Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Inequalities between Neumann and Dirichlet eigenvalues of Schr\"odinger
  operators   Given a Schr\"odinger operator with a real-valued potential on a bounded,
convex domain or a bounded interval we prove inequalities between the
eigenvalues corresponding to Neumann and Dirichlet boundary conditions,
respectively. The obtained inequalities depend partially on monotonicity and
convexity properties of the potential. The results are counterparts of
classical inequalities for the Laplacian but display some distinction between
the one-dimensional case and higher dimensions.

30
The categories of this paper are: q-alg
26
The categories of this paper are: bayes-an
4
The categories of this paper are: cond-mat
21
The categories of this paper are: adap-org
22
The categories of this paper are: chao-dyn
16
The categories of this paper are: stat
6
The categories of this paper are: astro-ph
24
The categories of this paper are: dg-ga
0
The categories of this paper are: hep-ph
12
The categories of this paper are: nucl-th
23
The categories of this paper are: solv-int
17
The categories of this paper are: q-fin
35
The categories of this paper are: mtrl-th
7
The categories of this paper are: hep-th
13
The categories of this paper are: quant-ph
31
The categories of this paper are: ao-sci