Self-adjoint Extensions Of Symmetric Operators Research Articles

Hofstadter butterflies for square and honeycomb periodic arrays of quantum dots with Aharonov-Bohm solenoids

Periodic array of quantum dots in a magnetic field is considered. The magnetic field consists of two components: uniform and periodic. The periodic field is presented by the Aharonov-Bohm solenoids inserted in the quantum dots. The model based on the theory of self-adjoint extensions of symmetric operators is suggested. Two types of “flux-energy” diagrams (“Hofstadter butterflies”) are constructed and discussed. Diagrams of the first type show the spectrum of the system for different values of the uniform magnetic field (flux) and fixed AB flux. Diagrams of the second type show the spectrum of the system for different values of the AB magnetic field flux and fixed flux of the uniform magnetic field. “Flux-energy” diagrams for honeycomb lattice demonstrates splitting of bands in comparison with the case of square lattice. One can see one more peculiarity: butterflies for honeycomb case don't contain wide band except from zero.

3D Helmholtz resonator with two close point-like windows: Regularisation for Dirichlet case

In this paper, a model of 3D Helmholtz resonator with two close point-like windows is considered. The Dirichlet condition is assumed at the boundary. The model is based on the theory of self-adjoint extensions of symmetric operators in Pontryagin space. The model is explicitly solvable and allows one to obtain the equation for resonances (quasi-eigenvalues) in an explicit form. A proper choice of the model parameter leads to the coincidence of the model solution with the main term of the asymptotics (in the window width) of the realistic solution, corresponding to small windows. A regularization is suggested to obtain a realistic limiting result for two merging windows.

On Distributions of Self-Adjoint Extensions of Symmetric Operators

On Distributions of Self-Adjoint Extensions of Symmetric Operators

Spectra of Electronic Excitations in Graphene Near Coulomb Impurities

We study the problem of the electron excitation spectrum in the presence of point-like and regularized Coulomb impurities in gapped graphene. To this end, we use the Dirac model and in the point-like case theory of self-adjoint extensions of symmetric operators. In the point-like case, we construct a family of self-adjoint Hamiltonians describing the excitations for any charge of an impurity. Spectra and (generalized) eigenfunctions for all such Hamiltonians are found. Then, we consider the spectral problem in the case of a regularized Coulomb potential of impurities for a special regularization. We study exact equations for charges of impurities that may generate bound states with energy that coincides with the upper boundary of the negative branch of the continuous spectrum (supercritical charges) and calculate these charges.

A model of a boundary composed of the Helmholtz resonators

ABSTRACT A model of resonators for which boundary is composed of the Helmholtz resonators is constructed. It is based on the theory of self-adjoint extensions of symmetric operators. We consider a limiting procedure when the number of resonators tends to infinity and look after the spectrum of the Neumann Laplacian. It is shown that there is a sequence of the model systems and, correspondingly, a sequence of eigenfunctions of the model operators which converge to an eigenfunction of the Laplacian with a specific boundary condition.

Operators on Anti-dual pairs: Self-adjoint Extensions and the Strong Parrott Theorem

AbstractThe aim of this paper is to develop an approach to obtain self-adjoint extensions of symmetric operators acting on anti-dual pairs. The main advantage of such a result is that it can be applied for structures not carrying a Hilbert space structure or a normable topology. In fact, we will show how hermitian extensions of linear functionals of involutive algebras can be governed by means of their induced operators. As an operator theoretic application, we provide a direct generalization of Parrott’s theorem on contractive completion of 2 by 2 block operator-valued matrices. To exhibit the applicability in noncommutative integration, we characterize hermitian extendibility of symmetric functionals defined on a left ideal of a $C^{\ast }$-algebra.

Point-like perturbation of Rashba Hamiltonian

Quantum system with spin–orbit interaction in three-dimensional space is considered. Point-like perturbation for Rashba Hamiltonian is constructed in the framework of the theory of self-adjoint extensions of symmetric operators. Boundary triplet approach is used. The Weyl function and the scattering matrix are constructed.

