Self-adjoint Extensions of Restrictions (original) (raw)
Related papers
International Journal of Geometric Methods in Modern Physics, 2015
This is a series of five 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.
Self-adjoint extensions by additive perturbations
Let A N be the symmetric operator given by the restriction of A to N , where A is a self-adjoint operator on the Hilbert space H and N is a linear dense set which is closed with respect to the graph norm on D(A), the operator domain of A. We show that any self-adjoint extension A Θ of A N such that D(A Θ) ∩ D(A) = N can be additively decomposed by the sum A Θ =Ā + T Θ , where both the operatorsĀ and T Θ take values in the strong dual of D(A). The operatorĀ is the closed extension of A to the whole H whereas T Θ is explicitly written in terms of a (abstract) boundary condition depending on N and on the extension parameter Θ, a self-adjoint operator on an auxiliary Hilbert space isomorphic (as a set) to the deficiency spaces of A N. The explicit connection with both Kreȋn's resolvent formula and von Neumann's theory of self-adjoint extensions is given.
On semibounded restrictions of self-adjoint operators
Integral Equations and Operator Theory, 1998
After the yon Neumann's remark [t0] about ':pathologies" of unbounded symmetric operators and an abstract theorem about stability domain [9], we devel-.ope here a general theory allowing to construct semibounded restrictions of selfadjoint operators in explidt form. We apply this theory to quantum-mechanical momentum (position) operator to describe corresponding stability domains. Generalization to the case of measurabIe functions of these operators is considered. In conclusion we discuss spectral properties of self-adjoint extensions of constructed self-adjoint restrictions.
On Self-Adjoint Extensions and Symmetries in Quantum Mechanics
Annales Henri Poincaré, 2014
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 using both 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 rep-resentation theorem for closed and semi-bounded quadratic forms. The previous results are applied to the study of G-invariant self-adjoint exten-sions 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.
Restrictions and Extensions of Semibounded Operators
Complex Analysis and Operator Theory, 2014
We study restriction and extension theory for semibounded Hermitian operators in the Hardy space H 2 of analytic functions on the disk D. Starting with the operator z d dz , we show that, for every choice of a closed subset F ⊂ T = ∂D of measure zero, there is a densely defined Hermitian restriction of z d dz corresponding to boundary functions vanishing on F. For every such restriction operator, we classify all its selfadjoint extension, and for each we present a complete spectral picture. We prove that different sets F with the same cardinality can lead to quite different boundary-value problems, inequivalent selfadjoint extension operators, and quite different spectral configurations. As a tool in our analysis, we prove that the von Neumann deficiency spaces, for a fixed set F , have a natural presentation as reproducing kernel Hilbert spaces, with a Hurwitz zeta-function, restricted to F × F , as reproducing kernel.
Triplet extensions I: Semibounded operators in the scale of Hilbert spaces
Journal d'Analyse Mathématique, 2009
The extension theory for semibounded symmetric operators is generalized by including operators acting in a triplet of Hilbert spaces. We concentrate our attention on the case where the minimal operator is essentially self-adjoint in the basic Hilbert space and construct a family of its self-adjoint extensions inside the triplet. All such extensions can be described by certain boundary conditions and a natural counterpart of Krein's resolvent formula is obtained.
On self-adjoint extensions and Quantum symmetries. ArXiv:1402.5537
2014
Abstract. 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-equivariant version of Kato’s 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 L2-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 and only if U and V commut...
On extensions of symmetric operators
Operators and Matrices, 2020
We give an explicit description of all minimal self-adjoint extensions of a densely defined, closed symmetric operator in a Hilbert space with deficiency indices (1,1).
On self-adjoint extensions and quantum symmetries
2014
The previous results are applied to the study of GGG-invariant self-adjoint extensions of the Laplace-Beltrami operator on a smooth Riemannian manifold with boundary on which the group GGG acts. These extensions are labeled by admissible unitaries UUU acting on the L2L^2L2-space at the boundary and having spectral gap at −1-1−1. It is shown that if the unitary representation VVV of the symmetry group GGG is traceable, then the self-adjoint extension of the Laplace-Beltrami operator determined by UUU is GGG-invariant if UUU and VVV commute at the boundary. Various significant examples are discussed at the end.