Constructing a Hamiltonian (as a polynomial of $q_i$ and $p_i$) from its spectrum For a countable sequence of positive numbers $S=\{\lambda_i\}_{i\in N}$ is there a construction producing a Hamiltonian with spectrum $S$ (or at least having the same eigenvalues for $i\leq s$ for some $s$)?
Here by 'Hamiltonian' I understand a polynomial of $p_i$ and $q_i$ (or equivalently - $a_i$ and $a_i^\dagger$) of $k$ pairs of variables and of order $2n$. Both $k$ and $n$ can be functions of $S$ and $s$.
For example, for the spectrum
$$S =\{3,5,7,9,\ldots\}$$
one of Hamiltonians working for any $s$ is
$$H = 3+a^\dagger a.$$
 A: It seems there is a simply way to do this for polynomials with finite degree $d$. Since $a^\dagger a$ is the number operator, we can take $a^\dagger a = N$, where $N$ is the number of excitations corresponding to a particular level. Then if the Hamiltonian has the general form $H = \sum_{k=0}^d c_k (a^\dagger a)^k$, the energy corresponding to a particular state is $E_N = \sum_{k=0}^d c_k N^k$. Since $N$ is constant for a given eigenstate of the Hamiltonian, the equation for $E_N$ is just a linear equation in the variables $\{c_k\}_k$. Since you have the spectrum, you have $E_N$, and hence you can solve using Guassian elimination (or whatever your preferred technique is for solving linear systems of equations). Even if the spectrum is infinite, you will only need $d+1$ equations to fix the values of $c_k$, so this is a simple calculation.
A: It was already mentioned in a comment above about a topic in physics.SE, that it may be related with inverse scattering problem. I only may add, that there is precise method of construction of Shroedinger operator $p^2 + V(q)$ with arbitrary finite spectrum for one-dimensional case. It is very well developed due to application to soliton theory. The polynomial operators $V(q)$ have some problems due to infinities for $q \to \pm \infty$, but for $V(q)$ with fast convergence to zero there are hundreds papers. Especially simple construction may be found, e.g., in M. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, 1981.
The idea, that if you have some $L_1 = p^2 + u(q)$ with given spectrum, there is an elegant method to add yet another eigenvalue $\zeta$. You should consider equation $L_1 g = \zeta g$, find the solution $g(q)$. Then $L_2 = p^2 + u + 2 (\ln (g))''$ saves all eigenvalues of $L_1$, but also has new eigenvalue $\zeta$ with eigenfunction $1/g(q)$. So you may start with $u=0$ and add eigenvalues one after another.
Yet, as I mentioned here is a problem with polynomial case, it is one-dimensional and may be only useful thing is idea to search for trick like $g \to 1/g$ with adding eigenvalues...
[ADD] (1). I think, it is possible also to consider products $L_k(q_k)$, etc.
(2). It is not unique, e.g., isospectral deformation of $L$ is described by KdV equation (yet, it was also mentioned in a reference in Physics.SE thread)
A: Here's a new paper which may be related this problem, and appeared today on the archive:
Recovering the Hamiltonian from spectral data
Cyrille Heriveaux, Thierry Paul
http://arxiv.org/abs/1202.5102
