Self-adjoint Conditions Research Articles

Generalized Trigonometric Integral Transforms

Some generalizations of the Fourier and Hartley transforms on ℝ and ℝ n are studied. These transforms involve functional coefficients. The Parseval identities, convolution formulas, conditions of self-adjointness and unitary property for new transforms are obtained.

On point interactions realised as Ter-Martirosyan–Skornyakov Hamiltonians

For quantum systems of zero-range interaction we discuss the mathematical scheme within which modelling the two-body interaction by means of the physically relevant ultra-violet asymptotics known as the "Ter-Martirosyan-Skornyakov condition" gives rise to a self-adjoint realisation of the corresponding Hamiltonian. This is done within the self-adjoint extension scheme of Krein, Visik, and Birman. We show that the Ter-Martirosyan-Skornyakov asymptotics is a condition of self-adjointness only when is imposed in suitable functional spaces, and not just as a pointwise asymptotics, and we discuss the consequences of this fact on a model of two identical fermions and a third particle of different nature.

1-D Schrödinger Operators with Local Interactions on a Discrete Set with Unbounded Potential

We study spectral properties of the one-dimensional Schrodinger operators \( {\mathrm{H}}_{\mathrm{X},\alpha, \mathrm{q}}:=-\frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}+\mathrm{q}(x)+{\varSigma_x}_{{}_n}\in X{\alpha}_n\delta \left(x-{x}_n\right) \) with local interactions, d* = 0, and an unbounded potential q being a piecewise constant function, by using the technique of boundary triplets and the corresponding Weyl functions. Under various sufficient conditions for the self-adjointness and discreteness of Jacobi matrices, we obtain the condition of self-adjointness and discreteness for the operator H X,α,q.

Dirac–Krein Systems on Star Graphs

We study the spectrum of a self-adjoint Dirac-Krein operator with potential on a compact star graph $\mathcal G$ with a finite number $n$ of edges. This operator is defined by a Dirac-Krein differential expression with summable matrix potentials on each edge, by self-adjoint boundary conditions at the outer vertices, and by a self-adjoint matching condition at the common central vertex of $\mathcal G$. Special attention is paid to Robin matching conditions with parameter $\tau \in\mathbb R\cup\{\infty\}$. Choosing the decoupled operator with Dirichlet condition at the central vertex as a reference operator, we derive Krein's resolvent formula, introduce corresponding Weyl-Titchmarsh functions, study the multiplicities, dependence on $\tau$, and interlacing properties of the eigenvalues, and prove a trace formula. Moreover, we show that, asymptotically for $R\to \infty$, the difference of the number of eigenvalues in the intervals $[0,R)$ and $[-R,0)$ deviates from some integer $\kappa_0$, which we call dislocation index, at most by $n+2$.

Magnetic fields on resistance spaces

On a metric measure space X that supports a regular, strongly local resistance form we consider a magnetic energy form that corresponds to the magnetic Laplacian for a particle confined to X . We provide sufficient conditions for closability and self-adjointness in terms of geometric conditions on the reference measure without assuming energy dominance.

Nonlinear Self-Adjointness and Generalized Conserved Quantities of the Inviscid Burgers’ Equation with Nonlinear Source

The conservation laws of the inviscid Burgers’ equation under consideration have been studied recently using the concept of quasi self-adjointness and self-adjointness. These two concepts have been extended to the notion of nonlinear self-adjointness to enable more conservation laws of differential equations, that are not achievable through them, be constructed. We explore this avenue in the present study and establish the generalized nonlinearly self-adjoint condition for the inviscid Burgers’ equation. This condition not only gives rise to new nontrivial independent conserved vectors, but also includes results of previous work as a particular case.

