Basis Of Hilbert Space Research Articles

DYNAMIC TRACTION VECTOR FIELD ESTIMATION IN A STRUCTURE USING HYBRID STRAIN ANALYSIS

The three-dimensional dynamic traction vector of a vibrating structure is obtained with the aid of the so-called hybrid strain analysis. Hybrid strain analysis is a method where vibration response measured at a limited number of points and numerically approximated continuous Hilbert space basis functions combined with spatial differentiation yields the frequency-dependent strain tensor field in a vibrating structure.Here, an extension and special application of hybrid strain analysis is proposed. In a special case with a built-up structure with unknown dynamic properties in the interfaces between the parts, the frequency- and spatial-dependent stress tensors, and thus also the traction vector, are obtained for the structural parts using the proposed technique.The method is validated using numerical simulation of measured vibration responses, with very good agreement between the calculated and the true traction vector. The calculated traction vector is shown to converge towards the true traction vector in an arbitrary small area on the boundary of the structure.The method is demonstrated for isotropic elastic material properties. This is, however, no limitation for the method; it can be also applied to a structure with anelastic material properties.

Strong-coupling-expansion analysis of the false-vacuum decay rate of the lattice ϕ4model in 1 + 1 dimensions

Strong-coupling expansion is performed for the lattice phi^4 model in 1+1 dimensions. Because the strong-coupling limit itself is not solvable, we employed numerical calculations so as to set up unperturbed eigensystems. Restricting the number of Hilbert-space bases, we performed linked-cluster expansion up to eleventh order. We carried out alternative simulation by means of the density-matrix renormalization group. Thereby, we confirmed that our series-expansion data with a convergence-acceleration trick are in good agreement with the simulation result. Through the analytic continuation to the domain of negative biquadratic interaction, we obtain the false-vacuum decay rate. Contrary to common belief that tunnelling phenomenon lies out of perturbative treatments, our series expansion reproduces the instanton-theory behaviour for high potential barrier. For shallow barrier, on the contrary, our result tells that the relaxation is no more described by instanton, but the decay rate acquires notable enhancement.

Symmetric approximation of frames and bases in Hilbert spaces

We introduce the symmetric approximation of frames by normalized tight frames extending the concept of the symmetric orthogonalization of bases by orthonormal bases in Hilbert spaces. We prove existence and uniqueness results for the symmetric approximation of frames by normalized tight frames. Even in the case of the symmetric orthogonalization of bases, our techniques and results are new. A crucial role is played by whether or not a certain operator related to the initial frame or basis is Hilbert-Schmidt.

CALCULATING THE DYNAMIC STRAIN TENSOR FIELD USING MODAL ANALYSIS AND NUMERICAL DIFFERENTIATION

Based on vibration response measured at a limited number of points and continuous Hilbert space basis functions combined with spatial differentiation, a new method for strain tensor field assessment, in the frequency domain, is proposed. Essentially, results from hybrid modal analysis (HMA), are transformed from the displacement space to the strain space by use of finite difference schemes. This approach will produce a continuous strain (second order) tensor, except on a number of surfaces where material discontinuities are present.Hybrid strain analysis (HSA), interpreted as dynamic strain analysis based on HMA, may be done in several ways. The method proposed here is the first realization of HSA and it has been verified by an experimental test case. The predicted results correspond well with measured strain responses.The method is applicable to a vibrating structure, regardless of the material properties of the body. Hence, any possible damping, e.g., material damping, damping in joints between structural parts, etc., need not be known a priori. Moreover, the zero frequency elastic properties and mass distribution of the structure need not be the true ones; any (meaningful) arbitrary values of the elastic properties and mass distribution can be assumed in order to obtain the Hilbert space basis functions. Material inhomogeneities present, i.e., in a piecewise continuous structure, will cause the method (when applied with unknown material properties) to be somewhat inaccurate in the vicinity around the discontinuity. The derived model is valid for a specific excitation, not necessarily known.

Continued fraction representation of the Coulomb Green’s operator and unified description of bound, resonant and scattering states

If a quantum-mechanical Hamiltonian has an infinite symmetric tridiagonal (Jacobi) matrix form in some discrete Hilbert-space basis representation, then its Green's operator can be constructed in terms of a continued fraction. As an illustrative example we discuss the Coulomb Green's operator in a Coulomb-Sturmian basis representation. Based on this representation, a quantum-mechanical approximation method for solving Lippmann-Schwinger integral equations can be established, which is equally applicable for bound-, resonant- and scattering-state problems with free and Coulombic asymptotics as well. The performance of this technique is illustrated with a detailed investigation of a nuclear potential describing the interaction of two $\ensuremath{\alpha}$ particles.

Grid methods and Hilbert space basis for simulations of quantum dynamics

We discuss spatial grid methods adapted to the structure of Hilbert spaces, used to simulate quantum mechanical systems. We review the construction of Finite Basis Representation (FBR) and the Discrete Variable Representation (DVR). A mixed representation (pseudo-spectral method) is constructed through a quadrature relation linking both bases.

