Self-adjoint Conditions Research Articles

Lie symmetries with optimal subalgebra, quasi self-adjointness condition, conservation laws and power series solutions for the Mikhailov-Shabat model in quantum theory

Lie symmetries with optimal subalgebra, quasi self-adjointness condition, conservation laws and power series solutions for the Mikhailov-Shabat model in quantum theory

Closed form expressions for the Green’s function of a quantum graph—a scattering approach

In this work we present a three step procedure for generating a closed form expression of the Green’s function on both closed and open finite quantum graphs with general self-adjoint matching conditions. We first generalize and simplify the approach by Barra and Gaspard (2001 Phys. Rev. E 65 016205) and then discuss the validity of the explicit expressions. For compact graphs, we show that the explicit expression is equivalent to the spectral decomposition as a sum over poles at the discrete energy eigenvalues with residues that contain projector kernel onto the corresponding eigenstate. The derivation of the Green’s function is based on the scattering approach, in which stationary solutions are constructed by treating each vertex or subgraph as a scattering site described by a scattering matrix. The latter can then be given in a simple closed form from which the Green’s function is derived. The relevant scattering matrices contain inverse operators which are not well defined for wave numbers at which bound states in the continuum exists. It is shown that the singularities in the scattering matrix related to these bound states or perfect scars can be regularised. Green’s functions or scattering matrices can then be expressed as a sum of a regular and a singular part where the singular part contains the projection kernel onto the perfect scar.

Deficiency Indices of Block Jacobi Matrices That Do Not Satisfy the Carleman Condition, and Operators with Point Interactions

We study two classes of Jacobi matrices that have arisen in connection with the investigation of operators with point interactions. New conditions for self-adjointness and maximal deficiency indices of block Jacobi matrices are obtained. New conditions for the discreteness of the spectrum of the Schrödinger and Dirac operators with point interactions are found.

$$j$$-Self-Adjointness Conditions for Jacobi Matrices and Schrödinger and Dirac Operators with Point Interactions

$$j$$-Self-Adjointness Conditions for Jacobi Matrices and Schrödinger and Dirac Operators with Point Interactions

Brown measures of free circular and multiplicative Brownian motions with self-adjoint and unitary initial conditions

Let Z_N be a Ginibre ensemble and let A_N be a Hermitian random matrix independent of Z_N such that A_N converges in distribution to a self-adjoint random variable x_0 in a W^* -probability space (\mathscr{A},\tau) . For each t>0 , the random matrix A_N+\sqrt{t}\,Z_N converges in \ast -distribution to x_0+c_t , where c_t is a circular variable of variance t , freely independent of x_0 . We use the Hamilton–Jacobi method to compute the Brown measure \rho_t of x_0+c_t . The Brown measure has a density that is constant along the vertical direction inside the support. The support of the Brown measure of x_0+c_t is related to the subordination function of the free additive convolution of x_0+s_t , where s_t is a semicircular variable of variance t , freely independent of x_0 . Furthermore, the push-forward of \rho_t by a natural map is the law of x_0+s_t . Let G_N(t) be the Brownian motion on the general linear group and let U_N be a unitary random matrix independent of G_N(t) such that U_N converges in distribution to a unitary random variable u in (\mathscr{A},\tau) . The random matrix U_NG_N(t) converges in \ast -distribution to ub_t where b_t is the free multiplicative Brownian motion, freely independent of u . We compute the Brown measure \mu_t of ub_t , extending the recent work by Driver–Hall–Kemp, which corresponds to the case u=I . The measure has a density of the special form \frac{1}{r^2}w_t(\theta) in polar coordinates in its support. The support of \mu_t is related to the subordination function of the free multiplicative convolution of uu_t where u_t is the free unitary Brownian motion, freely independent of u . The push-forward of \mu_t by a natural map is the law of uu_t . In the special case that u is Haar unitary, the Brown measure \mu_t follows the annulus law . The support of the Brown measure of ub_t is an annulus with inner radius e^{-t/2} and outer radius e^{t/2} . In its support, the density in polar coordinates is given by \frac{1}{2\pi t}\,\frac{1}{r^2}.

On the symmetrization and composition of nonstandard finite difference schemes as an alternative to Richardson's extrapolation

From the classical explicit Euler scheme of first order, nonstandard finite difference (NSFD) schemes were envisioned to mimic the essential properties of the governing differential equation model for every time step-size. In the context of compartmental epidemiological models, these properties are generally concerned with the positivity of subpopulations, conservation laws (dynamics of the total population), and stability. However, for autonomous systems, the symmetry (self-adjoint) condition is not preserved. Compartmental epidemiological models are Poisson systems, so methods from geometric numerical integration should be applicable. It is found that symmetrization of NSFD schemes does not respect positivity for every step-size, though other characteristics are maintained and second order is reached. Composition through the Lie formalism can then be applied to obtain higher-order schemes. This is a more efficient and consistent alternative to Richardson's extrapolation, which has often been used to go beyond order one.

