@phdthesis{14374,
  abstract     = {Superconductivity has many important applications ranging from levitating trains over qubits to MRI scanners. The phenomenon is successfully modeled by Bardeen-Cooper-Schrieffer (BCS) theory. From a mathematical perspective, BCS theory has been studied extensively for systems without boundary. However, little is known in the presence of boundaries. With the help of numerical methods physicists observed that the critical temperature may increase in the presence of a boundary. The goal of this thesis is to understand the influence of boundaries on the critical temperature in BCS theory and to give a first rigorous justification of these observations. On the way, we also study two-body Schrödinger operators on domains with boundaries and prove additional results for superconductors without boundary.

BCS theory is based on a non-linear functional, where the minimizer indicates whether the system is superconducting or in the normal, non-superconducting state. By considering the Hessian of the BCS functional at the normal state, one can analyze whether the normal state is possibly a minimum of the BCS functional and estimate the critical temperature. The Hessian turns out to be a linear operator resembling a Schrödinger operator for two interacting particles, but with more complicated kinetic energy. As a first step, we study the two-body Schrödinger operator in the presence of boundaries.
For Neumann boundary conditions, we prove that the addition of a boundary can create new eigenvalues, which correspond to the two particles forming a bound state close to the boundary.

Second, we need to understand superconductivity in the translation invariant setting. While in three dimensions this has been extensively studied, there is no mathematical literature for the one and two dimensional cases. In dimensions one and two, we compute the weak coupling asymptotics of the critical temperature and the energy gap  in the translation invariant setting. We also prove that their ratio is independent of the microscopic details of the model in the weak coupling limit; this property is referred to as universality.

In the third part, we study the critical temperature of superconductors in the presence of boundaries. We start by considering the one-dimensional case of a half-line with contact interaction. Then, we generalize the results to generic interactions and half-spaces in one, two and three dimensions. Finally, we compare the critical temperature of a quarter space in two dimensions to the critical temperatures of a half-space and of the full space.},
  author       = {Roos, Barbara},
  issn         = {2663 - 337X},
  pages        = {206},
  publisher    = {Institute of Science and Technology Austria},
  title        = {{Boundary superconductivity in BCS theory}},
  doi          = {10.15479/at:ista:14374},
  year         = {2023},
}

@article{14542,
  abstract     = {It is a remarkable property of BCS theory that the ratio of the energy gap at zero temperature Ξ
 and the critical temperature Tc is (approximately) given by a universal constant, independent of the microscopic details of the fermionic interaction. This universality has rigorously been proven quite recently in three spatial dimensions and three different limiting regimes: weak coupling, low density and high density. The goal of this short note is to extend the universal behavior to lower dimensions d=1,2 and give an exemplary proof in the weak coupling limit.},
  author       = {Henheik, Sven Joscha and Lauritsen, Asbjørn Bækgaard and Roos, Barbara},
  issn         = {1793-6659},
  journal      = {Reviews in Mathematical Physics},
  publisher    = {World Scientific Publishing},
  title        = {{Universality in low-dimensional BCS theory}},
  doi          = {10.1142/s0129055x2360005x},
  year         = {2023},
}

@article{13207,
  abstract     = {We consider the linear BCS equation, determining the BCS critical temperature, in the presence of a boundary, where Dirichlet boundary conditions are imposed. In the one-dimensional case with point interactions, we prove that the critical temperature is strictly larger than the bulk value, at least at weak coupling. In particular, the Cooper-pair wave function localizes near the boundary, an effect that cannot be modeled by effective Neumann boundary conditions on the order parameter as often imposed in Ginzburg–Landau theory. We also show that the relative shift in critical temperature vanishes if the coupling constant either goes to zero or to infinity.},
  author       = {Hainzl, Christian and Roos, Barbara and Seiringer, Robert},
  issn         = {1664-0403},
  journal      = {Journal of Spectral Theory},
  number       = {4},
  pages        = {1507–1540},
  publisher    = {EMS Press},
  title        = {{Boundary superconductivity in the BCS model}},
  doi          = {10.4171/JST/439},
  volume       = {12},
  year         = {2023},
}

@article{10850,
  abstract     = {We study two interacting quantum particles forming a bound state in d-dimensional free
space, and constrain the particles in k directions to (0, ∞)k ×Rd−k, with Neumann boundary
conditions. First, we prove that the ground state energy strictly decreases upon going from k
to k+1. This shows that the particles stick to the corner where all boundary planes intersect.
Second, we show that for all k the resulting Hamiltonian, after removing the free part of the
kinetic energy, has only finitely many eigenvalues below the essential spectrum. This paper
generalizes the work of Egger, Kerner and Pankrashkin (J. Spectr. Theory 10(4):1413–1444,
2020) to dimensions d > 1.},
  author       = {Roos, Barbara and Seiringer, Robert},
  issn         = {0022-1236},
  journal      = {Journal of Functional Analysis},
  keywords     = {Analysis},
  number       = {12},
  publisher    = {Elsevier},
  title        = {{Two-particle bound states at interfaces and corners}},
  doi          = {10.1016/j.jfa.2022.109455},
  volume       = {282},
  year         = {2022},
}

@article{12276,
  abstract     = {Ongoing development of quantum simulators allows for a progressively finer degree of control of quantum many-body systems. This motivates the development of efficient approaches to facilitate the control of such systems and enable the preparation of nontrivial quantum states. Here we formulate an approach to control quantum systems based on matrix product states (MPSs). We compare counterdiabatic and leakage minimization approaches to the so-called local steering problem that consists in finding the best value of the control parameters for generating a unitary evolution of the specific MPS in a given direction. In order to benchmark the different approaches, we apply them to the generalization of the PXP model known to exhibit coherent quantum dynamics due to quantum many-body scars. We find that the leakage-based approach generally outperforms the counterdiabatic framework and use it to construct a Floquet model with quantum scars. We perform the first steps towards global trajectory optimization and demonstrate entanglement steering capabilities in the generalized PXP model. Finally, we apply our leakage minimization approach to construct quantum scars in the periodically driven nonintegrable Ising model.},
  author       = {Ljubotina, Marko and Roos, Barbara and Abanin, Dmitry A. and Serbyn, Maksym},
  issn         = {2691-3399},
  journal      = {PRX Quantum},
  keywords     = {General Medicine},
  number       = {3},
  publisher    = {American Physical Society},
  title        = {{Optimal steering of matrix product states and quantum many-body scars}},
  doi          = {10.1103/prxquantum.3.030343},
  volume       = {3},
  year         = {2022},
}