A simplified method of solving Birkhoffian function and Lagrangian

By using the calculus of variations, the conservative mechanical systems can be formulated by Lagrange's equations or Hamilton's equations, which are the basis of establishing, simplifying and integrating the equations of motion. Thus it is important to find the solutions of inverse problems for different dynamical systems so as to construct the most of the Lagrange's equations and Hamilton's equations. However, the Lagrangian or Hamiltonian formulation for a dynamical system, limited by the conditions of self-adjointness, is not directly universal if the physical variables remain without using Darboux transformations. Fortunately, Refs. [7, 11] show that based on the Cauchy-Kovalevsky theorem of the integrability conditions for partial differential equations and the converse of the Poincar lemma, it can be proved that there exists a direct universality of Birkhoff's equation for local Newtonian system by reducing the Newton's equations to a first-order form, which means that all local, analytic, regular, finite-dimensional, unconstrained or holonomic, conservative or non-conservative forms always admit, in a star-shaped neighborhood of a regular point of their variables, a representation in terms of first-order Birkhoff's equations in the coordinate and time variables of the experiment. The systems whose equations of motion are represented by the first-order Birkhoff's equations on a symplectic or a contact manifold spanned by the physical variables are called Birkhoffian systems. At present, one of the most important tasks of Birkhoffian mechanics is to study the method of constructing the Birkhoffian and Birkhoffian functions. However, due to the complexity of Birkhoffian system, there exist only a few of results in the literature. Among them, the most famous main methods in this problem are achieved by Santilli[Santilli R M 1983 Foundations of Theoretical Mechanics II (New York: Springer-Verlag) pp12-15]. But the redundant term in Santilli's second method which is used as the classical construction method, is always neglected. As a result, the calculation procedure is tedious and complicated, and the efficiency is not high. Therefore, it is necessary to simplify the Santilli's second method. In Section 2, we will review the first-order standard form of holonomic system in the frame of Cartesian coordinates, which is the starting point of our studying the Birkhoffian systems. In Section 3, the Birkhoff's equations and the key role of Birkhoffian dynamics functions for deriving Birkhoff's equations are introduced. In Section 4, the redundant items are eliminated by using some mathematical operation skills, and then a more simplified constructing method is put forward. In Section 5, the findings in this study are summarized. Through simplifying the Santilli's second method, we realize that the determining of the Birkhoff's equations by constructing the Birkhoffian functions is equivalent to the determining of its symplectic matrix. This view provides a new perspective for solving the problem of constructing the Birkhoffian functions. Finally, the simplified method is applied to Lagrangian inverse problem, and a simplified method of solving Lagrangian function is obtained.

A control strategy of Hamilton realization and mechanics Lagrangization in doubly-fed wind power generation system

Abstract With the nonlinear characteristic of doubly-fed wind power generation (DFWPG) system, Hamilton realization based on analytical mechanics plays a significant role in the system problems of stability analysis and controller design. Hamilton system expression of grid-side converter (GSC) and machine-side converter (MSC) is obtained and the stability control is designed with Lagrange theory of analytical mechanics in this paper. Firstly, to obtain the differential equations which satisfy satisfies self-adjoint conditions, coordinate transformations of dynamic equations of GSC and MSC are conducted. The system Hamilton function and Hamilton realization expression are determined based on Euler–Lagrange process and generalized force method. On the basis of feedback control theory the controller is designed, which can make the system tend to be asymptotically stable in the neighborhood of the equilibrium point. With Hessian matrix positive-definite the Hamilton system is determined to be stable. Additionally, within Matlab/Simulink environment, the transient simulation of DFWPG system is carried out and the effectiveness of controller derived in this paper is verified, by comparisons of response effects with PI control and regulation performances under different wind speeds. The rapid response of the dc bus voltage control, grid-side unit power factor control, a maximum capture of wind energy can be realized, using control design process of Hamilton realization. The system analysis and control design process of Hamilton realization have broad prospects of applications and developments.

Structures on self-adjoint vertex conditions of local Sturm-Liouville operators on graphs

We study the vertex conditions of local Sturm-Liouville operators on metric graphs. Our aim is to give a new description of vertex conditions defining the self-adjoint Sturm-Liouville operators and to clarify the natural geometric structure on the space of complex vertex conditions. Based on this description, we give the self-adjointness results for local Sturm-Liouville operators on finite graphs and the Povzner-Wienholtz-type self-adjointness results for local Sturm-Liouville operators on infinite graphs.

Lie symmetry analysis, nonlinear self-adjointness and conservation laws to an extended (2+1)-dimensional Zakharov–Kuznetsov–Burgers equation

This paper addresses an extended (2+1)-dimensional Zakharov–Kuznetsov–Burgers (ZKB) equation. The Lie symmetry analysis leads to many plethora of solutions to the equation. The nonlinear self-adjointness condition for the ZKB equation is established and subsequently used to construct simplified but infinitely many nontrivial and independent conserved vectors. In particular, we also get conservation laws of the equation with the corresponding Lie symmetry.

A study on the adjoint of pseudo-differential operators on $$\mathbb {S}^{1}$$ S 1 and $$\mathbb {Z}$$ Z

In this paper we find a formula for the adjoint of bounded pseudo-differential operators on $$L^2(\mathbb {S}^{1})$$ and also a formula for the adjoint of some bounded pseudo-differential operators on $$L^2(\mathbb {Z})$$ , which yealds a necessary and sufficient condition for self-adjointness of these operators.

