Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Invariant subspaces of biconfluent Heun operators and special solutions
  of Painlev\'e IV   We show that there is a full correspondence between the parameters space of
the degenerate biconfluent Heun connection (BHC) and that of Painlev\'{e} IV
that admits special solutions. The BHC degenerates when either the Stokes' data
for the irregular singularity at $\infty$ degenerates or the regular singular
point at the origin becomes an apparent singularity. We show that if the BHC is
written as isomonodromy family of biconfluent Heun equations (BHE), then the
BHE degenerates precisely when it admits eigen-solutions of the biconfluent
Heun operators, after choosing appropriate accessory parameter, of specially
constructed invariant subspaces of finite dimensional solution spaces spanned
by parabolic cylinder functions. We have found all eigen-solutions over this
parameter space apart from three exceptional cases after choosing the right
accessory parameters. These eigen-solutions are expressed as certain finite sum
of parabolic cylinder functions. We extend the above sum to new convergent
series expansion in terms of parabolic cylinder functions to the BHE. The
infinite sum solutions of the BHE terminates precisely when the parameters of
the BHE assumes the same values as those of the degenerate biconfluent Heun
connection except at three instances after choosing the right accessory
parameter.

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The categories of this paper are: patt-sol
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The categories of this paper are: acc-phys
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The categories of this paper are: plasm-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: dg-ga
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The categories of this paper are: adap-org
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The categories of this paper are: solv-int
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The categories of this paper are: cond-mat
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The categories of this paper are: nucl-th
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The categories of this paper are: cmp-lg
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The categories of this paper are: quant-ph
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The categories of this paper are: nlin
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The categories of this paper are: alg-geom
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The categories of this paper are: physics
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The categories of this paper are: comp-gas
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The categories of this paper are: cs