Inverse Spectral Theory Research Articles

On inverse spectral theory for singularly perturbed operators: point spectrum

Let A be a self-adjoint operator in a separable Hilbert space . Let {ψk:k ⩾ 1} be a given (finite or infinite) orthonormal system such that and let Λ ≔ {λk:k ⩾ 1} be an arbitrary sequence of real numbers. Conditions for the existence of a pure singular perturbation of A solving the inverse eigenvalue problem are given.

Direct and Inverse Spectral Theory of One-dimensional Schrödinger Operators with Measures

We present a direct and rather elementary method for defining and analyzing one-dimensional Schrodinger operators H = −d2/dx2 + μ with measures as potentials. The basic idea is to let the (suitably interpreted) equation −f′′ + μ f = zf take center stage.We show that the basic results from direct and inverse spectral theory then carry over to Schrodinger operators with measures.

Predicting rogue waves in random oceanic sea states

Using the inverse spectral theory of the nonlinear Schrödinger (NLS) equation we correlate the development of rogue waves in oceanic sea states characterized by the Joint North Sea Wave Project (JONSWAP) spectrum with the proximity to homoclinic solutions of the NLS equation. We find in numerical simulations of the NLS equation that rogue waves develop for JONSWAP initial data that are “near” NLS homoclinic data, while rogue waves do not occur for JONSWAP data that are “far” from NLS homoclinic data. We show the nonlinear spectral decomposition provides a simple criterium for predicting the occurrence and strength of rogue waves.

Inverse spectral theory for symmetric operators with several gaps: scalar-type Weyl functions

Let S be the orthogonal sum of infinitely many pairwise unitarily equivalent symmetric operators with non-zero deficiency indices. Let J be an open subset of R . If there exists a self-adjoint extension S 0 of S such that J is contained in the resolvent set of S 0 and the associated Weyl function of the pair { S , S 0 } is monotone with respect to J, then for any self-adjoint operator R there exists a self-adjoint extension S ˜ such that the spectral parts S ˜ J and R J are unitarily equivalent. It is shown that for any extension S ˜ of S the absolutely continuous spectrum of S 0 is contained in that one of S ˜ . Moreover, for a wide class of extensions the absolutely continuous parts of S ˜ and S are even unitarily equivalent.

The reconstruction of an analytic string from two spectra

We would like to introduce a new method to solve the inverse spectral problem associated with the string. We show that we can recover the first 2n Taylor coefficients of the density of an analytic string from the knowledge of 2n eigenvalues. This provides an alternative method to the well-known M G Krein's continued fraction algorithm. Our algorithm is fast since it amounts to solving a sequence of 2 by 2 linear systems. We prove that the approximations converge to the exact solution provided a condition on the density of the string holds. This brings about a new and simple constructive method in the inverse spectral theory of the string.

A note on gaps of Hill's equation

is defined in terms of the discrete Fouriertransform uˆ of u.The following result was first proven by Mar cenko & Ostrowskiˇ ˘“ [12], using in-verse spectral theory. Their approach was later simplified by Garnett & Trubowitz [3]and generalized in [7]. For a more elementary proof see also [4, 5].Theorem 1.1 The gap lengths satisfyX

An Inverse Spectral Result for the Periodic Euler-Bernoulli Equation

The Floquet (direct spectral) theory of the periodic Euler-Bernoulli equation has been developed by the author in [19], [21], and [20]. Here we begin a systematic study of the inverse periodic spectral theory, in the spirit of the corresponding theory of the second-order operator, namely the Hill’s operator. Our main result is that, if there are no pseudogaps (equivalently, if the BlochFloquet variety is reducible in a certain sense), then the Euler-Bernoulli operator is the square of a second-order (Hill-type) operator. This result had been conjectured by the author, in his earlier works.

On the Spectral Problem Associated with the Camassa-Holm Equation

We give a basic uniqueness theorem in the inverse spectral theory for a Sturm-Liouville equation with a weight which is not of one sign. It is shown that the theorem may be applied to the spectral problem associated with the Camassa-Holm integrable system which models shallow water waves.

