Inverse Resonance Problem Research Articles

Inverse resonance problem with partial information on the interval

We consider the inverse resonance problem in scattering theory. In one-dimensional setting, the scattering matrix consists of entries of meromorphic functions. The resonances are defined as the poles of the meromorphic determinant. For the compactly supported perturbation, we are able to quantitatively estimate the zeros and poles of each meromorphic entry. The size of potential support is connected to the zero density of scattered wave field due to the form of Fourier transform. We will investigate certain properties of Fourier transforms in scattering theory and derive the inverse uniqueness on scattering source given certain knowledge on the perturbation and all the given resonances.

Finite range perturbations of finite gap Jacobi and CMV operators

Necessary and sufficient conditions are presented for a measure to be the spectral measure of a finite range perturbation of a Jacobi or CMV operator from a finite gap isospectral torus. The special case of eventually periodic operators solves an open problem of Simon [28, D.2.7].We also solve the inverse resonance problem: it is shown that an operator is completely determined by the set of its eigenvalues and resonances, and we provide necessary and sufficient conditions on their configuration for such an operator to exist.

The inverse resonance problem for left-definite Sturm–Liouville operators

We address inverse spectral and scattering problems for the half-line, left-definite Sturm–Liouville equation, −u″+qu=λwu. These problems have been considered recently as they are critical in integrating the Camassa–Holm equation. Previous results required that the support of w be free of gaps, relatively open intervals on which w=0 almost everywhere. We relax this condition and prove an inverse spectral theorem that tells to what extent the Weyl–Titchmarsh m-function or the spectral measure determines the coefficients of the equation. Note that, unlike the Schrödinger equation, knowing the spectral measure is not the same as knowing the m-function. We also prove an inverse resonance theorem that explains to what extent the eigenvalues and resonances determine the spectral measure (and, thus, the coefficients). Again, unlike the Schrödinger case, these data are not sufficient; the presence of w multiplying the spectral parameter complicates the analysis. However, we show that, in most cases, only one other number is needed to fully recover the spectral measure.

Stability of the Inverse Resonance Problem for Jacobi Operators

When the coefficients of a Jacobi operator are finitely supported perturbations of the 1 and 0 sequences, respectively, the left reflection coefficient is a rational function whose poles inside, respectively outside, the unit disk correspond to eigenvalues and resonances. By including the zeros of the reflection coefficient, we have a set of data that determines the Jacobi coefficients up to a translation as long as there is at most one half-bound state. We prove that the coefficients of two Jacobi operators are pointwise close assuming that the zeros and poles of their left reflection coefficients are ε-close in some disk centered at the origin.

Stability of the inverse resonance problem on the line

In the absence of a half-bound state, a compactly supported potential of a Schrödinger operator on the line is determined up to a translation by the zeros and poles of the meromorphically continued left (or right) reflection coefficient. The poles are the eigenvalues and resonances, while the zeros also are physically relevant. We prove that all compactly supported potentials (without half-bound states) that have reflection coefficients whose zeros and poles are ε-close in some disc centered at the origin are also close (in a suitable sense). In addition, we prove the stability of small perturbations of the zero potential (which has a half-bound state) from only the eigenvalues and resonances of the perturbation.

On the inverse resonance problemfor Jacobi operators—uniqueness and stability

We estimate the difference of the coefficients of two Jacobi operators (from a certain class) from knowledge about their eigenvalues and resonances. More specifically, we prove that if eigenvalues and resonances of the two operators in a sufficiently large disk are respectively close, then the coefficients are close too. A uniqueness result for the inverse resonance problem follows as a corollary.

Periodic Jacobi operator with finitely supported perturbation on the half-lattice

We consider a periodic Jacobi operator J with finitely supported perturbations on the half-lattice. We describe all eigenvalues and resonances of J and give their properties. We solve the inverse resonance problem: we prove that the mapping from finitely supported perturbations to the Jost functions is one-to-one and onto, we show how the Jost functions can be reconstructed from all eigenvalues, resonances and from the set of zeros of S(λ) − 1, where S(λ) is the scattering matrix.

