Periodic Reporting for period 1 - COCONUT (Computational Complexity in Quantum Mechanics)
Berichtszeitraum: 2020-10-01 bis 2022-09-30
This project studies computational problems in quantum mechanics in the framework of the Solvability Complexity (SCI) Hierarchy. The SCI Hierarchy is a novel theory concerned with the fundamental boundaries of computability. It started out as a way to rigorously track the "number of successive limits" that are needed to solve a computational problem using a computer algorithm. Since the beginning of the project, the field has evolved into a sophisticated and comprehensive classification hierarchy that combines information about solvability, convergence and error estimation. While the theoretical concepts are deliberately kept generic enough to encompass almost any computational problem in mathematics, the main focus of applications so far has been in the mathematical fields of spectral theory and partial differential equations.
These equations are of particular relevance in Quantum Mechanics, where they describe the time evolution of particle states. The famous Schrödinger equation forms the basis of modern quantum mechanics. It describes nonrelativistic particles in a force field described by a so-called potential function V. As a prime example, the reader may consider the movement of an electron in the electric field of a proton. Together, the two particles form a hydrogen atom, which can have different states of excitation (depending on the amount of energy the system absorbs from its environment). These excited states are discrete (or quantised) - they only occur at very specific energies. The Schrödinger equation is able to correctly predict these energies: they are given by the so-called eigenvalues (or the spectrum) of the equation. Applications of these quantised energy states are vast: for example, they form the basis for the entire field of spectroscopy - a branch of Physics, which allows us to learn what the stars in our galaxy are made of, or to determine the chemical composition of our atmosphere via its absorption of sunlight. Each chemical element has its own energy spectrum, which identifies it like a fingerprint.
These equations are of particular relevance in Quantum Mechanics, where they describe the time evolution of particle states. The famous Schrödinger equation forms the basis of modern quantum mechanics. It describes nonrelativistic particles in a force field described by a so-called potential function V. As a prime example, the reader may consider the movement of an electron in the electric field of a proton. Together, the two particles form a hydrogen atom, which can have different states of excitation (depending on the amount of energy the system absorbs from its environment). These excited states are discrete (or quantised) - they only occur at very specific energies. The Schrödinger equation is able to correctly predict these energies: they are given by the so-called eigenvalues (or the spectrum) of the equation. Applications of these quantised energy states are vast: for example, they form the basis for the entire field of spectroscopy - a branch of Physics, which allows us to learn what the stars in our galaxy are made of, or to determine the chemical composition of our atmosphere via its absorption of sunlight. Each chemical element has its own energy spectrum, which identifies it like a fingerprint.
The computer algorithms that we developed as part of the MSCA project allow the reliable computation of the Schrödinger equation's spectrum in a wide range of physical contexts. More precisely, we obtained nontrivial results in three areas:
WP1: nonrelativistic quantum mechanics,
WP2: scattering resonances,
WP3: relativistic quantum mechanics.
The work carried out throughout the project is as follows:
WP1:
- Objective: Study the computability of spectra of periodic Schrödinger operators.
Result: The computational problem can be solved in one limit and practical implementations are possible.
Publication: Universal Algorithms for Computing Spectra of Periodic Operators (published in Numerische Mathematik volume 150, pages 719–767 (2022))
Blog Post: https://frank-roesler.github.io/research_periodic/
- Objective: Study the computational complexity of inverse spectral problems (i.e. given the spectrum, compute the potential function).
Result: The computational problem can be solved in at most one limit (in certain circumstances even in finitely many arithmetic operations). Practical implementations are demonstrated to be feasible.
Publication: Universal algorithms for solving inverse spectral problems (preprint: arXiv:2203.13078)
Blog Post: https://frank-roesler.github.io/research_inverse/
- Objective: Study the influence of domain regularity on computability, i.e. if the boundary of the domain is "fuzzy", can we still compute eigenvalues on it?
Result: Under very general assumptions on the domain, reliable computation of eigenvalues is still possible. Examples and implementations are provided.
Publication: Computing Eigenvalues of the Laplacian on Rough Domains (accepted for publication in Mathematics of Computation. preprint: arXiv:2104.09444)
Blog Post: https://frank-roesler.github.io/research_rough_domains/
WP2:
- Objective: Study the computability of scattering resonances in quantum mechanics.
Result: Scattering resonances can be reliably computed in one limit. An explicit algorithm is provided, alongside an implementation.
Publication: Computing scattering resonances (published in the Journal of the European Mathematical Society, DOI 10.4171/JEMS/1258 (2022))
Blog Post: https://frank-roesler.github.io/research_resonances/
- Objective: Study the computational complexity of scattering problems in so-called Helmholtz resonator chambers.
