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Compressed modes for variational problems in mathematics and physics. (English) Zbl 1292.81024
Summary: (Significance). Intuition suggests that many interesting phenomena in physics, chemistry, and materials science are “short-sighted” – that is, perturbation in a small spatial region only affects its immediate surroundings. In mathematical terms, near-sightedness is described by functions of finite range. As an example, the so-called Wannier functions in quantum mechanics are localized functions, which contain all the information about the properties of the system, including its spectral properties. This work’s main research objective is to develop theory and numerical methods that can systematically derive functions that span the energy spectrum of a given quantum-mechanical system and are nonzero only in a finite spatial region. These ideas hold the key for developing efficient methods for solving partial differential equations of mathematical physics.

MSC:
81P68 Quantum computation
35Q40 PDEs in connection with quantum mechanics
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