Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Quasi-exactly solvable quartic potential   A new two-parameter family of quasi-exactly solvable quartic polynomial
potentials $V(x)=-x^4+2iax^3+(a^2-2b)x^2+2i(ab-J)x$ is introduced. Until now,
it was believed that the lowest-degree one-dimensional quasi-exactly solvable
polynomial potential is sextic. This belief is based on the assumption that the
Hamiltonian must be Hermitian. However, it has recently been discovered that
there are huge classes of non-Hermitian, ${\cal PT}$-symmetric Hamiltonians
whose spectra are real, discrete, and bounded below [physics/9712001].
Replacing Hermiticity by the weaker condition of ${\cal PT}$ symmetry allows
for new kinds of quasi-exactly solvable theories. The spectra of this family of
quartic potentials discussed here are also real, discrete, and bounded below,
and the quasi-exact portion of the spectra consists of the lowest $J$
eigenvalues. These eigenvalues are the roots of a $J$th-degree polynomial.

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The categories of this paper are: cs
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The categories of this paper are: adap-org
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The categories of this paper are: econ
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The categories of this paper are: q-bio
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The categories of this paper are: solv-int
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The categories of this paper are: hep-lat
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The categories of this paper are: q-alg
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The categories of this paper are: astro-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: funct-an
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The categories of this paper are: hep-ex
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The categories of this paper are: nucl-ex
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The categories of this paper are: chem-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: gr-qc