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Hence the transformed equation is (2.5).

The eigenvalues of the matrix M ˜ become λ ˜ = μ λ and therefore, the solution y ˜ of the transformed equation is smoother than the solution y of the original one.

Imposition of boundary conditions of the transformed equations is a major drawback that has to be tackled, especially for three-dimensional problems.

The important point about the transformed equations is that they are decoupled, with the unique stable solution ((varSigma_{0}, 0)), which is equivalent to (n_{E} = n_{I}) in the original variables.

These transformed equations are then expressed in suitable dimensionless form.

The transformed equations are numerically integrated using fourth fifth order Runge Kutta Fehlberg method.

The transformed equations are as follows.

(7) can be found such that the transformed equations are numerically equivalent to the initial value approach.

Choosing an invertible transformation matrix assures that the system's dynamics are preserved, and the original states can be retrieved from the new ones at any time as long as none of the transformed equations are eliminated.

These transforming equations are termed observation operators.

The transformed Schrödinger equation is solved with a 4th order splitting method.