Many-Particle Quantum Graphs: A Review

Bölte, Jens; Kerner, Joachim

doi:10.1007/978-3-030-44097-8_2

Cited by 1 publication

(2 citation statements)

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“…At the beginning of this section, we derive an explicit expression of the resolvent, λ / ∈ σ(−∆ ρ ), (12) R ρ (λ) := (−∆ ρ − λ) −1 .…”

Section: The Resolvent Kernelmentioning

confidence: 99%

“…The motivation of studying contact interactions in many-particle physics for one-dimensional systems dates back to the famous Lieb-Liniger model, [38], used to test Bogoliubov's perturbation theory for Bose gases. Since then manyparticle contact interactions are well established in one dimensions and graphs for testing and modeling famous phenomenons such as superconductivity or Bose-Einstein condensation, [20,51,11,10,30,12,18]. For the importance of one-dimensional contact interactions in physics and recent experimental implementations we refer to [14].…”

Section: Introductionmentioning

confidence: 99%

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An asymptotic expansion of the trace of the heat kernel of a singular two-particle contact interaction in one dimension

Egger¹

2018

Preprint

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The regularized trace of the heat kernel of a one-dimensional Schrödinger operator with a singular two-particle contact interaction being of Lieb-Liniger type is considered. We derive a complete small-time asymptotic expansion in (fractional) powers of the time, t. Most importantly, we do not invoke standard parametrix constructions for the heat kernel. Instead, we first derive the large-energy expansion of the regularized trace of the resolvent for the considered operator. Then, we exploit that the resolvent may be obtained by a Laplace transformation of the heat semi-group, and an application of a suitable inverse Watson lemma eventually yields the small-t asymptotic expansion of the heat-kernel trace.

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“…At the beginning of this section, we derive an explicit expression of the resolvent, λ / ∈ σ(−∆ ρ ), (12) R ρ (λ) := (−∆ ρ − λ) −1 .…”

Section: The Resolvent Kernelmentioning

confidence: 99%