Periodic Jacobi operator with finitely supported perturbations: The inverse resonance problem

We consider a periodic Jacobi operator H with finitely supported perturbations on Z . We solve the inverse resonance problem: we prove that the mapping from finitely supported perturbations to the scattering data, the inverse of the transmission coefficient and the Jost function on the right half-axis, is one-to-one and onto. We consider the problem of reconstruction of the scattering data from all eigenvalues, resonances and the set of zeros of R − ( λ ) + 1 , where R − is the reflection coefficient.

Inverse Scattering Results for Manifolds Hyperbolic Near Infinity

We study the inverse resonance problem for conformally compact manifolds which are hyperbolic outside a compact set. Our results include compactness of isoresonant metrics in dimension two and of isophasal negatively curved metrics in dimension three. In dimensions four or higher we prove topological finiteness theorems under the negative curvature assumption.

The inverse resonance problem for CMV operators

We consider the class of CMV operators with super-exponentially decaying Verblunsky coefficients. For these we define the concept of a resonance. Then we prove the existence of Jost solutions and a uniqueness theorem for the inverse resonance problem: given the location of all resonances, taking multiplicities into account, the Verblunsky coefficients are uniquely determined.

On the Inverse Resonance Problem for Schrödinger Operators

We consider Schrodinger operators on [0, ∞) with compactly supported, possibly complex-valued potentials in L1([0, ∞)). It is known (at least in the case of a real-valued potential) that the location of eigenvalues and resonances determines the potential uniquely. From the physical point of view one expects that large resonances are increasingly insignificant for the reconstruction of the potential from the data. In this paper we prove the validity of this statement, i.e., we show conditional stability for finite data. As a by-product we also obtain a uniqueness result for the inverse resonance problem for complex-valued potentials.

The Inverse Resonance Problem for Hermite Operators

In this paper the inverse resonance problem for the Hermite operator is investigated. The Hermite operator \(H=\mathfrak{a}+\mathfrak{a}^{*}+b\) with the creation operator \(\mathfrak{a}\) , the annihilation operator \(\mathfrak{a}^{*}\) , and a finitely supported multiplication operator b, is an unbounded operator on l2(ℕ0) having finitely many eigenvalues and infinitely many resonances (except for b=0, when there are no eigenvalues or resonances). It is shown that knowing the location of eigenvalues and resonances determines the potential b uniquely.

Stability for the inverse resonance problem for a Jacobi operator with complex potential

The potential of a discrete half-line Schrödinger operator is uniquely determined by the location of its Dirichlet eigenvalues and resonances. In a practical setting one can expect only to know a finite number of these. In this paper we give an estimate for the difference of two potentials (one of them finitely supported) for which eigenvalues and small resonances are the same but which may differ with respect to their large resonances.

Numerical technique for the inverse resonance problem

Motivated by the work of Regge (Nuovo Cimento 8 (1958) 671; 9 (1958) 491) we are interested in the problem of recovering a radial potential in R 3 from its resonance parameters, which are zeros of the appropriately defined Jost function. For a potential of compact support these may in turn be identified as the complex eigenvalues of a nonselfadjoint Sturm–Liouville problem with an eigenparameter dependent boundary condition. In this paper we propose and study a particular computational technique for this problem, based on a moment problem for a function g( t) which is related to the boundary values of the corresponding Gelfand–Levitan kernel.

The inverse resonance problem for perturbations of algebro-geometric potentials

We prove that compactly supported perturbations of algebro-geometric potentials for the one-dimensional Schrödinger equation are uniquely determined by the location of all their eigenvalues and resonances.

Stability for inverse resonance problem

For the Schrodinger operator on the half line, we show that if ϰ0={ϰ0}1∞ is a sequence of zeroes (eigenvalues and resonances) of the Jost function for some real compactly supported potential q 0 and ϰ−ϰ0∈le2 for some e > 1, then ϰ is the sequence of zeroes of the Jost function for some unique real compactly supported potential. Moreover, we show that the measure associated with the zeroes of the Jost function is a Carleson measure, and the sum ∑(1+|ϰn0|)−p, p > 1, is estimated in terms of the Jost function.