On the M-function and Borg–Marchenko theorems for vector-valued Sturm–Liouville equations

We will consider a vector-valued Sturm–Liouville equation of the form R[U]≔−(PU′)′+QU=λWU, x∈[0,b), with P−1, W, Q∈Lloc1([0,b))m×m being Hermitian and under some additional conditions on P−1 and W. We give an elementary deduction of the leading order term asymptotics for the Titchmarsh–Weyl M-function corresponding to this equation. In the special case of P=W=I, Q∈L1([0,∞))m×m and the Neumann boundary conditions at 0, we will also prove that M=(1/−λ)(I+R)(I−R)−1, where R=limn→∞ Rn=∑n=1∞Qn, for recursively defined sequences {Rn} and {Qn}. If Q∈Lloc1([0,b))m×m, 0<b⩽∞, the same formula is valid with an exponentially small error for large λ. It is clear that expansions of this type are helpful in finding representatives of the KdV invariants. For P=W=I, we prove that the spectral measure corresponding to the equation R[U]=λU uniquely determines Q as well as b and the boundary conditions at 0 and b. We finally give a new proof of a local form of the Borg–Marchenko theorem (cf. Gesztesy and Simon, “On local Borg–Marchenko uniqueness results,” Commun. Math. Phys. 211, 273–287 (2000), Chap. 3); a theorem which is due to Simon [see Simon, “A new approach to inverse spectral theory, I. fundamental formalism,” Ann. Math. 150, 1–29 (1999)] in the scalar case. For applications to physics, it is worth mentioning that vector-valued Sturm–Liouville equations appear in some problems in magneto-hydro-dynamics.

Inverse spectral theory for one-dimensional Schr�dinger operators: The A function

We link recently developed approaches to the inverse spectral problem (due to Simon and myself, respectively). We obtain a description of the set of Simon’s A functions in terms of a positivity condition. This condition also characterizes the solubility of Simon’s fundamental equation.

Some Constructions in the Inverse Spectral Theory of Cyclic Groups

The results of this paper concern the ‘large spectra’ of sets, by which we mean the set of points in ${\bb F}_p^{\times}$ at which the Fourier transform of a characteristic function $\chi_A$, $A\subseteq {\bb F}_p$, can be large. We show that a recent result of Chang concerning the structure of the large spectrum is best possible. Chang's result has already found a number of applications in combinatorial number theory.We also show that if $|A|=\lfloor {p/2}\rfloor$, and if $R$ is the set of points $r$ for which $|\hat{\chi}_A(r)|\geqslant \alpha p$, then almost nothing can be said about $R$ other than that $|R|\ll \alpha^{-2}$, a trivial consequence of Parseval's theorem.

Digitally graded GaAs/Al 0.44Ga 0.56As quantum-cascade laser

A method for the optimal design and realization of a GaAs/Al 0.44Ga 0.56As quantum-cascade laser (QCL) is presented. Firstly, an optimal, smooth active region profile is derived, using inverse spectral theory. A digitally graded laser structure is then designed, having an approximately equivalent potential profile. The gain and threshold current of the optimized device are calculated using a 15-level self-consistent rate equation model, and are shown to represent substantial improvements over the figures obtained, using the same calculation method, for the recently reported room temperature GaAs/Al 0.44Ga 0.56As QCL of Page et al. (Appl. Phys. Lett. 78 (2001) 3529).

Sampling for the fourth-order Sturm–Liouville differential operator

We extend the sampling method to compute the eigenvalues of a fourth-order differential operator. We show that the Shannon sampling theorem is not applicable due to the growth of the solutions on the real line. The inverse spectral theory, as developed by McLaughlin, and Kramer's theorem allow us to express the characteristic function explicitly by a new sampling formula.

