it is sufficient to prove the claim in this thread, since a wavefunction that doesn't go to zero at infinity can't possibly be normalized.
Last fiddled with by jinydu on 20051223 at 00:36 
20060104, 04:08  #6 
Dec 2003
Hopefully Near M48
2·3·293 Posts 
Bump. Please?

20060105, 20:17  #7 
∂^{2}ω=0
Sep 2002
República de California
2^{2}×3^{2}×17×19 Posts 
Ah  I thought you were indicating in your 22 Dec post that you'd solved the problem to your own satisfaction.
I suggest assuming a separable forcing function, i.e. replace f(r,theta,phi) by F(r)*G(theta)*H(phi), and a separable solution psi(r,theta,phi) = a(r)*b(theta)*c(phi). Plugging into the equation should yield a fairly simple ODE for a(r) in terms of F(r) and the radial terms of the spherical Laplacian. If you assume F(r) can be expanded as a power series in r, you should be able to derive general expressions for the coefficients of the series that then represents the solution function a(r), and say something pretty general about decay at infinity. Turning that into a general bound on generic (i.e. not necessarily separable f and psi) is trickier, but perhaps your radialonly analysis will allow you to bound a general solution, e.g. show that if a general forcing function f(r,theta,phi) can be bounded above by a separable function F(r)*G(theta)*H(phi), then the corresponding (usually unknown, since it would require an exact solution of the nonseparable equation) solution psi(r,theta,phi) can similarly be bounded above by the separablecase solution function a(r)*b(theta)*c(phi). Sorry I don't have more time to help with the details, but I really think in terms of asymptoticdecayatinfinity behavior, it should suffice to study a radialonly ODE analog of the full equation  angular variations must always remain bounded (at any constant r) by the mere fact of their periodicity in the angular coordinates (i.e. psi(r,theta+2*k*pi,phi) = psi(r, theta, phi), similar for f, similarly for the second angle phi.) 
20060121, 20:58  #8 
Dec 2003
Hopefully Near M48
1758_{10} Posts 
Unfortunately, even the 1D case has me stumped:
Now prove that is not finite and nonzero. Last fiddled with by jinydu on 20060121 at 21:00 
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