Here is where the landscape function comes in. This function started as a purely mathematical discovery: when solving a Schrödinger equation such as (where we have now suppressed all physical constants for simplicity), it turns out that the behavior of the eigenfunctions at various energy level...
for all . In some applications, one also wants to impose positive definiteness, which asserts that for all . These hypotheses are sufficient in the case when is bounded, and in particular when is finite dimensional. However, as it turns out, for unbounded operators these conditions are not,...
for all . For simplicity let us restrict to be compact. There is an obvious necessary condition for this embeddability to occur, which comes from energy conservation law for the Euler equations; unpacking everything, this implies that the bilinear form in (2) has to obey a cancellation conditi...
In practice, when is a disordered potential, the effective potential tends to be behave like a somewhat “smoothed out” or “homogenized” version of that exhibits superior numerical performance. For instance, the classical Weyl law predicts (assuming a smooth confining potential ) that the ...