The Curse of Dimensions in Schrödinger's Equation

The basis of modern physics is viewed to be Schrödinger's Equation SE as a linear time-evolution equation in $3N$ spatial dimensions for a system with $N$ electrons. Numerical digital solution with a resolution of 100 in each spatial variable involves $100^{3N}=10^{6N}$ mesh points in space, already with $N=4$ beyond thinkable computational power. 

When SE was formulated in 1926 when digital computation was not an issue, and so the fact that SE effectively is uncomputable did not enter the minds of its creators Born-Heisenberg-Schrödinger, although Schrödinger was not happy with the many dimensions lacking physicality. It was sufficient that an analytical solution was found for $N=1$ leaving $N>1$ into terra incognita waiting to be explored until digital computation became available, but then was found hit the wall from the curse of dimensions.

This is where we stand today: SE is the basic mathematical model of atom physics but SE is not a computable model. It is thus impossible to make a prediction of the evolution in time of an atomic system with more than 3 electrons by computational solution of SE. It is thus impossible to check if SE correctly models physics by comparing SE predictions with observations of real physics.

Yet SE serves as the canonical model of atom physics in its original formulation, as uncomputable today  as 100 years ago, because of its many dimensions also without physical meaning. 

What can be the value of an uncomputable mathematical model of some physics?  A physicist will tell that it still has a value because SE can be (drastically) dimensionally reduced to computable form and so allow computation of (drastically) simplified approximate solutions. SE would then serve as a suitable starting point for dimensional reduction into a computable model with physical meaning. But it would be the dimensional reduction which would carry the physics.

The alternative would be to start instead directly with a dimensionally reduced model with physical meaning, and thus leave SE to history as no longer useful. This possibility is explored as RealQM. 

Physicists speak with large ease about multi-dimensional wave functions $\Psi$ as solutions to SE, as if they are computable and have physical meaning. The consensus is the "SE works but nobody understands why". Philosophers of physics study the (lack of) meaning of SE, theoretical physicists have turned to more fundamental models such as QED and String Theory, chemists seek to understand what SE offers for molecules, while computational physicists solve other equations, and there is no synthesis in sight.