Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Boundary pairs associated with quadratic forms   We introduce an abstract framework for elliptic boundary value problems in a
variational form. Given a non-negative quadratic form in a Hilbert space, a
boundary pair consists of a bounded operator, the boundary operator, and an
auxiliary Hilbert space, the boundary space, such that the boundary operator is
bounded from the quadratic form domain into the auxiliary Hilbert space. These
data determine a Neumann and Dirichlet operator, a Dirichlet solution and a
Dirichlet-to-Neumann operator. The basic example we have in mind is a manifold
with boundary, where the quadratic form is the integral over the squared
derivative, and the boundary map is the restriction of a function to (a subset
of) the boundary of the manifold. As one of the main theorems, we derive a
resolvent formula relating the difference of the resolvents of the Neumann and
Dirichlet operator with the Dirichlet solution and the Dirichlet-to-Neumann
operator. From this, we deduce a spectral characterisation for a point being in
the spectrum of the Neumann operator in terms of the family of
Dirichlet-to-Neumann operators. The relation can be generalised also to
Robin-type boundary conditions. We identify conditions expressed purely in
terms of boundary pairs, which allow us to relate our concept with existing
concepts such as boundary triples. We illustrate the theory by many examples
including Jacobi operators, Laplacians on spaces with (non-smooth) boundary and
the Zaremba (mixed boundary conditions) problem.

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The categories of this paper are: hep-th
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The categories of this paper are: funct-an
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The categories of this paper are: nucl-th
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The categories of this paper are: nucl-ex
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The categories of this paper are: chao-dyn
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The categories of this paper are: supr-con
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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: hep-ex
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The categories of this paper are: bayes-an
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The categories of this paper are: q-fin
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The categories of this paper are: alg-geom
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-ph
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The categories of this paper are: q-bio
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The categories of this paper are: cs