Boundary contributions to the hypervirial theorem

Esteve, J. G.; Falceto, Fernando; Giri, Pulak Ranjan

doi:10.1103/physreva.85.022104

Cited by 11 publications

(21 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Thus, for instance, quasi-periodic boundary conditions will break the symmetry P . Another, less trivial, example in the same spirit is provided by a result by Falceto and Esteve on a generalization of the Virial Theorem [Es12]. Consider a free particle in one dimension restricted to move in [0, ∞) and subject to Robin boundary conditions, i.e.…”

Section: Spontaneous Symmetry Breaking and Self-adjoint Extensionsmentioning

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.

View full text Add to dashboard Cite

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

show abstract

Section: Spontaneous Symmetry Breaking and Self-adjoint Extensionsmentioning

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.

View full text Add to dashboard Cite

show abstract

“…But, in addition, if the boundary conditions are also imposed, it may be that this restriction fails. For details see [8]. In principle, it is very difficult determine beforehand if this restriction is violated or no.…”

Section: Time Derivative Of the Operator's Mean Valuementioning

confidence: 99%

“…Remarkably enough that in [8] the modification of the Heisenberg equation is suggestedforeseeing this result, the authors propose inclusion of the extra terms into the operator equation in advance as follows…”

Section: Time Derivative Of the Operator's Mean Valuementioning

confidence: 99%

“…During decades there have been many studies in which the authors try to reconsider some fundamental relations of quantum mechanics on strong mathematical properties of the operators and their matrix elements appeared. For example, the Ehrenfest theorem or/and, in general, mean values of time derivatives of the operators have been encountered [1][2][3][4][5][6][7][8].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Hypervirial and Ehrenfest Theorems in Spherical Coordinates: Systematic Approach

Khelashvili

Nadareishvili

2020

Phys. Part. Nuclei

View full text Add to dashboard Cite

Elaboration of some fundamental relations in 3-dimensional quantum mechanics is considered taking into account the restricted character of areas in radial distance. In such cases the boundary behavior of the radial wave function and singularity of operators at the origin of coordinates contribute to these relations. We derive the relation between the average value of the operator's time derivative and the time derivative of the mean value of this operator, which is usually considered to be the same by definition. The deviation from the known result is deduced and manifested by extra term, which depends on the boundary behavior mentioned above. The general form for this extra term takes place in the hypervirial-like theorems. As a particular case, the virial theorem for Coulomb and oscillator potentials is considered and correction to the Kramers' sum rule is derived. Moreover the corrected Ehrenfest theorem is deduced and its consistency with real physical picture is demonstrated. потенциалов и получено поправки к правилам сумм Крамерса. Кроме того, выведена исправленная теорема Еренфеста и продемонстрирована ее соответствие с реальной физической картиной

show abstract

“…The algebraic relations of their conserved quantities reveal the dynamical symmetries of the two systems are described by the symmetry groups SO(3) and SU (2) respectively. This algebraic structure is called Higgs Algebra, and has received attention from a variety of literatures [14,15,16,17,18,19,20]. The Bertrand's theorem has been extended in different directions [21,22,23,24].…”

Section: Introductionmentioning

confidence: 99%

Higgs Algebraic Symmetry of Screened System in a Spherical Geometry

Li¹,

Zhang²,

Gu³

et al. 2013

Commun. Theor. Phys.

View full text Add to dashboard Cite

Abstract. The orbits and the dynamical symmetries for the screened Coulomb potentials and isotropic harmonic oscillators have been studied by Wu and Zeng [Z. B. Wu and J. Y. Zeng, Phys. Rev. A 62,032509 (2000)]. We find the similar properties in the responding systems in a spherical space, whose dynamical symmetries are described by Higgs Algebra. There exists a conserved aphelion and perihelion vector, which, together with angular momentum, constitute the generators of the geometrical symmetry group at the aphelia and perihelia points (ṙ = 0).

show abstract

Boundary contributions to the hypervirial theorem

Cited by 11 publications

References 18 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

Hypervirial and Ehrenfest Theorems in Spherical Coordinates: Systematic Approach

Higgs Algebraic Symmetry of Screened System in a Spherical Geometry

Contact Info

Product

Resources

About