A number of longstanding open problems in mathematical physics, such as the Haldane conjecture or the 2D AKLT gap conjecture, concern spectral gaps of particular many-body models. In the related setting of quantum field theories, determining if Yang-Mills theory is gapped is one of the Millennium Prize Problems. Numerous important results about many-body Hamiltonians only apply to gapped systems. Determining when a model is gapped or gapless is therefore one of the primary goals of theoretical condensed matter physics.
The motivation came from quantum mechanics. Eugene Wigner had the idea that spectra of large atoms could be modeled by random Hamiltonian matrices. This is a very active field connected with some of the most challenging open problems in mathematical physics and in pure mathematics.
open problems in mathematical physics [Si]
Btw, one might like to visit the website which contains a collection of open problems in mathematical physics. Coming to a problem related to this post (though somewhat remotely), one might be surprised to find through this link that the perturbative result of O(N > 2) model in d=2 being always in a single disordered phase at all temperatures has actually not been rigourosly proved. Infact, it is even disputed by some numerical results (though I don’t claim any understanding of these). See for example, .