# Analytically solve the eigenvalue problem with infinite dimensions by Mathematica?

If I am given a symbolic expression of all the matrix elements in an infinite-dimensional space, e.g., the Hamiltonian of a quantum mechanical system, is it possible to get the symbolic expression for all of the eigenvalues by Mathematica?

One of the simplest case is the Harmonic Oscillator in quantum mechanics, where the elements of Hamiltonian matrix can be written as:

$$H_{mn}=(n+\frac{1}{2})\delta_{mn}$$

where we all know that the eigenvalues are: $$E_n=n+\frac{1}{2},( n=0,1,2,3,...)$$ but Mathematica seems not having any functions to find the eigenvalues given such conditions, since the dimension of the matrix is infinity.

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Mathematica doesn't really have built-in support for operators, as opposed to finite sections of them... if you don't mind numerical solutions, you can, however, take eigenvalues of finite sections of the operators, and use SequenceLimit[] on a sequence of approximations. – J. M. Jun 1 '13 at 16:35
Thanks! But I wonder if it's possible to write a program to cope with such algebraic problems, I tried to program with Mathematica but failed at calculating the infinite-dimensional determinant... – REX Jun 1 '13 at 18:58
If you can write the Hamiltonian in terms of the generators of some Lie group, you could try converting to a differential equation and solving that instead. But the chances of getting an analytical solution depend very much on the details of your actual problem. – Jens Jun 1 '13 at 20:10