The Krein\u2013von Neumann Extension for Schr\xf6dinger Operators on Metric Graphs

The Krein–von Neumann extension is studied for Schrödinger operators on metric graphs. Among other things, its vertex conditions are expressed explicitly, and its relation to other self-adjoint vertex conditions (e.g. continuity-Kirchhoff) is explored. A variational characterisation for its positive eigenvalues is obtained. Based on this, the behaviour of its eigenvalues under perturbations of the metric graph is investigated, and so-called surgery principles are established. Moreover, isoperimetric eigenvalue inequalities are obtained.

Boundary Problems for Three-Dimensional Dirac Operators and Generalized MIT Bag Models for Unbounded Domains

We consider operators of boundary value problems for 3D- Dirac operators in unbounded domains with the uniformly regular boundary. We give effective conditions of self-adjointness of operators under consideration and a description of their essential spectra. We also give applications to operators of the MIT bag problems for unbounded domains

Hydrogenoid Spectra with Central Perturbations

Through the Kre\u{\i}n-Vi\v{s}ik-Birman extension scheme, unlike the classical analysis based on von Neumann's theory, we reproduce the construction and classification of all self-adjoint realisations of three-dimensional hydrogenoid-like Hamiltonians with singular perturbation supported at the Coulomb centre (the nucleus), as well as of Schr\"{o}dinger operators with Coulomb potentials on the half-line. These two problems are technically equivalent, albeit sometimes treated by their own in the the literature. Based on such scheme, we then recover the formula to determine the eigenvalues of each self-adjoint extension, as corrections of the non-relativistic hydrogenoid energy levels. We discuss in which respect the Kre\u{\i}n-Vi\v{s}ik-Birman scheme is somewhat more natural in yielding the typical boundary condition of self-adjointness at the centre of the perturbation.

Limits of quantum graph operators with shrinking edges

We address the question of convergence of Schrödinger operators on metric graphs with general self-adjoint vertex conditions as lengths of some of graph's edges shrink to zero. We determine the limiting operator and study convergence in a suitable norm resolvent sense. It is noteworthy that, as edge lengths tend to zero, standard Sobolev-type estimates break down, making convergence fail for some graphs. We use a combination of functional-analytic bounds on the edges of the graph and Lagrangian geometry considerations for the vertex conditions to establish a sufficient condition for convergence. This condition encodes an intricate balance between the topology of the graph and its vertex data. In particular, it does not depend on the potential, on the differences in the rates of convergence of the shrinking edges, or on the lengths of the unaffected edges.

On unbounded commuting Jacobi operators and some related issues

Abstract We consider the situations, when two unbounded operators generated by infinite Jacobi matrices, are self-adjoint and commute. It is found that if two Jacobi matrices formally commute, then two corresponding operators are either self-adjoint and commute, or admit a commuting self-adjoint extensions. In the latter case such extensions are explicitly described. Also, some necessary and sufficient conditions for self-adjointness of Jacobi operators are studied.

Compressed Resolvents and Reduction of Spectral Problems on Star Graphs

In this paper a two-step reduction method for spectral problems on a star graph with n+1 edges e_{0}, e_{1}, ldots , e_{n} and a self-adjoint matching condition at the central vertex v is established. The first step is a reduction to the problem on the single edge e_0 but with an energy depending boundary condition at v. In the second step, by means of an abstract inverse result for Q-functions, a reduction to a problem on a path graph with two edges e_0, widetilde{e}_1 joined by continuity and Kirchhoff conditions is given. All results are proved for symmetric linear relations in an orthogonal sum of Hilbert spaces. This ensures wide applicability to various different realizations, in particular, to canonical systems and Krein strings which include, as special cases, Dirac systems and Stieltjes strings. Employing two other key inverse results by de Branges and Krein, we answer e.g. the following question: If all differential operators are of one type, when can the reduced system be chosen to consist of two differential operators of the same type?

Polymeric quantum mechanics and the zeros of the Riemann zeta function

We analyze the Berry–Keating model and the Sierra and Rodríguez-Laguna Hamiltonian within the polymeric quantization formalism. By using the polymer representation, we obtain for both models, the associated polymeric quantum Hamiltonians and the corresponding stationary wave functions. The self-adjointness condition provides a proper domain for the Hamiltonian operator and the energy spectrum, which turned out to be dependent on an introduced scale parameter. By performing a counting of semiclassical states, we prove that the polymer representation reproduces the smooth part of the Riemann–von Mangoldt formula, and also introduces a correction depending on the energy and the scale parameter. This may shed some light on the understanding of the fluctuation behavior of the zeros of the Riemann function from a purely quantum point of view.

Self-adjoint realisations of the Dirac-Coulomb Hamiltonian for heavy nuclei

We derive a classification of the self-adjoint extensions of the three-dimensional Dirac-Coulomb operator in the critical regime of the Coulomb coupling. Our approach is solely based upon the Kreĭn-Višik-Birman extension scheme, or also on Grubb’s universal classification theory, as opposite to previous works within the standard von Neumann framework. This let the boundary condition of self-adjointness emerge, neatly and intrinsically, as a multiplicative constraint between regular and singular part of the functions in the domain of the extension, the multiplicative constant giving also immediate information on the invertibility property and on the resolvent and spectral gap of the extension.