Unconditional bases of Hilbert spaces composed of values of entire vector functions of exponential type

Unconditional bases of Hilbert spaces composed of values of entire vector functions of exponential type

Complex Hamiltonian evolution equations and field theory

The form of the complex Hamiltonian evolution equations (linear and nonlinear) is deduced here from that of the classical Hamiltonian systems. Their properties and those of the associated continuous Poisson bracket are studied. It is shown that if any space-localized solution of the above type equations is expanded in any Hilbert space basis the expansion coefficients are canonically conjugate variables and obey Hamiltonian systems of complex equations. The last is proved also for sets of “canonically conjugate” functionals of general type. As an application, commutation relations which are equivalent in form to Dirac’s quantization rules (the constant in place of ℏ is undetermined) are obtained. The question of how a nonlinear field theory may contain quantum mechanics as a special case is discussed in view of this paper’s results.

Unified treatment of the Coulomb and harmonic oscillator potentials inDdimensions

Quantum mechanical models and practical calculations often rely on some exactly solvable models like the Coulomb and the harmonic oscillator potentials. The D dimensional generalized Coulomb potential contains these potentials as limiting cases, thus it establishes a continuous link between the Coulomb and harmonic oscillator potentials in various dimensions. We present results which are necessary for the utilization of this potential as a model and practical reference problem for quantum mechanical calculations. We define a Hilbert space basis, the generalized Coulomb–Sturmian basis, and calculate the Green’s operator on this basis and also present an SU(1,1) algebra associated with it. We formulate the problem for the one-dimensional case, too, and point out that the complications arising due to the singularity of the one-dimensional Coulomb problem can be avoided with the use of the generalized Coulomb potential.

Real modes of vibration and hybrid modal analysis

Based on continuous Hilbert space basis functions and mean square convergence properties of spatial generalised Fourier series a new technique for hybrid modal analysis (HMA) is proposed. The technique utilises a mix of experimental, measured vibration responses and good numerical approximations of well defined, three-dimensional, real, normal modes. The modes are assumed to be solutions to an elastic, eigenvalue problem corresponding to the true geometry of the analysed body or structure. Starting from a known geometry and a suitable set of modes, it is shown that the measured data may be supplemented with spatial information about the whole vibrational displacement field which is correctly represented by the chosen set of complete (Hilbert space), modal basis vector fields. The spatial information is extracted from the measured vibration responses by curve fitting a truncated modal response model to the data. As a result of the curve fit a number of generalised Fourier coefficient spectra are obtained which together with the corresponding modes may be used to predict or simulate responses which were not measured or used in the curve fit. Given also the true mass density field of the body, the time average of the kinetic energy of the whole body may be approximated using only a restricted set of measured vibration responses. The method is completely general and applicable in all cases where the vibrational displacements are small enough compared to the characteristic dimensions of the analysed body. The effect of modal truncation and means to approximate corresponding dynamic displacement residuals are discussed for linear, anisotropic solid bodies and structures with general damping. Modal coupling due to damping and unknown elastic properties are also discussed. The method has been tested, with very good results, on experimental data measured with a laser doppler vibrometer on a plexiglas plate excited with a shaker in the frequency range 25–500 Hz.

A representation theorem for Schauder bases in Hilbert space

A sequence of vectors { f 1 , f 2 , f 3 , … } \{f_1,f_2,f_3,\dotsc \} in a separable Hilbert space H H is said to be a Schauder basis for H H if every element f ∈ H f\in H has a unique norm-convergent expansion \[ f = ∑ c n f n . f=\sum c_nf_n. \] If, in addition, there exist positive constants A A and B B such that \[ A ∑ | c n | 2 ≤ ‖ ∑ c n f n ‖ 2 ≤ B ∑ | c n | 2 , A\sum |c_n|^2\le \left \|\sum c_nf_n\right \|^2\le B\sum |c_n|^2, \] then we call { f 1 , f 2 , f 3 , … } \{f_1,f_2,f_3,\dotsc \} a Riesz basis. In the first half of this paper, we show that every Schauder basis for H H can be obtained from an orthonormal basis by means of a (possibly unbounded) one-to-one positive self adjoint operator. In the second half, we use this result to extend and clarify a remarkable theorem due to Duffin and Eachus characterizing the class of Riesz bases in Hilbert space.

Conditions for the completeness of the spectral domain of a harmonizable process

We generalize a theorem of Köthe and Toeplitz on unconditional bases in Hilbert spaces to Hilbert space-valued measures. This leads to a necessary and sufficient condition for the completeness of the spectral domain of a weakly harmonizable process whose shift operator exists and is invertible. A process in this class has a complete spectral domain if and only if it is the image of a stationary process under a topological isomorphism.