Result: Helmholtz resonances can be computed reliably in one limit for resonator chambers of arbitrary shape. Implementation is demonstrably feasible.
Publication: Computing the Sound of the Sea in a Seashell (published in Foundations of Computational Mathematics volume 22, pages 697–731 (2022))
Blog Post: https://frank-roesler.github.io/research_helmholtz/
WP3:
- Objective: Study the computational complexity of the eigenvalue problem for the Klein-Gordon Equation (which describes relativistic quantum particles).
Result: Under suitable assumptions on the potential, the eigenvalues of the Klein-Gordon Equation can be computed in one limit and certain error bounds are possible. A computer algorithm, together with an implementation are provided.
Publication: Computing Klein-Gordon Eigenvalues (preprint: arXiv:2210.12516)
Blog Post: https://frank-roesler.github.io/research_kg/
WP1: nonrelativistic quantum mechanics,
WP2: scattering resonances,
WP3: relativistic quantum mechanics.
The work carried out throughout the project is as follows:
WP1:
- Objective: Study the computability of spectra of periodic Schrödinger operators.
Result: The computational problem can be solved in one limit and practical implementations are possible.
Publication: Universal Algorithms for Computing Spectra of Periodic Operators (published in Numerische Mathematik volume 150, pages 719–767 (2022))
Blog Post: https://frank-roesler.github.io/research_periodic/
- Objective: Study the computational complexity of inverse spectral problems (i.e. given the spectrum, compute the potential function).
Result: The computational problem can be solved in at most one limit (in certain circumstances even in finitely many arithmetic operations). Practical implementations are demonstrated to be feasible.
Publication: Universal algorithms for solving inverse spectral problems (preprint: arXiv:2203.13078)
Blog Post: https://frank-roesler.github.io/research_inverse/
- Objective: Study the influence of domain regularity on computability, i.e. if the boundary of the domain is "fuzzy", can we still compute eigenvalues on it?
Result: Under very general assumptions on the domain, reliable computation of eigenvalues is still possible. Examples and implementations are provided.
Publication: Computing Eigenvalues of the Laplacian on Rough Domains (accepted for publication in Mathematics of Computation. preprint: arXiv:2104.09444)
Blog Post: https://frank-roesler.github.io/research_rough_domains/
WP2:
- Objective: Study the computability of scattering resonances in quantum mechanics.
Result: Scattering resonances can be reliably computed in one limit. An explicit algorithm is provided, alongside an implementation.
Publication: Computing scattering resonances (published in the Journal of the European Mathematical Society, DOI 10.4171/JEMS/1258 (2022))
Blog Post: https://frank-roesler.github.io/research_resonances/
- Objective: Study the computational complexity of scattering problems in so-called Helmholtz resonator chambers.
Result: Helmholtz resonances can be computed reliably in one limit for resonator chambers of arbitrary shape. Implementation is demonstrably feasible.
Publication: Computing the Sound of the Sea in a Seashell (published in Foundations of Computational Mathematics volume 22, pages 697–731 (2022))
Blog Post: https://frank-roesler.github.io/research_helmholtz/
WP3:
- Objective: Study the computational complexity of the eigenvalue problem for the Klein-Gordon Equation (which describes relativistic quantum particles).
Result: Under suitable assumptions on the potential, the eigenvalues of the Klein-Gordon Equation can be computed in one limit and certain error bounds are possible. A computer algorithm, together with an implementation are provided.
Publication: Computing Klein-Gordon Eigenvalues (preprint: arXiv:2210.12516)
Blog Post: https://frank-roesler.github.io/research_kg/
The impact of the project is twofold:
First, it has significantly advanced our understanding of the fundamental limits of computability in practically relevant contexts. In particular, we have identified previously unknown classes of computational problems that can be reliably solved on a computer. Some of these results are surprising: in one of our more recent studies we discovered that inverse spectral problems are no more difficult to solve than classical (forward) spectral problems. A fact that is counterintuitive to most mathematicians.
Second, the numerical algorithms we developed and implemented will allow physicists, chemists and computer scientists to perform reliable computations in quantum mechanical applications and further advance our understanding of quantum mechanics itself.
First, it has significantly advanced our understanding of the fundamental limits of computability in practically relevant contexts. In particular, we have identified previously unknown classes of computational problems that can be reliably solved on a computer. Some of these results are surprising: in one of our more recent studies we discovered that inverse spectral problems are no more difficult to solve than classical (forward) spectral problems. A fact that is counterintuitive to most mathematicians.
Second, the numerical algorithms we developed and implemented will allow physicists, chemists and computer scientists to perform reliable computations in quantum mechanical applications and further advance our understanding of quantum mechanics itself.