On the theory of self-adjoint extensions of symmetric operators and its applications to quantum physics

Ibort, Alberto; Pérez-Pardo, Juan Manuel

doi:10.1142/s0219887815600051

Cited by 15 publications

(13 citation statements)

References 15 publications

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“…Before making explicit the previous structures in concrete examples we notice that the previous discussion works in a similar way with the covariant Laplacian ∆ A discussed in Section 3. Thus if we are given a group acting by unitary bundle isomorphisms on an Hermitean bundle E → Ω (and by isometric diffeomorphisms on the Riemannian manifold Ω), then any unitary operator U at the boundary, (that in addition satisfies the conditions of possessing gap and being admissible, [Ib14c], that guarantee that the quadratic form constructed from the operator ∇ with boundary conditions dictated by U , read more about self-adjoint extensions determined by quadratic forms in [Ib14] and [Ib15] this volume), and that verifies the commutation relations of Theorem 19 describes a G-invariant quadratic form. The closure of this quadratic form characterizes uniquely a G-invariant self-adjoint extension of the Laplace-Beltrami operator.…”

Section: Examples: Groups Acting By Isometriesmentioning

confidence: 99%

The topology and geometry of self-adjoint and elliptic boundary conditions for Dirac and Laplace operators

Asorey

Ibort

Marmo

2015

Int. J. Geom. Methods Mod. Phys.

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The theory of self-adjoint extensions of first and second order elliptic differential operators on manifolds with boundary is studied via its most representative instances: Dirac and Laplace operators.The theory is developed by exploiting the geometrical structures attached to them and, by using an adapted Cayley transform on each case, the space M of such extensions is shown to have a canonical group composition law structure.The obtained results are compared with von Neumann's Theorem characterising the self-adjoint extensions of densely defined symmetric operators on Hilbert spaces. The 1D case is thoroughly investigated.The geometry of the submanifold of elliptic self-adjoint extensions M ellip is studied and it is shown that it is a Lagrangian submanifold of the universal Grassmannian Gr. The topology of M ellip is also explored and it is shown that there is a canonical cycle whose dual is the Maslov class of the manifold. Such cycle, called the Cayley surface, plays a relevant role in the study of the phenomena of topology change.Self-adjoint extensions of Laplace operators are discussed in the path integral formalism, identifying a class of them for which both treatments leads to the same results.A theory of dissipative quantum systems is proposed based on this theory and a unitarization theorem for such class of dissipative systems is proved.The theory of self-adjoint extensions with symmetry of Dirac operators is also discussed and a reduction theorem for the self-adjoint elliptic Grasmmannian is obtained.Finally, an interpretation of spontaneous symmetry breaking is offered from the

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Section: Examples: Groups Acting By Isometriesmentioning

confidence: 99%

The topology and geometry of self-adjoint and elliptic boundary conditions for Dirac and Laplace operators

Asorey

Ibort

Marmo

2015

Int. J. Geom. Methods Mod. Phys.

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“…(iii) It still remains open to determine the deficiency spaces of H AB = H A ⊗ I + I ⊗ H B in the case when both H A and H B are only assumed to be symmetric. As stated in [4], it is quite natural to conjecture that…”

Section: Computing the Deficiency Spacesmentioning

confidence: 99%

Self-adjoint extensions of bipartite Hamiltonians

Lenz¹,

Timon²,

Wirth³

2021

Proceedings of the Edinburgh Mathematical Society

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We compute the deficiency spaces of operators of the form $H_A{\hat {\otimes }} I + I{\hat {\otimes }} H_B$ , for symmetric $H_A$ and self-adjoint $H_B$ . This enables us to construct self-adjoint extensions (if they exist) by means of von Neumann's theory. The structure of the deficiency spaces for this case was asserted already in Ibort et al. [Boundary dynamics driven entanglement, J. Phys. A: Math. Theor.47(38) (2014) 385301], but only proven under the restriction of $H_B$ having discrete, non-degenerate spectrum.

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“…If the Riemannian manifold has boundaries those operators are in general symmetric operators but not self-adjoint, cf. [41] or [29] and references therein for an introduction to the topic. Each self-adjoint extension describes a different physical situation.…”

Section: Control Of Quantum Systemsmentioning

confidence: 99%

Quantum Control at the Boundary

Balmaseda

Pérez-Pardo

2019

Springer Proceedings in Physics

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We introduce a scheme for controlling the state of a quantum system by manipulating its boundary conditions. This contrasts with the usual approach based on direct interactions with the system, that is, by adding interaction terms to the Hamiltonian of the system. We address this infinite-dimensional control problem, providing conditions to the existence of dynamics and approximate controllability for a family of quantum one-dimensional systems.

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On the theory of self-adjoint extensions of symmetric operators and its applications to quantum physics

Cited by 15 publications

References 15 publications

The topology and geometry of self-adjoint and elliptic boundary conditions for Dirac and Laplace operators

The topology and geometry of self-adjoint and elliptic boundary conditions for Dirac and Laplace operators

Self-adjoint extensions of bipartite Hamiltonians

Quantum Control at the Boundary

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