Fractional wave-function revivals in the infinite square well

We describe the time evolution of a wave function in the infinite square well using a fractional revival formalism, and show that at all times the wave function can be described as a superposition of translated copies of the initial wave function. Using the model of a wave form propagating on a dispersionless string from classical mechanics to describe these translations, we connect the reflection symmetry of the square-well potential to a reflection symmetry in the locations of these translated copies, and show that they occur in a ``parity-conserving'' form. The relative phases of the translated copies are shown to depend quadratically on the translation distance along the classical path. We conclude that the time-evolved wave function in the infinite square well can be described in terms of translations of the initial wave-function shape, without approximation and without any reference to its energy eigenstate expansion. That is, the set of translated initial wave functions forms a Hilbert space basis for the time-evolved wave functions.

Wavelet bases for a unitary operator

Let T be a unitary operator on a complex Hilbert space ℋ, and X, Y be finite subsets of ℋ. We give a necessary and sufficient condition for TZ(X): {Tnx: n ∈ Z, x ∈ X} to be a Riesz basis of its closed linear span 〈TZ(X)〉. If TZ(X) and TZ(Y) are Riesz bases, and 〈TZ(X)〉⊂〈TZ(Y)〉, then X is extendable to X′ such that TZ(X′) is a Riesz basis of TZ(Y) The proof provides an algorithm for the construction of Riesz bases for the orthogonal complement of 〈TZ(X)〉 in 〈TZ(Y)〉. In the case X consists of a single B-spline, the algorithm gives a natural and quick construction of the spline wavelets of Chui and Wang [2, 3]. Further, the duality principle of Chui and Wang in [3] and [4] is put in the general setting of biorthogonal Riesz bases in Hilbert space.

Spinon bases, Yangian symmetry and fermionic representations of Virasoro characters in conformal field theory

We study the description of the SU(2), level k = 1, Wess-Zumino-Witten conformal field theory in terms of the modes of the spin- 1 2 affine primary field φ α . These are shown to satisfy generalized ‘canonical commutations relations’, which we use to construct a basis of Hilbert space in terms of representations of the Yangian Y( sl 2). Using this description, we explicitly derive so-called ‘fermionic representations’ of the Virasoro characters, which were first conjectured by R. Kedem, T. Klassen, B. McCoy and E. Melzer [Phys. Lett. B 307 (1993) 68, and references therein]. We point out that similar results are expected for a wide class of rational conformal field theories.

Pre-frame operators, Besselian frames, and near-Riesz bases in Hilbert spaces

A problem of enduring interest in connection with the study of frames in Hubert space is that of characterizing those frames which can essentially be regarded as Riesz bases for computational purposes or which have certain desirable properties of Riesz bases. In this paper we study several aspects of this problem using the notion of a pre-frame operator and a model theory for frames derived from this notion. In particular, we show that the deletion of a finite set of vectors from a frame { x n } n = 1 ∞ \{ {x_n}\} _{n = 1}^\infty leaves a Riesz basis if and only if the frame is Besselian (i.e., ∑ n = 1 ∞ a n x n {\sum } _{n = 1}^\infty \,{a_n}{x_n} converges ⇔ ( a n ) ∈ l 2 \Leftrightarrow ({a_n}) \in {l^2} ).

Basis of the Ponzano-Regge-Turaev-Viro-Ooguri quantum-gravity model is the loop representation basis.

We show that the Hilbert space basis that defines the Ponzano-Regge- Turaev-Viro-Ooguri combinatorial definition of 3-d Quantum Gravity is the same as the one that defines the Loop Representation. We show how to compute lengths in Witten's 3-d gravity and how to reconstruct the 2-d geometry from a state of Witten's theory. We show that the non-degenerate geometries are contained in the Witten's Hilbert space. We sketch an extension of the combinatorial construction to the physical 4-d case, by defining a modification of Regge calculus in which areas, rather than lengths, are taken as independent variables. We provide an expression for the scalar product in the Loop representation in 4-d. We discuss the general form of a nonperturbative quantum theory of gravity, and argue that it should be given by a generalization of Atiyah's topological quantum field theories axioms.

Basis set reduction in Hilbert space.

We present Hilbert-space basis set reduction as a novel approach to reduce the computational effort of accurate correlation calculations for large basis sets. We motivate the method by an examination of the perturbative corrections in scaling theory and present a criterion for the choice of the internal basis. The method is illustrated with two calculations for the ground state of carbon and the energy curve of the beryllium dimer: In either case the errors introduced are less than 1% of the correlation energy. The method scales as the second power of the number of the basis functions and can exploit the benefits of massive parallelization.

On one class of bases in Hilbert spaces and the similarity problem for volterra operators

On one class of bases in Hilbert spaces and the similarity problem for volterra operators

WAVE FUNCTION REPRESENTATION FOR THE GENERALIZED EIGENSTATES OF SU(1, 1) NONCOMPACT GENERATORS

The SU(1, 1) algebra is represented by a two bosonic operator realization. This allows one to obtain a remarkable simplification of the SU(1, 1) noncompact generator eigenvectors by representing the Hilbert space basis in terms of two harmonic oscillator wave functions.