• Corpus ID: 219401589

# Computing Scattering Resonances

@article{BenArtzi2020ComputingSR,
title={Computing Scattering Resonances},
author={Jonathan Ben-Artzi and Marco Marletta and Frank Rosler},
journal={arXiv: Spectral Theory},
year={2020}
}
• Published 5 June 2020
• Mathematics, Computer Science
• arXiv: Spectral Theory
The question of whether it is possible to compute scattering resonances of Schrodinger operators - independently of the particular potential - is addressed. A positive answer is given, and it is shown that the only information required to be known a priori is the size of the support of the potential. The potential itself is merely required to be $\mathcal{C}^1$. The proof is constructive, providing a universal algorithm which only needs to access the values of the potential at any requested…
9 Citations

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## References

SHOWING 1-10 OF 28 REFERENCES
On the Solvability Complexity Index for Unbounded Selfadjoint and Schrödinger Operators
We study the solvability complexity index (SCI) for unbounded selfadjoint operators on separable Hilbert spaces and perturbations thereof. In particular, we show that if the extended essential
RESONANCES IN PHYSICS AND GEOMETRY
R esonances are most readily associated with musical instruments or with the Tacoma bridge disaster. The latter is described in many physics and ODE books, and at the Ontario Science Center one can
On the Solvability Complexity Index Hierarchy and Towers of Algorithms
• Mathematics, Computer Science
• 2015
The Solvability Complexity Index (SCI) hierarchy is established, yielding a classification hierarchy for all types of problems in computational mathematics that determines the boundaries of what computers can achieve in scientific computing.
Solving the quintic by iteration
• Mathematics
• 1989
Equations that can be solved using iterated rational maps are characterized: an equation is ‘computable’ if and only if its Galois group is within A5 of solvable. We give explicitly a new solution to
Energies and widths of quasibound levels (orbiting resonances) for spherical potentials
• Physics
• 1978
Various methods for calculating the energies and widths of quasibound levels (orbiting or shape resonances) for spherical potentials are critically compared. A derivation for the previously‐proposed
Guaranteed resonance enclosures and exclosures for atoms and molecules
• Physics
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
• 2014
This paper confirms, with absolute certainty, a conjecture on a certain oscillatory behaviour of higher auto-ionizing resonances of atoms and molecules beyond a threshold, and allows one, for the first time, to enclose and to exclude resonances with guaranteed certainty.
Scattering Theory
Quantum Theory of Scattering Processes.(International Encyclopedia of Physical Chemistry and Chemical Physics.) By J. E. G. Farina. Pp. xi + 152. (Pergamon: Oxford and New York, February 1973.) £4.50.