Constructing quantum observables and self-adjoint extensions of symmetric operators. II. Differential operators

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. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 3–36, August, 2007.     

A vertical eigenfunction expansion for the propagation of sound in a downward-refracting atmosphere over a complex impedance plane

The propagation of sound in a stratified downward-refracting atmosphere over a complex impedance plane is studied. The problem is solved by separating the wave equation into vertical and horizontal parts. The vertical part has non-self-adjoint boundary conditions, so that the well-known expansion in orthonormal eigenfunctions cannot be used. Instead, a less widely known eigenfunction expansion for non-self-adjoint ordinary differential operators is employed. As in the self-adjoint case, this expansion separates the acoustic field into a ducted part, expressed as a sum over modes which decrease exponentially with height, and an upwardly propagating part, expressed as an integral over modes which are asymptotically (with height) plane waves. The eigenvalues associated with the modes in this eigenfunction expansion are, in general, complex valued. A technique is introduced which expresses the non-self-adjoint problem as a perturbation of a self-adjoint one, allowing one to efficiently find the complex eigenvalues without having to resort to searches in the complex plane. Finally, an application is made to a model for the nighttime boundary layer.     

Generalized uncertainty principle and self-adjoint operators

In this work we explore the self-adjointness of the GUP-modified momentum and Hamiltonian operators over different domains. In particular, we utilize the theorem by von-Neumann for symmetric operators in order to determine whether the momentum and Hamiltonian operators are self-adjoint or not, or they have self-adjoint extensions over the given domain. In addition, a simple example of the Hamiltonian operator describing a particle in a box is given. The solutions of the boundary conditions that describe the self-adjoint extensions of the specific Hamiltonian operator are obtained.     

AN EXACT METHOD FOR THE FREE VIBRATION ANALYSIS OF TIMOSHENKO-KELVIN BEAMS WITH OSCILLATORS

In this paper, a modern exact method is proposed for solving the problem of free vibrations of a Timoshenko-type viscoelastic beam with discrete rigid bodies, connected to the beam by means of viscoelastic constraints. The phenomenon of free vibrations of this discrete-continuous system is described by a set of three partial and two subsystem ordinary differential equations with generalized boundary conditions and initial conditions. Vector notation of the equations allows one to identify the self-adjoint linear operators of inertia, stiffness and damping. In this case, these operators are not homothetic hence a separation of variables in this set of equations is possible only in a complex Hilbert space. Such separation of variables leads to ordinary differential equations of motion with respect to time as well as to a set of three ordinary differential equations with respect to a spatial variable and two subsystem algebraical equations. Solution of the boundary-value problem was carried out in the classical way, but its results are complex conjugated. Using these results and the fundamental principle, describing the orthogonality property of complex eigenvectors, the problem of free vibrations of the system with arbitrary initial conditions has been finally solved exactly.     

A geometrical representation of the interacting boson model of nuclei

The representation of the interacting boson model hamiltonian as a second-order differential operator in geometrical variables is studied in detail. It is shown that, with appropriate boundary conditions and biorthogonal weight functions, it reproduces exactly both the spectrum and matrix elements of operators of the algebraic boson model. It can be written in self-adjoint form and expanded in a symmetrized moment expansion, allowing the identification of collective mass parameters and energy surfaces, but differs in detail from the conventional geometrical collective model.     

Bisection algorithm to calculate hybrid modes of birefringent planar graded index waveguides

Guided modes of a planar dielectric waveguide which encounter a nondiagonal permittivity tensor are neither TE nor TM, but hybrid. They are described by a pair of coupled second-order differential equations for the transversal electric and magnetic field components. We construct a real-valued function which plays the role of the transversal electric or magnetic field in the uncoupled Sturm-Liouville differential equation for TE or TM modes. The number of zeroes, or nodes, of this function labels the modes. The nodes increase with the prospective propagation constant. This fact is proven by constructing suitable self-adjoint operators and referring to the minimax principle. The nodal properties allow to formulate an efficient bisection algorithm for effective indices and field distributions of guided hybrid modes.     

Self-adjoint extensions and unitary operators on the boundary

We establish a bijection between the self-adjoint extensions of the Laplace operator on a bounded regular domain and the unitary operators on the boundary. Each unitary encodes a specific relation between the boundary value of the function and its normal derivative. This bijection sets up a characterization of all physically admissible dynamics of a nonrelativistic quantum particle confined in a cavity. Moreover, this correspondence is discussed also at the level of quadratic forms. Finally, the connection between this parametrization of the extensions and the classical one, in terms of boundary self-adjoint operators on closed subspaces, is shown.     

