We construct in this article a class of closed semi-bounded quadratic forms on the space of square integrable functions over a smooth Riemannian manifold with smooth compact boundary. Each of these quadratic forms specifies a semibounded self-adjoint extension of the Laplace-Beltrami operator. These quadratic forms are based on the Lagrange boundary form on the manifold and a family of domains parametrized by a suitable class of unitary operators on the boundary that will be called admissible. The corresponding quadratic forms are semi-bounded below and closable. Finally, the representing operators correspond to semi-bounded self-adjoint extensions of the Laplace-Beltrami operator. This family of extensions is compared with results existing in ✩ 635 Boundary conditions the literature and various examples and applications are discussed.

The dynamics of the magnetic field in a superconducting phase is described by an effective massive bosonic field theory. If the superconductor is confined in a domain M with boundary ∂M , the boundary conditions of the electromagnetic fields are predetermined by physics. They are time-reversal and also parity invariant for adapted geometry. They lead to edge excitations while in comparison, the bulk energies have a large gap. A similar phenomenon occurs for topological insulators where appropriate boundary conditions for the Dirac Hamiltonian also lead to similar edge states and an "incompressible bulk". They give spin-momentum locking as well. In addition time-reversal and parity invariance emerge for adapted geometry. Similar edge states appear in QCD bag models with chiral boundary conditions.

Given a unitary representation of a Lie group G on a Hilbert space H, we develop the theory of G-invariant self-adjoint extensions of symmetric operators both using von Neumann's theorem and the theory of quadratic forms. We also analyze the relation between the reduction theory of the unitary representation and the reduction of the G-invariant unbounded operator. We also prove a G-invariant version of the representation theorem for quadratic forms.The previous results are applied to the study of G-invariant self-adjoint extensions of the Laplace-Beltrami operator on a smooth Riemannian manifold with boundary on which the group G acts. These extensions are labeled by admissible unitaries U acting on the L 2 -space at the boundary and having spectral gap at −1. It is shown that if the unitary representation V of the symmetry group G is traceable, then the self-adjoint extension of the Laplace-Beltrami operator determined by U is G-invariant if U and V commute at the boundary. Various significant examples are discussed at the end.

The search for a potential function S allowing us to reconstruct a given metric\ud tensor g and a given symmetric covariant tensor T on a manifold M is formulated\ud as the Hamilton-Jacobi problem associated with a canonically defined\ud Lagrangian on TM. The connection between this problem, the geometric structure\ud of the space of pure states of quantum mechanics, and the theory of contrast functions\ud of classical information geometry are outlined

We analyse the effects of Robin-like boundary conditions on different quantum field theories of spin 0, 1/2 and 1 on manifolds with boundaries. In particular, we show that these conditions often lead to the appearance of edge states. These states play a significant role in physical phenomena like quantum Hall effect and topological insulators. We prove in a rigorous way the existence of spectral lower bounds on the kinetic term of different Hamiltonians, even in the case of abelian gauge fields where it is a non-elliptic differential operator. This guarantees the stability and consistency of massive field theories with masses larger than the lower bound of the kinetic term. Moreover, we find an upper bound for the deepest edge state. In the case of Abelian gauge theories we analyse a generalisation of Robin boundary conditions. For Dirac fermions we analyse the cases of Atiyah-Patodi-Singer and chiral bag boundary conditions. The explicit dependence of the bounds on the boundary conditions and the size of the system is derived under general assumptions.

A numerical algorithm to solve the spectral problem for arbitrary self-adjoint extensions of one-dimensional regular Schrödinger operators is presented. It is shown that the set of all self-adjoint extensions of one-dimensional regular Schrödinger operators is in one-to-one correspondence with the group of unitary operators on the finite-dimensional Hilbert space of boundary data, and they are characterized by a generalized class of boundary conditions that include the well-known Dirichlet, Neumann, Robin, and (quasi-)periodic boundary conditions. The numerical algorithm is based on a nonlocal boundary extension of the finite element method and their convergence is proved. An appropriate basis of boundary functions must be introduced to deal with arbitrary boundary conditions and the conditioning of its computation is analyzed. Some significant numerical experiments are also discussed as well as the comparison with some standard algorithms. In particular it is shown that appropriate perturbations of standard boundary conditions for the free particle lead to the theoretically predicted result of very large absolute values of the negative groundlevels of the system as well as the localization of the corresponding eigenvectors at the boundary (edge states). Introduction. The study of the self-adjointness of Schrödinger operators hasbeen a fundamental mathematical problem since the beginning of quantum mechanics and there is a vast literature on the subject. (See, for instance, [Re75], the review [Si00], and references therein.) In spite of this, there is a continuous flow of new results and even surprises. (See, for instance, the recent papers where some apparent paradoxical aspects of the spectrum of certain self-adjoint extensions of the Schrödinger operator in two dimensions are analyzed [Be08], [Ma09], [Be09].)Consider the evolution of a quantum system on a D-dimensional Riemannian manifold M with boundary ∂M under the influence of a potential V which is given by the Schrödinger equation i ∂Ψ ∂t = HΨ, with H the Hamiltonian operator of the system given by

Abstract. This is a series of 5 lectures around the common subject of the construction of self-adjoint extensions of symmetric operators and its applications to Quantum Physics. We will try to offer a brief account of some recent ideas in the theory of self-adjoint extensions of symmetric operators on Hilbert spaces and their applications to a few specific problems in Quantum Mechanics.

We show how to use boundary conditions to drive the evolution on a Quantum Mechanical system. We will see how this problem can be expressed in terms of a time-dependent Schr\"{o}dinger equation. In particular we will need the theory of self-adjoint extensions of differential operators in manifolds with boundary. An introduction of the latter as well as meaningful examples will be given. It is known that different boundary conditions can be used to describe different topologies of the associated quantum systems. We will use the previous results to study how this topology change can be accomplished in a dynamical way