Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1912.03670 · Proceedings of the Edinburgh Mathematical Society

Self-adjoint extensions of bipartite Hamiltonians

Daniel H. Lenz, Timon Weinmann, Melchior Wirth

Published Jun 22, 2021• 2 citations
arXivHTMLPDF

Abstract

Abstract 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.

DOI
10.1017/s0013091521000080

Related papers

J-self-adjoint operators with \mathcal{C} -symmetries: an extension theory approach

Sergio A Albeverio, U Günther, Sergiusz Kużel

A well-known tool in conventional (von Neumann) quantum mechanics is the self-adjoint extension technique for symmetric operators. It is used, e.g., for the construction of Dirac–Hermitian Hamiltonians with point-interaction potentials. Here we reshape this technique to allow for the construction of pseudo-Hermitian ( J -self-adjoint) Hamiltonians with complex point interactions. We demonstrate that the resulting Hamiltonians are bijectively related to the so-called hypermaximal neutral subspaces of the defect Krein space of the symmetric operator. This symmetric operator is allowed to have arbitrary but equal deficiency indices ⟨ n , n ⟩. General properties of the operators for these Hamiltonians are derived. A detailed study of -operator parametrizations and Krein type resolvent formulae is provided for J -self-adjoint extensions of symmetric operators with deficiency indices ⟨2, 2⟩. The technique is exemplified on 1D pseudo-Hermitian Schrödinger and Dirac Hamiltonians with complex point-interaction potentials.

arXiv:0811.0365•Journal of Physics A Mathematical and Theoretical•Feb 13, 2009•36 citations

Boundary dynamics driven entanglement

Alberto Ibort, Giuseppe Marmo, Juan Manuel Pérez-Pardo

We will show how to generate entangled states out of unentangled states on a bipartite system by means of dynamical boundary conditions. The auxiliary system is defined by a symmetric but not self-adjoint Hamiltonian, and we will also study the space of self-adjoint extensions of the bipartite system. We will show that only a small set of these extensions leads to separable dynamics, and we will characterize these extensions. Various simple examples illustrating this phenomenon are discussed; in particular, we will analyze the hybrid system consisting of a planar quantum rotor and a spin system under a wide class of boundary conditions.

arXiv:1212.2260•Journal of Physics A Mathematical and Theoretical•Sep 5, 2014•13 citations

Self-adjoint extensions of $k$-photon light-matter Hamiltonians

Felix Fischer, Felix Knapp, Daniel Burgarth +1 more

Multiphoton light-matter interactions, in which a bosonic mode exchanges $k$ excitations at a time with a quantum system, are a source of genuine nonlinearity in quantum optics and are increasingly accessible experimentally. Here we study the class of operators $H = H_{\rm mat}\otimes I + I\otimesωa^\ast a + Σ\otimes(a^\ast)^k + Σ^\ast\otimes a^k$ on $\mathcal{H}\otimes L^2(\mathbb{R})$, coupling a single bosonic mode to an arbitrary matter system through a bounded operator $Σ$. When $Σ$ is normal and nonzero, we prove that $H$ is self-adjoint if and only if $k\leq2$; for $k\geq3$ we compute the deficiency indices, parametrise all self-adjoint extensions, and show that every extension has purely discrete spectrum whenever the matter system is finite-dimensional. Our analysis rests on a block Jacobi decomposition paired with a suitable unitary transformation depending on the polar decomposition of $Σ$. The normality of $Σ$ is optimal: a $k$-photon Jaynes-Cummings model, with non-normal coupling, remains self-adjoint for every $k$. We illustrate our results on the $k$-photon Rabi and Dicke models.

math-phCross-listedarXiv:2607.22378•Jul 27, 2026•0 citations

On Distributions of Self-Adjoint Extensions of Symmetric Operators

Franco Fagnola, Zheng Li

In quantum probability a self-adjoint operator on a Hilbert space determines a real random variable and one can define a probability distribution with respect to a given state.In this paper we consider self-adjoint extensions of certain symmetric operators, such as momentum and Hamiltonian operators, with various boundary conditions, explicitly compute their probability distributions in some state and study dependence of these probability distributions on boundary conditions.