Continued fractions and integrable systems

Stieltjes’ solution of the classical moment problem is the forerunner of inverse spectral theory. In the modern theory of completely integrable systems, the method of inverse scattering is used to obtain explicit solutions of a class of Hamiltonian systems which arise as isospectral deformations of a linear operator. In this lecture we explain how Stieltjes's formulas are used to obtain explicit solutions to a number of completely integrable systems, including discrete reductions of some nonlinear partial differential equations, and the isospectral deformations of Jacobi matrices a.k.a. Toda flows.

Gain optimization in electrically pumped AlGaAs quantum cascade lasers

The design of the active region of a current-pumped quantum-cascade laser to achieve maximal gain is proposed. Starting with an arbitrary smooth potential, a family of isospectral Hamiltonians with a predefined energy spectrum is generated by use of the inverse spectral theory. By varying the relevant control parameter, one varies the potential shape, inducing changes in transition dipole moments and electron–phonon scattering times, and thus finds the potential that gives the largest gain. As an example, a simple step quantum-well structure with just a few layers is then designed such that, in postgrowth heating-induced layer interdiffusion, it will acquire a shape that is as close as possible to that of the optimal smooth potential.

Matrix Finite-Zone Dirac-Type Equations

Weyl–Titchmarsh matrix functions play an essential role in the spectral theory of Dirac-type equations ( Oper. Theory: Adv. Appl. 107 (1999)). In this paper, we have constructed a class of Weyl–Titchmarsh matrix-functions generating potentials of finite-zone type. It has turned out that the corresponding potentials have derivatives of an arbitrary order. Using the above-mentioned results, we deduce the matrix analogue of the trace formula for finite-zone matrix potentials. In the last part of the paper, we consider separately the scalar case of Dirac-type equations. For this case, we have constructed finite-zone potentials in explicit forms and proved that these potentials are quasiperiodical. We note that for scalar Schrödinger equations the corresponding results are well known (see Invent. Math. 30 (1975), 217–274; “Soliton and the Inverse Scattering Transform,” SIAM, Philadelphia, 1981; Rev. Sci. Technol. 23 (1983), 20–50; “Theory of Solitons, The Method of Inverse Problem,” New York, 1984; “Inverse Sturm–Liouville Problems,” VSP, Zeist, 1987; “Inverse Spectral Theory,” Academic Press, New York, 1987).

On the interdiffusion-based quantum cascade laser

Design procedure for the active region of current pumped quantum cascade laser is proposed, so to achieve maximal gain. Starting with an arbitrary smooth potential, a family of isospectral Hamiltonians with predefined energy spectrum is generated using the inverse spectral theory. By varying the relevant control parameter the potential shape is varied, inducing changes in transition dipole moments and electron-phonon scattering times, and the optimal potential which gives the largest gain is thus found. For purpose of realization, a simple step quantum-well structure with just a few layers is then designed so that in the post-growth heating-induced layer interdiffusion, it will acquire a shape as close as possible to the optimal smooth potential.

Digitally graded active region for optically pumped intersubband lasers and nonlinear wavelength convertors

A strategy is proposed for the realization of quantum-well structures optimized for devices based on intersubband optical transitions. It relies on the recently established techniques for the design of optimal smooth quantum-well profile by using the inverse spectral theory, followed by the design of an (almost) equivalent digitally graded structure, comprising just two different alloy compositions. Digital grading greatly simplifies the structure growth while essentially fully retaining the properties achieved in smoothly graded optimized structures. Example designs are presented for an optically pumped quantum-well laser and for a quantum well intended for the second-harmonic generation.

Extension of Ambarzumyan's Theorem to General Boundary Conditions

We extend the classical Ambarzumyan's theorem for the Sturm–Liouville equation (which is concerned only with Neumann boundary conditions) to the general boundary conditions, by imposing an additional condition on the potential function. Our result supplements the Pöschel–Trubowitz inverse spectral theory. We also have parallel results for vectorial Sturm–Liouville systems.

On the inverse spectral theory of Schrödinger and Dirac operators

On the inverse spectral theory of Schrödinger and Dirac operators