Inverse eigenvalue problems for layered media

Hald, Ole H.

doi:10.1002/cpa.3160300105

Cited by 12 publications

(10 citation statements)

References 33 publications

(27 reference statements)

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“…The efforts of many authors have been concentrated on the inverse eigenvalue problems on the determination of q(x) and v(x) from the eigen values of the operator [43][44][45][46][47][48], One can cite more recent investigations, related to a polynomial form of potential coefficients [49,52,53] in the second order differential operator and as well as the intertwining technique for wave equations with linearly energy dependent potentials [34].…”

Section: Introductionmentioning

confidence: 99%

“…Wave equations with energy dependent potentials are widely used in relativistic and nonrelativistic quan tum mechanics for a long time [42][43][44][45][46][47][48][49][50][51][52][53], in particular, for investigation of waveguide devices in such modern fields of physics as optoelectronics [33,54] and photo electronics [57]. The efforts of many authors have been concentrated on the inverse eigenvalue problems on the determination of q(x) and v(x) from the eigen values of the operator [43][44][45][46][47][48], One can cite more recent investigations, related to a polynomial form of potential coefficients [49,52,53] in the second order differential operator and as well as the intertwining technique for wave equations with linearly energy dependent potentials [34].…”

Section: Introductionmentioning

confidence: 99%

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Intertwining relations and Darboux transformations for the wave equations

Suzko

Velicheva

2012

Phys. Part. Nuclei

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Using the intertwining operator technique we construct Darboux transformations for the wave equation with position dependent effective mass and with linearly energy dependent potentials. The for mally adjoint generators of supersymmetry and two formally self adjoint superpartner Hamiltonians are con structed and they close a quadratic pseudo superalgebra. The Darboux transformations are constructed in differential and integral forms and an interrelation is established between them. The approach is applied to generation of isospectral potentials with additional or removal bound states or construction of new partner potentials without changing the spectrum, i.e. fully isospectral potentials. The method is illustrated by some concrete examples. The influence of distance between levels on the form of potentials is investigated. In par ticular, asymmetric double well potentials are generated.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Intertwining relations and Darboux transformations for the wave equations

Suzko

Velicheva

2012

Phys. Part. Nuclei

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“…Regularity and uniform positivity conditions (19) easily follow from the corresponding properties of the coefficients p and ρ. Let us check (20).…”

Section: Reduction To Impedance-type Formmentioning

confidence: 99%

“…Under the assumption that both the undamaged and damaged systems are symmetric, that is p, p and ρ are even functions with respect to the mid-point of the rod, we show how to construct the stiffness coefficient p such that the damaged rod (p, ρ) has exactly the prescribed (measured) values of the first N eigenfrequencies of the Dirichlet spectrum. Therefore, one expects to recover information about the damage from the behavior of the reconstructed coefficient p. It should be noticed that the analysis is restricted to symmetric rods since, in this case, the knowledge of the full Dirichlet spectrum determines uniquely the function p, provided that ρ is given, see [19] and [24] for uniqueness results for more general type of second-order Sturm-Liouville operators.…”

Section: Introductionmentioning

confidence: 99%

The use of quasi-isospectral operators for damage detection in rods

2017

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We consider the inverse problem of reconstructing the axial stiffness of a damaged rod from the knowledge of a finite number of resonant frequencies of the free axial vibration under supported end conditions. The damage is described as a reduction of the axial stiffness, and the undamaged and damaged configurations of the rod are assumed to be symmetric. The method is based on repeated determination of quasi-isospectral rod operators, that is rods which have the same spectrum of a given rod with the exception of a single resonant frequency which is free to move in a prescribed interval. The reconstruction procedure is explicit and it is numerically implemented and tested for the identification of single and multiple localized damages. The sensitivity of the technique to the number of frequencies used and to the shape, intensity and position of the damages, as well as to the presence of noise in the data, is evaluated and discussed. The effect of suitable filtering of the results based on a priori information on the physics of the problem is proposed. An experimental application to the identification of localized damage in a free-free steel rod is also presented.

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“…This formula leads to uniqueness theories, as shown in [3,7,8,14]. However, taking into account the relationship between the lowest eigenvalue and the boundary conditions, it is clearly stated that if the boundary conditions are given and mixed, the lowest eigenvalue is neglected.…”

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confidence: 99%