Journal of Stochastic Analysis•Jun 3, 2021•0 citations

Constructing quantum observables and self-adjoint extensions of symmetric operators. III. Self-adjoint boundary conditions

Boris Leonidovich Voronov, Dmitri M Gitman, Igor Viktorovich Tyutin

Russian Physics Journal•Feb 1, 2008•13 citations

Self-adjoint approximations of degenerate Schrodinger operator

Vsevolod Zhanovich Sakbaev, I. V. Volovich

The problem of construction a quantum mechanical evolution for the Schrodinger equation with a degenerate Hamiltonian which is a symmetric operator that does not have self-adjoint extensions is considered. Self-adjoint regularization of the Hamiltonian does not lead to a preserving probability limiting evolution for vectors from the Hilbert space but it is used to construct a limiting evolution of states on a C*-algebra of compact operators and on an abelian subalgebra of operators in the Hilbert space. The limiting evolution of the states on the abelian algebra can be presented by the Kraus decomposition with two terms. Both of this terms are corresponded to the unitary and shift components of Wold's decomposition of isometric semigroup generated by the degenerate Hamiltonian. Properties of the limiting evolution of the states on the C*-algebras are investigated and it is shown that pure states could evolve into mixed states.

arXiv:1701.02777•Jan 10, 2017•0 citations
Abstract We compute the deficiency spaces of operators of the form HA⊗^I+I⊗^HBH_A{\hat {\otimes }} I + I{\hat {\otimes }} H_BHA​⊗^​I+I⊗^​HB​ , for symmetric HAH_AHA​ and self-adjoint HBH_BHB​ . 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 HBH_BHB​ having discrete, non-degenerate spectrum.
A well-known tool in conventional (von Neumann) quantum mechanics is the self-adjoint extension technique for symmetric operators. It is used, e.g., for the construction of Dirac–Hermitian Hamiltonians with point-interaction potentials. Here we reshape this technique to allow for the construction of pseudo-Hermitian ( J -self-adjoint) Hamiltonians with complex point interactions. We demonstrate that the resulting Hamiltonians are bijectively related to the so-called hypermaximal neutral subspaces of the defect Krein space of the symmetric operator. This symmetric operator is allowed to have arbitrary but equal deficiency indices ⟨ n , n ⟩. General properties of the operators for these Hamiltonians are derived. A detailed study of -operator parametrizations and Krein type resolvent formulae is provided for J -self-adjoint extensions of symmetric operators with deficiency indices ⟨2, 2⟩. The technique is exemplified on 1D pseudo-Hermitian Schrödinger and Dirac Hamiltonians with complex point-interaction potentials.
We will show how to generate entangled states out of unentangled states on a bipartite system by means of dynamical boundary conditions. The auxiliary system is defined by a symmetric but not self-adjoint Hamiltonian, and we will also study the space of self-adjoint extensions of the bipartite system. We will show that only a small set of these extensions leads to separable dynamics, and we will characterize these extensions. Various simple examples illustrating this phenomenon are discussed; in particular, we will analyze the hybrid system consisting of a planar quantum rotor and a spin system under a wide class of boundary conditions.
Self-adjoint extensions of kkk-photon light-matter Hamiltonians
Multiphoton light-matter interactions, in which a bosonic mode exchanges kkk excitations at a time with a quantum system, are a source of genuine nonlinearity in quantum optics and are increasingly accessible experimentally. Here we study the class of operators H=Hmat⊗I+I⊗ωa∗a+Σ⊗(a∗)k+Σ∗⊗akH = H_{\rm mat}\otimes I + I\otimesωa^\ast a + Σ\otimes(a^\ast)^k + Σ^\ast\otimes a^kH=Hmat​⊗I+I⊗ωa∗a+Σ⊗(a∗)k+Σ∗⊗ak on H⊗L2(R)\mathcal{H}\otimes L^2(\mathbb{R})H⊗L2(R), coupling a single bosonic mode to an arbitrary matter system through a bounded operator ΣΣΣ. When ΣΣΣ is normal and nonzero, we prove that HHH is self-adjoint if and only if k≤2k\leq2k≤2; for k≥3k\geq3k≥3 we compute the deficiency indices, parametrise all self-adjoint extensions, and show that every extension has purely discrete spectrum whenever the matter system is finite-dimensional. Our analysis rests on a block Jacobi decomposition paired with a suitable unitary transformation depending on the polar decomposition of ΣΣΣ. The normality of ΣΣΣ is optimal: a kkk-photon Jaynes-Cummings model, with non-normal coupling, remains self-adjoint for every kkk. We illustrate our results on the kkk-photon Rabi and Dicke models.
The problem of construction a quantum mechanical evolution for the Schrodinger equation with a degenerate Hamiltonian which is a symmetric operator that does not have self-adjoint extensions is considered. Self-adjoint regularization of the Hamiltonian does not lead to a preserving probability limiting evolution for vectors from the Hilbert space but it is used to construct a limiting evolution of states on a C*-algebra of compact operators and on an abelian subalgebra of operators in the Hilbert space. The limiting evolution of the states on the abelian algebra can be presented by the Kraus decomposition with two terms. Both of this terms are corresponded to the unitary and shift components of Wold's decomposition of isometric semigroup generated by the degenerate Hamiltonian. Properties of the limiting evolution of the states on the C*-algebras are investigated and it is shown that pure states could evolve into mixed states.