Combined optical solitary waves and conservation laws for nonlinear Chen–Lee–Liu equation in optical fibers

This paper obtains a combined optical solitary wave solution that is modeled by nonlinear Chen–Lee–Liu equation (NCLE) which arises in the context of temporal pulses along optical fibers associated with the self-steepening nonlinearity using the complex envelope function ansatz. The novel combined solitary wave describes bright and dark solitary wave properties in the same expression. The intensity and the nonlinear phase shift of the combined solitary wave solution are reported. Moreover, the Lie point symmetry generators or vector fields of a system of partial differential equations (PDEs) which is acquired by transforming the NCLE to a real and imaginary parts are derived. It is observed that the obtained system is nonlinearly self-adjoint with an explicit form of a differential substitution satisfying the nonlinear self-adjoint condition. Then we use these facts to establish a set of conservation laws (Cls) for the system using the general Cls theorem. Numerical simulation and physical interpretations of the obtained results are demonstrated with interesting figures showing the meaning of the acquired results. It is hoped that the results reported in this paper can enrich the nonlinear dynamical behaviors of the NCLE.

Optical solitons, conservation laws and modulation instability analysis for the modified nonlinear Schrödinger’s equation for Davydov solitons

In this paper, the optical solitons to the modified nonlinear Schrödinger’s equation for davydov solitons are investigate. The modified F-expansion method is the integration technique employed to achieve this task. This yielded a combined and other soliton solutions. The Lie point symmetry generators of a system of partial differential equations acquired by decomposing the equation into real and imaginary components are derived. We prove that the system is nonlinearly self-adjoint with an explicit form of a differential substitution satisfying the nonlinear self-adjoint condition. Then we use these facts to construct a set of local conservation laws (Cls) for the system using the general Cls theorem presented by Ibragimov. Furthermore, the modulation instability (MI) is analyzed based on the standard linear-stability analysis and the MI gain spectrum is got. Numerical simulation of the obtained results are analyzed with interesting figures showing the physical meaning of the solutions.

Optical solitary waves, conservation laws and modulation instability analysis to the nonlinear Schrödinger's equation in compressional dispersive Alvèn waves

In this paper, the sine-Gordon equation expansion method (SGEM) is used to acquire the optical solitary waves to the nonlinear Schrödinger's equation (NLSE) that arises from compressional dispersive Alvèn (CDA) waves. As a result of the operations, dark, bright, dark–bright and singular optical solitary waves are derived. The solitary waves appear with all necessary constraint conditions which guarantee their existence. The Lie point symmetry generators of a system of partial differential equations (PDEs) obtained by transforming the equation into real and imaginary parts are derived. We prove that the system is nonlinearly self-adjoint with an explicit form of a differential substitution satisfying the nonlinear self-adjoint condition. Then we use these facts to construct a set of conservation laws (Cls) for the system using the general Cls theorem presented by Ibragimov. Furthermore, the modulation instability (MI) is studied based on the standard linear-stability analysis and the MI gain spectrum is got. Numerical simulation of the obtained results are analyzed with interesting figures showing the physical meaning of the solutions.

A study on the adjoint of pseudo-differential operators on compact lie groups

In this paper we find a formula for the adjoint of bounded pseudo-differential operators on compact Lie groups, which yields a necessary and sufficient condition for self-adjointness of these operators. By a similar method we find necessary and sufficient conditions for bounded pseudo-differential operators on compact Lie groups to be normal or unitary and we give some examples of our results on . Then we show that similar results are true for some bounded pseudo-differential operators on .

Absolutely Continuous Spectrum for Laplacians on Radial Metric Trees and Periodicity

On an infinite, radial metric tree graph we consider the corresponding Laplacian equipped with self-adjoint vertex conditions from a large class including $$\delta $$ - and weighted $$\delta '$$ -couplings. Assuming the numbers of different edge lengths, branching numbers and different coupling conditions to be finite, we prove that the presence of absolutely continuous spectrum implies that the sequence of geometric data of the tree as well as the coupling conditions are eventually periodic. On the other hand, we provide examples of self-adjoint, non-periodic couplings which admit absolutely continuous spectrum.

Optical solitons, nonlinear self-adjointness and conservation laws for the cubic nonlinear Shrödinger's equation with repulsive delta potential

In this paper, the complex envelope function ansatz method is used to acquire the optical solitons to the cubic nonlinear Shrödinger's equation with repulsive delta potential (δ−NLSE). The method reveals dark and bright optical solitons. The necessary constraint conditions which guarantee the existence of the solitons are also presented. We studied the δ−NLSE by analyzing a system of partial differential equations (PDEs) obtained by decomposing the equation into real and imaginary components. We derive the Lie point symmetry generators of the system and prove that the system is nonlinearly self-adjoint with an explicit form of a differential substitution satisfying the nonlinear self-adjoint condition. Then we use these facts to establish a set of conserved vectors for the system using the general Cls theorem presented by Ibragimov. Some interesting figures for the acquired solutions are also presented.