On Z -Invariant Self-Adjoint Extensions of the Laplacian on Quantum Circuits

An analysis of the invariance properties of self-adjoint extensions of symmetric operators under the action of a group of symmetries is presented. For a given group G, criteria for the existence of G-invariant self-adjoint extensions of the Laplace–Beltrami operator over a Riemannian manifold are illustrated and critically revisited. These criteria are employed for characterising self-adjoint extensions of the Laplace–Beltrami operator on an infinite set of intervals, Ω , constituting a quantum circuit, which are invariant under a given action of the group Z . A study of the different unitary representations of the group Z on the space of square integrable functions on Ω is performed and the corresponding Z -invariant self-adjoint extensions of the Laplace–Beltrami operator are introduced. The study and characterisation of the invariance properties allows for the determination of the spectrum and generalised eigenfunctions in particular examples.

Point-like perturbation for Lamé operator

ABSTRACTA model of scattering of elastic wave by point-like scatterers on the plane is suggested. Point-like perturbation is constructed for the two-dimensional Lamé operator in the framework of the theory of self-adjoint extensions of symmetric operators. The eigenvalues of the model operator are found. The scattering matrix is obtained. Examples of scattering diagrams for several scatterers are presented.

Model of tunnelling through quantum dot and spin–orbit interaction

Solvable mathematical model is suggested for tunnelling through quantum dot. The model is based on the theory of self-adjoint extensions of symmetric operators. The spin–orbit interaction is taken into account. The transmission coefficient is obtained. The result is compared with the case where spin–orbit interaction is absent.

Resonance states completeness for relativistic particle on a sphere with two semi-infinite lines attached

The paper is devoted to resonances playing an important role in direct and inverse scattering problems. A model of a relativistic particle on hybrid manifold consisting of a sphere with two semi-infinite wires attached is considered. The model is based on the theory of self-adjoint extensions of symmetric operators. Completeness of resonance states in the space of square integrable functions on the sphere is proved. The proof uses the relation between the completeness and the factorization of the characteristic function in Sz.-Nagy functional model.

The Maslov index and the spectra of second order elliptic operators

We consider second order elliptic differential operators on a bounded Lipschitz domain Ω. Firstly, we establish a natural one-to-one correspondence between their self-adjoint extensions, with domains of definition containing in H1(Ω), and Lagrangian planes in H1/2(∂Ω)×H−1/2(∂Ω). Secondly, we derive a formula relating the spectral flow of the one-parameter families of such operators to the Maslov index, the topological invariant counting the signed number of conjugate points of paths of Lagrangian planes in H1/2(∂Ω)×H−1/2(∂Ω). Furthermore, we compute the Morse index, the number of negative eigenvalues, in terms of the Maslov index for several classes of the second order operators: the θ→-periodic Schrödinger operators on a period cell Q⊂Rn, the elliptic operators with Robin-type boundary conditions, and the abstract self-adjoint extensions of the Schrödinger operators on star-shaped domains. Our work is built on the techniques recently developed by B. Booß-Bavnbek, K. Furutani, and C. Zhu, and extends the scope of validity of their spectral flow formula by incorporating the self-adjoint extensions of the second order operators with domains in the first order Sobolev space H1(Ω). In addition, we generalize the results concerning relations between the Maslov and Morse indices quite recently obtained by G. Cox, J. Deng, C. Jones, J. Marzuola, A. Sukhtayev and the authors. Finally, we describe and study a link between the theory of abstract boundary triples and the Lagrangian description of self-adjoint extensions of abstract symmetric operators.

Quasi boundary triples and semi-bounded self-adjoint extensions

In this note semi-bounded self-adjoint extensions of symmetric operators are investigated with the help of the abstract notion of quasi boundary triples and their Weyl functions. The main purpose is to provide new sufficient conditions on the parameters in the boundary space to induce self-adjoint realizations, and to relate the decay of the Weyl function to estimates on the lower bound of the spectrum. The abstract results are illustrated with uniformly elliptic second-order partial differential equations on domains with non-compact boundaries.

