Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Quasi-exactly solvable Schr\"odinger equations, symmetric polynomials,
  and functional Bethe ansatz method   For applications to quasi-exactly solvable Schr\"odinger equations in quantum
mechanics, we consider the general conditions that have to be satisfied by the
coefficients of a second-order differential equation with at most $k+1$
singular points in order that this equation has particular solutions that are
$n$th-degree polynomials. In a first approach, we show that such conditions
involve $k-2$ integration constants, which satisfy a system of linear equations
whose coefficients can be written in terms of elementary symmetric polynomials
in the polynomial solution roots whenver such roots are all real and distinct.
In a second approach, we consider the functional Bethe ansatz method in its
most general form under the same assumption. Comparing the two approaches, we
prove that the above-mentioned $k-2$ integration constants can be expressed as
linear combinations of monomial symmetric polynomials in the roots, associated
with partitions into no more than two parts. We illustrate these results by
considering a quasi-exactly solvable extension of the Mathews-Lakshmanan
nonlinear oscillator corresponding to $k=4$.

1
The categories of this paper are: cs
34
The categories of this paper are: chem-ph
33
The categories of this paper are: plasm-ph
24
The categories of this paper are: dg-ga
4
The categories of this paper are: cond-mat
27
The categories of this paper are: comp-gas
23
The categories of this paper are: solv-int
32
The categories of this paper are: atom-ph
15
The categories of this paper are: nucl-ex
37
The categories of this paper are: supr-con
12
The categories of this paper are: nucl-th
35
The categories of this paper are: mtrl-th
31
The categories of this paper are: ao-sci
8
The categories of this paper are: hep-ex
6
The categories of this paper are: astro-ph
7
The categories of this paper are: hep-th