Alexander Turbiner
Solvability of
integrable systems: from geometry to algebra
Exact solutions of non-trivial Schroedinger equations are crucially important
for applications. Almost unique source of these solutions is integrable
Olshanetsky-Perelomov quantum Hamiltonians (rational and trigonometric)
emerging
in the Hamiltonian Reduction Method. As alternative a Lie-algebraic theory of
these solutions can be developed. It can be shown that all A-B-C-D
Olshanetsky-Perelomov Hamiltonians (rational and trigonometric) come from
a single quadratic polynomial in generators of the maximal affine subalgebra of
the gl(n)-algebra but unusually realized by differential operators.
The memory about A-B-C-D origin is kept in coefficients of the polynomial. For
the case of exceptional E Olshanetsky-Perelomov Hamiltonians unknown
infinite-dimensional
algebras admitting finite-dimensional irreps appear.
Lie-algebraic theory allows to construct the 'quasi-exactly-solvable'
generalizations of the above Hamiltonians where a finite number of eigenstates
is known exactly.
A general notion of (quasi)-exactly-solvable spectral problem is introduced.