Idealizations, essential self-adjointness, and minimal model explanation in the Aharonov–Bohm effect

Two approaches to understanding the idealizations that arise in the Aharonov–Bohm (AB) effect are presented. It is argued that a common topological approach, which takes the non-simply connected electron configuration space to be an essential element in the explanation and understanding of the effect, is flawed. An alternative approach is outlined. Consequently, it is shown that the existence and uniqueness of self-adjoint extensions of symmetric operators in quantum mechanics have important implications for philosophical issues. Also, the alleged indispensable explanatory role of said idealizations is examined via a minimal model explanatory scheme. Last, the idealizations involved in the AB effect are placed in a wider philosophical context via a short survey of part of the literature on infinite and essential idealizations.

Symmetric Pairs of Unbounded Operators in Hilbert Space, and Their Applications in Mathematical Physics

In a previous paper, the authors introduced the idea of a symmetric pair of operators as a way to compute self-adjoint extensions of symmetric operators. In brief, a symmetric pair consists of two densely defined linear operators $A$ and $B$, with $A \subseteq B^\star$ and $B \subseteq A^\star$. In this paper, we will show by example that symmetric pairs may be used to deduce closability of operators and sometimes even compute adjoints. In particular, we prove that the Malliavin derivative and Skorokhod integral of stochastic calculus are closable, and the closures are mutually adjoint. We also prove that the basic involutions of Tomita-Takesaki theory are closable and that their closures are mutually adjoint. Applications to functions of finite energy on infinite graphs are also discussed, wherein the Laplace operator and inclusion operator form a symmetric pair.

Model of tunnelling through periodic array of quantum dots

Several explicitly solvable models of electron tunnelling in a system of single and double two-dimensional periodic arrays of quantum dots with two laterally coupled leads in a homogeneous magnetic field are constructed. First, a model of single layer formed by periodic array of zero-range potentials is described. The Landau operator (the Schrodinger operator with a magnetic field) with point-like interactions is the system Hamiltonian. We deal with two types of the layer lattices: square and honeycomb. The periodicity condition gives one an invariance property for the Hamiltonian in respect to magnetic translations group. The consideration of double quantum layer reduces to the replacement of the basic cell for the single layer by a cell including centers of different layers. Two variants of themodel for the double layer are suggested: with direct tunneling between the layers and with the connecting channels (segments in the model) between the layers. The theory of self-adjoint extensions of symmetric operators is a mathematical background of the model. The third stage of the construction is the description of leads connection. It is made by the operator extensions theory method too. Electron tunneling from input lead to the output lead through the double quantum layer is described. Energy ranges with extremely small (practically, zero) transmission were found. Dependencies of the transmission coefficient (particularly, “zero transmission bands” positions) on the magnetic field, the energy of electron and the distance between layers are investigated. The results are compared with the corresponding single-layer transmission.

Electron spectrum for aligned SWNT array in a magnetic field

Solvable model of single-walled nanotube array of cross-bar architecture in a magnetic field is constructed. The model is based on the theory of self-adjoint extensions of symmetric operators. Results of numerical analysis of the energy spectrum for the system is presented. The ”flux-energy” and ”angle-energy” diagrams for the Bloch bands are obtained. Evidence for a fractal structure of these diagrams is given.

Periodic chain of resonators: gap control and geometry of the system

A periodic system of coupled resonators is considered. It has a form of zigzag line. A band structure of the spectrum is studied. Solvable model based on the theory of self-adjoint extensions of symmetric operators is used. The position of gaps depends on the zigzag line angle. It opens a way to control the gap position by geometrical parameters. Particularly, it is possible to construct a system with predetermined gap position.

Self-adjoint elliptic operators with boundary conditions on not closed hypersurfaces

The theory of self-adjoint extensions of symmetric operators is used to construct self-adjoint realizations of a second-order elliptic differential operator on Rn with linear boundary conditions on (a relatively open part of) a compact hypersurface. Our approach allows to obtain Kreĭn-like resolvent formulae where the reference operator coincides with the “free” operator with domain H2(Rn); this provides an useful tool for the scattering problem from a hypersurface. Concrete examples of this construction are developed in connection with the standard boundary conditions, Dirichlet, Neumann, Robin, δ and δ′-type, assigned either on a (n−1) dimensional compact boundary Γ=∂Ω or on a relatively open part Σ⊂Γ. Schatten–von Neumann estimates for the difference of the powers of resolvents of the free and the perturbed operators are also proven; these give existence and completeness of the wave operators of the associated scattering systems.

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

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.