Is it possible to minimise a polynomial (of potential energy) to fit a set of any (random) eigenvalues/vectors?

I am trying to solve the 1-D Time-Independent Schrodinger Equation "backwards",

[-hbar^2/2m + U(x)]psi(x) = E(x)psi(x)

by using any random eigenvalues and their corresponding eigenvectors to get back to an original potential U(x). The polynomial should be in a form like this f(x) = a + b*x + c*x^(2) + d*x^(3) + e*x^(4)

For example, when f(x) = 1/2*x^(2), the eigenvalues are 0.5, 1.5, 2.5, 3.5... My question is, is there a way to put in eigenvalues such as 0.678, 1.234, 2.32, 4.95... and get some form of the polynomial back?

[John D'Errico]

If they are random numbers, then they are not eigenvalues.

[Christiaan]

A system can be in different states. In case of the Time-Independent Schrodinger Equation, these states are discrete. Therefore, for each possible state, there is a corresponding eigenvalue (=energylevel). And therefore, random eigenvalues are not possible.