About this Research Topic
Contact interactions, that is, potentials with support on a set of dimension lower than the dimension of the ambient space, have long been a source of important exactly solvable models in quantum mechanics. These models do not only allow us to develop new methods and investigate the foundations of the theory, but also have a wide range of applications, such as in atomic physics (e.g., the Lieb-Liniger and Tonks-Girardeau gases), graph theory, Casimir effect, and so on. As typical examples of these interactions, we mention Dirac deltas, finite or infinite combinations of Dirac deltas or its derivatives, and other types of interactions supported on more general curves, surfaces, and manifolds.
Mathematically, contact interactions are singular, since they are supported on sets of zero Lebesgue measure. Therefore, to unambiguously define a contact interaction it is necessary to establish the mathematical framework used to address the singularities. One method commonly used in the physics literature, inherited from quantum field theory, is to define singular potentials by means of a regularization procedure (often accompanied by renormalization), that is, as the limit of a sequence of regular functions converging in some sense to the singular potential. The regularization method has been particularly useful in investigating the limit of square potentials and the corresponding point interactions, both in non-relativistic and relativistic quantum mechanics.
Another, mathematically rigorous, method to properly define a singular interaction, is by using the theory of self-adjoint extensions (SAE) of symmetric operators. In this case, the self-adjoint Hamiltonian is usually completely defined by specifying the boundary conditions that the wave function must satisfy at the borders of the singularity. The SAE method has been used in a wide variety of models and applications.
Yet another approach to define singular interactions is by acknowledging that in this case the wave function should be considered as a distribution and, thus, the full apparatus of the distribution theory must be put to use to interpret the product of the potential and the wave function in the Schrödinger (or Dirac) wave equation. This approach has been recently used to investigate general point interactions.
This Research Topic is devoted to the study of singular interactions in quantum mechanics and welcomes contributions addressing the theoretical and mathematical aspects of these interactions in one, two or three dimensions, including regularization and renormalization, self-adjoint extensions, distribution theory or other mathematical methods, as well as contributions addressing physical models and applications. The Research Topic also welcomes contributions investigating more general interactions that, while not contact interactions, have singularities on a set of zero Lebesgue measure. Finally, while the focus of the Research Topic is on original contributions, review articles may be considered.
Mathematically, contact interactions are singular, since they are supported on sets of zero Lebesgue measure. Therefore, to unambiguously define a contact interaction it is necessary to establish the mathematical framework used to address the singularities. One method commonly used in the physics literature, inherited from quantum field theory, is to define singular potentials by means of a regularization procedure (often accompanied by renormalization), that is, as the limit of a sequence of regular functions converging in some sense to the singular potential. The regularization method has been particularly useful in investigating the limit of square potentials and the corresponding point interactions, both in non-relativistic and relativistic quantum mechanics.
Another, mathematically rigorous, method to properly define a singular interaction, is by using the theory of self-adjoint extensions (SAE) of symmetric operators. In this case, the self-adjoint Hamiltonian is usually completely defined by specifying the boundary conditions that the wave function must satisfy at the borders of the singularity. The SAE method has been used in a wide variety of models and applications.
Yet another approach to define singular interactions is by acknowledging that in this case the wave function should be considered as a distribution and, thus, the full apparatus of the distribution theory must be put to use to interpret the product of the potential and the wave function in the Schrödinger (or Dirac) wave equation. This approach has been recently used to investigate general point interactions.
This Research Topic is devoted to the study of singular interactions in quantum mechanics and welcomes contributions addressing the theoretical and mathematical aspects of these interactions in one, two or three dimensions, including regularization and renormalization, self-adjoint extensions, distribution theory or other mathematical methods, as well as contributions addressing physical models and applications. The Research Topic also welcomes contributions investigating more general interactions that, while not contact interactions, have singularities on a set of zero Lebesgue measure. Finally, while the focus of the Research Topic is on original contributions, review articles may be considered.
Keywords: Quantum Mechanics, Contact interactions, Singular potentials, Regularization and renormalization techniques, Low dimensional models
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