Realization of the three-dimensional quantum Euclidean space by differential operators

The three-dimensional quantum Euclidean space is an example of a non-commutative space that is obtained from Euclidean space by q-deformation. Simultaneously, angular momentum is deformed to , it acts on the q-Euclidean space that becomes a -module algebra this way. In this paper it is shown, that this algebra can be realized by differential operators acting on functions on . On a factorspace of a scalar product can be defined that leads to a Hilbert space, such that the action of the differential operators is defined on a dense set in this Hilbert space and algebraically self-adjoint becomes self-adjoint for the linear operator in the Hilbert space. The self-adjoint coordinates have discrete eigenvalues, the spectrum can be considered as a q-lattice. Received: 27 June 2000 / Published online: 9 August 2000     

Intertwining operators for solving differential equations,with applications to symmetric spaces

The use of intertwining operators to solve both ordinary and partial differential equations is developed. Classes of intertwining operators are constructed which transform between Laplacians which are self-adjoint with respect to different non-trivial measures. In the two-dimensional case, the intertwining operator transforms a non-separable partial differential operator to a separable one. As an application, the heat kernels on the rank 1 and rank 2 symmetric spaces are constructed.     

Simple explicit formulas for Gaussian path integrals with time-dependent frequencies

Quadratic fluctuations require an evaluation of ratios of functional determinants of second-order differential operators. We relate these ratios to the Green functions of the operators for Dirichlet, periodic and antiperiodic boundary conditions on a line segment. This permits us to take advantage of Wronski's construction method for Green functions without knowledge of eigenvalues. Our final formula expresses the ratios of functional determinants in terms of an ordinary 2 × 2 determinant of a constant matrix constructed from two linearly independent solutions of the homogeneous differential equations associated with the second-order differential operators. For ratios of determinants encountered in semiclassical fluctuations around a classical solution, the result can further be expressed in terms of this classical solution. In the presence of a zero mode, our method allows for a simple universal regularization of the functional determinants. For Dirichlet's boundary condition, our result is equivalent to Gelfand-Yaglom's. Explicit formulas are given for a harmonic oscillator with an arbitrary time-dependent frequency.     

Pseudo-differential projections and the topology of certain spaces of elliptic boundary value problems

We calculate the homotopy groups of the space of elliptic boundary value problems for an elliptic differential operatorA of a first order and of the space of elliptic self-adjoint boundary value problems whenA is a formally self-adjoint. In particular we show that the spectral flow of anS ¹ family of self-adjoint elliptic boundary value problems is well defined. This provides some information on spectral properties along the lines of the Vafa-Witten approach to spectral inequalities.     

Higher approximations for quasiclassical trajectory-coherent states of a nonrelativistic particle

A scheme is developed for constructing higher approximations for quasiclassical trajectory-coherent states of a nonrelativistic particle. The basic property of these states is that the quantum-mechanical mean of the coordinate and momentum operators in these states are accurate solutions of the Hamiltonian system in the limit as 0. The quantum-mechanical means of the energy, coordinate and momentum operators and the mean square deviations from the classical trajectory are calculated with an accuracy of order 0(²). Higher approximations are obtained by the Maslov complexgrowth method.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 14–18, October, 1987.     

Self-adjoint Operators and Conserved Currents

It is shown that there exists a conservedcurrent associated with any system of homogeneous linearpartial differential equations that can be written interms of a self-adjoint operator. By showing that the linearized Einstein-Maxwell,Einstein-Klein-Gordon and Einstein-Weyl equations can beexpressed in terms of self-adjoint operators, aconserved current is obtained in each case. Theconserved current associated with the perturbations of solutionsof the Einstein and the Einstein-Maxwell equationscoincides with the symplectic current found by otherauthors.     

Path integral solution of linear second order partial differential equations I: the general construction

A path integral is presented that solves a general class of linear second order partial differential equations with Dirichlet/Neumann boundary conditions. Elementary kernels are constructed for both Dirichlet and Neumann boundary conditions. The general solution can be specialized to solve elliptic, parabolic, and hyperbolic partial differential equations with boundary conditions. This extends the well-known path integral solution of the Schrödinger/diffusion equation in unbounded space. The construction is based on a framework for functional integration introduced by Cartier/DeWitt-Morette.     

A General Approximation of Quantum Graph Vertex Couplings by Scaled Schrödinger Operators on Thin Branched Manifolds

We demonstrate that any self-adjoint coupling in a quantum graph vertex can be approximated by a family of magnetic Schrödinger operators on a tubular network built over the graph. If such a manifold has a boundary, Neumann conditions are imposed at it. The procedure involves a local change of graph topology in the vicinity of the vertex; the approximation scheme constructed on the graph is subsequently ‘lifted’ to the manifold. For the corresponding operator a norm-resolvent convergence is proved, with the natural identification map, as the tube diameters tend to zero.     

A compatible and conservative spectral element method on unstructured grids

We derive a formulation of the spectral element method which is compatible on very general unstructured three-dimensional grids. Here compatible means that the method retains discrete analogs of several key properties of the divergence, gradient and curl operators: the divergence and gradient are anti-adjoints (the negative transpose) of each other, the curl is self-adjoint and annihilates the gradient operator, and the divergence annihilates the curl. The adjoint relations hold globally, and at the element level with the inclusion of a natural discrete element boundary flux term.     

Quantization postulate in quantum mechanics

The procedure of transition from classical observables to quantum (operator) observables in quantum mechanics is discussed. By an example it is shown that, even in simple cases, the method of self-adjoint extensions of formal differential expressions for defining physical observables as operators is not equivalent to the procedure of forming operator functions corresponding to these observables. This inequivalence is not a formal one but has physical consequences connected with the compatibility of observables.