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.

We develop a functional model for operators arising in the study of boundary-value problems of materials science and mathematical physics. We then provide explicit formulae for the resolvents of the associated extensions of symmetric operators in terms of the associated generalised Dirichlet-to-Neumann maps, which can be utilised in the analysis of the properties of parameter-dependent problems as well as in the study of their spectra.

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.

Abstract The 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.

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.

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 $(1, 1)$.

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.

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.

Spectral enclosures for non-self-adjoint extensions of symmetric operators

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.

We study the stability and the scattering properties of a spacetime with a topological defect along a spherical bubble. This bubble connects two flat spacetimes which are asymptotically Minkowski, so that the resulting universe may be regarded as containing a wormhole. Its distinguished feature is the absence of exotic matter, i.e., its matter content respects all the energy conditions. Although this wormhole is nontraversable, waves and quantum particles can tunnel between both universes. Interestingly enough, the wave equation alone does not uniquely determine the evolution of scalar waves on this background, and the theory of self-adjoint extensions of symmetric operators is required to find the relevant boundary conditions in this context. Here we show that, for a particular boundary condition, this spacetime is stable and gives rise to a scattering pattern which is identical to the more usual thin-shell wormhole composed of exotic matter. Other boundary conditions of interest are also analyzed, including an unstable configuration with sharp resonances at well-defined frequencies.

Abstract Given a symmetric linear transformation on a Hilbert space, a natural problem to consider is the characterization of its set of symmetric extensions. This problem is equivalent to the study of the partial isometric extensions of a fixed partial isometry. We provide a new function theoretic characterization of the set of all self-adjoint extensions of any symmetric linear transformation B with finite equal indices and inner Livšic characteristic function θ

An infinite bent chain of nanospheres connected by wires is considered. We assume that there are δ-like potentials at the contact points. A solvable mathematical model based on the theory of self-adjoint extensions of symmetric operators is constructed. The spectral equation for the model operator is derived in an explicit form. It is shown that the Hamiltonian has non-empty point spectrum. The positions of the eigenvalues for different values of the system parameters (the length of the connecting wires, the intensities of δ-interactions and the bent angle) are found.

The self-adjoint extensions of symmetric operators satisfying anticommutation relations are considered. It is proven that an anticommutative type of the Glimm-Jaffe-Nelson commutator theorem follows. Its application to an abstract Dirac operator is also considered.

We develop a general theory of $ \mathcal{P}\mathcal{T} $ -symmetric operators. Special attention is given to $ \mathcal{P}\mathcal{T} $ -symmetric quasiself-adjoint extensions of symmetric operator with deficiency indices : For these extensions, the possibility of their interpretation as self-adjoint operators in Krein spaces is investigated and the description of nonreal eigenvalues is presented. These abstract results are applied to the Schr¨odinger operator with Coulomb potential on the real axis.

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 discuss a problem of constructing self-adjoint ordinary differential operators starting from self-adjoint differential expressions based on the general theory of self-adjoint extensions of symmetric operators outlined in [1]. We describe one of the possible ways of constructing in terms of the closure of an initial symmetric operator associated with a given differential expression and deficient spaces. Particular attention is focused on the features peculiar to differential operators, among them on the notion of natural domain and the representation of asymmetry forms generated by adjoint operators in terms of boundary forms. Main assertions are illustrated in detail by simple examples of quantum-mechanical operators like the momentum or Hamiltonian.

Constructing physical observables as self-adjoint operators under quantum-mechanical description of systems with boundaries and/or singular potentials is a nontrivial problem. We present a comparative review of various methods for constructing ordinary self-adjoint differential operators associated with self-adjoint differential expressions based on the general theory of self-adjoint extensions of symmetric operators. The exposition is nontraditional and is based on the concept of asymmetry forms generated by adjoint operators. The main attention is given to a specification of self-adjoint extensions by self-adjoint boundary conditions. All the methods are illustrated by examples of quantum-mechanical observables like momentum and Hamiltonian. In addition to the conventional methods, we propose a possible alternative way of specifying self-adjoint differential operators by explicit self-adjoint boundary conditions that generally have an asymptotic form for singular boundaries. A comparative advantage of the method is that it allows avoiding an evaluation of deficient subspaces and deficiency indices. The effectiveness of the method is illustrated by a number of examples of quantum-mechanical observables.

A previous result of the author concerning almost unitary operators is applied to the spectral analysis of non-self-adjoint extensions of symmetric operators. For this purpose, the Cayley transform of such an extension is written as a perturbation of a unitary operator by a finite-rank operator of a special form in terms of the Weyl function. Bibliography: 3 titles.

The transmission coefficient of a nanodevice—a quantum ring with two one-dimensional conductors attached—is found. The Hamiltonian of a nanodevice is constructed in terms of the theory of self-adjoint extensions of symmetric operators. It is shown that, in this case, the transmission coefficient coincides with that determined by the Feynman sum rule for the probability amplitudes. The transmission coefficient of the nanodevice is studied as a function of the electron energy, magnetic field, and the relative positions of the conductor contacts and the ring.