Andreas Alexander Buchheit

• Dr. rer. nat
• Postdoctoral Researcher at Saarland University

The Epstein zeta function generalizes the Riemann zeta function to oscillatory lattice sums in higher dimensions. Beyond its numerous applications in pure mathematics, it has recently been identified as a key component in simulating exotic quantum materials. This work establishes the Epstein zeta function as a powerful tool in numerical analysis by...

This paper introduces a new method for the efficient computation of oscillatory multidimensional lattice sums in geometries with boundaries. Such sums are ubiquitous in both pure and applied mathematics, and have immediate applications in condensed matter physics and topological quantum physics. The challenge in their evaluation results from the co...

Continuum limits are a powerful tool in the study of many-body systems, yet their validity is often unclear when long-range interactions are present. In this paper, we rigorously address this issue and put forth an exact representation of long-range interacting lattices that separates the model into a term describing its continuous analog, the inte...

Continuum limits are a powerful tool in the study of many-body systems, yet their validity is often unclear when long-range interactions are present. In this work, we rigorously address this issue and put forth an exact representation of long-range interacting lattices that separates the model into a term describing its continuous analog, the integ...

We extend the classical Euler–Maclaurin (EM) expansion to sums over multidimensional lattices that involve functions with algebraic singularities. This offers a tool for a precise and fast evaluation of singular sums that appear in multidimensional long-range interacting systems. We find that the approximation error decays exponentially with the ex...

Continuum limits are a powerful tool in the study of many-body systems, yet their validity is often unclear when long-range interactions are present. In this work, we rigorously address this issue and put forth an exact representation of long-range interacting lattices that separates the model into a term describing its continuous analogue, the int...

We develop a new expansion for representing singular sums in terms of integrals and vice versa. This method provides a powerful tool for the efficient computation of large singular sums that appear in long-range interacting systems in condensed matter and quantum physics. It also offers a generalised trapezoidal rule for the precise computation of...

This thesis is concerned with the development of the singular Euler–Maclaurin expansion, a novel method that allows for the efficient evaluation of large sums over values of functions with singularities. The method offers an approximation to the sum whose runtime is independent of the number of summands and whose error falls of exponentially with t...

We extend the classical Euler-Maclaurin expansion to sums over multidimensional lattices that involve functions with algebraic singularities. This offers a tool for the precise quantification of the effect of microscopic discreteness on macroscopic properties of a system. First, the Euler-Maclaurin summation formula is generalised to lattices in hi...

We generalise the Euler-Maclaurin expansion and make it applicable to the product of a differentiable function and an asymptotically smooth singularity. The difference between sum and integral is written as a differential operator acting on the non-singular factor only plus a remainder integral. The singularity can be included in generalised Bernou...

The classical ground state of the Frenkel–Kontorova model is investigated for the case of a globally deformable substrate potential, whose amplitude is a function of the position of all particles. This system serves as an idealised description of a one-dimensional crystal of trapped ions in an optical cavity, where the deformable potential originat...

The progress in high-precision spectroscopy requires one to check the accuracy of theoretical models such as the master equation describing spontaneous emission of atoms. For this purpose, we systematically derive a master equation of an atom interacting with the modes of the electromagnetic field which naturally includes interference in the decay...

The progress in high-precision spectroscopy requires one to verify the accuracy of theoretical models such as the master equation describing spontaneous emission of atoms. For this purpose, we apply the coarse-graining method to derive a master equation of an atom interacting with the modes of the electromagnetic field. This master equation natural...