Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Singular traces and perturbation formulae of higher order   Let $H, V$ be self-adjoint operators such that $V$ belongs to the weak trace
class ideal. We prove higher order perturbation formula
$$\tau\big(f(H+V)-\sum_{j=0}^{n-1}\frac{1}{j!}\frac{d^j}{dt^j}
f(H+tV)\big|_{t=0}\big)=\int_{\mathbb{R}} f^{(n)}(t)\,dm_n(t),$$ where $\tau$
is a trace on the weak trace class ideal and $m_n$ is a finite measure that is
not necessarily absolutely continuous. This result extends the first and second
order perturbation formulas of Dykema and Shripka, who generalised the Krein
and Koplienko trace formulas to the weak trace class ideal. We also establish
the perturbation formulae when the perturbation $V$ belongs to the quasi-Banach
ideal weak-$L_n$ for any $n \geq 1$.

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The categories of this paper are: comp-gas
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The categories of this paper are: math-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: plasm-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: nlin
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The categories of this paper are: eess
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The categories of this paper are: econ
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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: adap-org
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The categories of this paper are: solv-int
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The categories of this paper are: hep-ex
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The categories of this paper are: physics
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The categories of this paper are: cond-mat
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The categories of this paper are: q-alg