Existence and asymptotic behaviour of weakly bounded states
| školitel: | prof Dirk Hundertmark, RNDr. Ing. Michal Jex, Ph.D. |
| e-mail: | zobrazit e-mail |
| typ práce: | dizertační práce |
| zaměření: | MI_MM |
| popis: | Stability of matter is one of the fundamental properties that governs the physics in our universe. On the microscopic scale the stability of quantum systems is directly related to the existence of eigenstates. The situation for eigenstates below the threshold of the essential spectrum is well studied. They exhibit exponential decay, their existence is linked to an energy gap, and they are stable under small perturbations of the system. However, the situation at the threshold of the essential spectrum is much more subtle. The existence of true bound states at this spectral phase transition is strongly correlated with the asymptotic behaviour of the potential part of the self-adjoint operator describing the quantum system. Moreover, these states are highly unstable under even small perturbations of the system. This makes it a challenging problem to tackle within the field of quantum mechanics and spectral theory. The objective of this PhD thesis is to deepen the understanding of weakly bounded states. The starting point will be recently developed method [HJL23a] which should be applied to more complicated systems and extended to other operators apart from standard Schrödinger operator. Especially useful would-be generalisation of the existence results for discrete and continuous Schrödinger operators [HJL23b, JŠ23] to Dirac operators. This would open a path to experimental verification of the properties of weakly bounded states on graphene-like structures for relativistic quantum mechanics setting. |
| literatura: | [HJL23a] D. Hundertmark, M. Jex, and M. Lange. „Quantum Systems at the Brink“. In: Springer INdAM Series. Springer Nature Singapore, 2023, pp. 259–273. ISBN: 9789819958948. DOI: 10.1007/978-981-99-5894-8_10. [HJL23b] D. Hundertmark, M. Jex, and M. Lange. „Quantum systems at the brink: existence of bound states, critical potentials, and dimensionality“. In: Forum of Mathematics, Sigma 11 (2023), e61. DOI: 10.1017/fms.2023.39. [JŠ23] M. Jex and F. Štampach. „On the ground state of lattice Schrödinger operators“. In: arXiv: 2312.08081 (2023), p. 20. |
| naposledy změněno: | 11.12.2024 10:03:28 |
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Ľubomíra Dvořáková | naposledy změněno: 12.9.2011