Nonlinear self-adjointness and conservation laws of Klein–Gordon–Fock equation with central symmetry

The concept of nonlinear self-adjointness, introduced by Ibragimov, has significantly extends approaches to constructing conservation laws associated with symmetries since it incorporates the strict self-adjointness, the quasi self-adjointness as well as the usual linear self-adjointness. Using this concept, the nonlinear self-adjointness condition for the Klein–Gordon–Fock equation was established and subsequently used to construct simplified but infinitely many nontrivial and independent conserved vectors. The Noether’s theorem was further applied to the Klein–Gordon–Fock equation to explore more distinct first integrals, result shows that conservation laws constructed through this approach are exactly the same as those obtained under strict self-adjointness of Ibragimov’s method.

Sharp essential self-adjointness of relativistic Schrödinger operators with a singular potential

This paper is devoted to the study of essential self-adjointness of a relativistic Schrödinger operator with a singular homogeneous potential. From an explicit condition on the coefficient of the singular term, we provide a sufficient and necessary condition for essential self-adjointness.

Zero Modes of Quantum Graph Laplacians and an Index Theorem

We study zero modes of Laplacians on compact and non-compact metric graphs with general self-adjoint vertex conditions. In the first part of the paper the number of zero modes is expressed in terms of the trace of a unitary matrix $\mathfrak{S}$ that encodes the vertex conditions imposed on functions in the domain of the Laplacian. In the second part a Dirac operator is defined whose square is related to the Laplacian. In order to accommodate Laplacians with negative eigenvalues it is necessary to define the Dirac operator on a suitable Kre\u{\i}n space. We demonstrate that an arbitrary, self-adjoint quantum graph Laplacian admits a factorisation into momentum-like operators in a Kre\u{\i}n-space setting. As a consequence, we establish an index theorem for the associated Dirac operator and prove that the zero-mode contribution in the trace formula for the Laplacian can be expressed in terms of the index of the Dirac operator.

Self-adjoint boundary conditions and interlacing of eigenvalues for the Sturm-Liouville equation on graphs

Applying the approach of Kostrykin and Schrader, [14], an explicit characterisation of self-adjoint boundary conditions at the nodes or vertices of a graph for the Sturm-Liouville equation is given. This is then proven to be equivalent to the conditions for self-adjointness of the corresponding system boundary value problem with separated boundary conditions. In addition, using an example, it is shown that the complete graph configuration is incorporated in the system boundary condition at the terminal end point. Making use of the separated system formulation, via matrix Prufer angle techniques, an interlacing property of the eigenvalues for a self-adjoint Sturm-Liouville boundary value problem on a graph is ascertained. Mathematics subject classification (2010): 34B45, 34L05, 34B24, 34L15.

The Lagrangian of Analytical Mechanics and its Application in Power System with TCSC

The research and development of Hamilton system have provided an effective approach for the nonlinear analysis and stability control of power system. It also plays an important role in the control of flexible AC transmission system. The application of Lagrange analytical mechanics can improve the realization of Hamilton system and the controller design. The model of single-machine infinite-bus system with TCSC is extended to an even-order system in this paper. On the basis of self-adjoint conditions the standard form of Hamilton realization is obtained with the theory of Lagrange mechanics. By non-conservative analytical mechanics theory, the system controller is designed and Matlab programming is realized on the single-machine infinite-bus system. The simulation effects of the designed controller and general controller under three-phase grounded short-circuit faults are compared, which verifies the effectiveness of the proposed control strategy in this paper. The construction of Hamilton function and design of the controller have a broad prospect of applications.

Quasi self-adjointness of a class of third order nonlinear dispersive equations

In this paper we find the conditions of self-adjointness and quasi self-adjointness for a class of third order PDEs. As an application, in some cases, we get conservation laws.

Construction of regular and singular Green's functions

The Green function of singular limit-circle problems is constructed directly for the problem, not as a limit of sequences of regular Green's functions. This construction is used to obtain adjointness and self-adjointness conditions which are entirely analogous to the regular case. As an application, a new and explicit formula for the Green function of the classical Legendre problem is found.

The method of undetermined tensor for constructing Birkhoffian functions

The method of constructing Birkhoffian and Birkhoffian funcations of mechanical system is studied. Based on the conditions of self-adjointness and integrability conditions of linear partial differential equations, a method of undetermined tensor for constructing Birkhoffian and Birkhoffian funcations of mechanical systems is given. Two examples are given to illustrate the application of the results.

Dirichlet forms and stochastic completeness of graphs and subgraphs

We study Laplacians on graphs and networks via regular Dirichlet forms. We give a sufficient geometric condition for essential selfadjointness and explicitly determine the generators of the associated semigroups on all ℓ p , 1 ≦ p < ∞, in this case. We characterize stochastic completeness thereby generalizing all earlier corresponding results for graph Laplacians. Finally, we study how stochastic completeness of a subgraph is related to stochastic completeness of the whole graph.