AMBARZUMYAN’S THEOREM WITH EIGENPARAMETER IN THE BOUNDARY CONDITIONS

In this paper, the classical Ambarzumyan’s theorem for the regular SturmLiouville problem is extended to the case in which the boundary conditions are eigenparameter dependent. Specifically, we show that if the spectrum of the operator D 2 +q with eigenparameter dependent boundary conditions is the same as the spectrum belonging to the zero potential, then the potential function q is actually zero.     

On a new class of boundary value problems for the Sturm-Liouville operator

We consider the spectral problem generated by the Sturm-Liouville operator with complex-valued potential q(x) ∈ L ₂(0, π) and degenerate boundary conditions. We show that the set of potentials q(x) for which there exist associated functions of arbitrarily high order in the system of root functions is everywhere dense in L ₁(0, π).     

Asymptotics of the spectrum of the Sturm-Liouville operator with regular boundary conditions

On the interval (0, π), we consider the spectral problem generated by the Sturm-Liouville operator with regular but not strongly regular boundary conditions. For an arbitrary potential q(x) ∈ L ₁ (0, π) [q(x) ∈ L ₂(0, π)], we establish exact asymptotic formulas for the eigenvalues of this problem.     

Asymptotic analysis of non-self-adjoint Hill operators

We obtain uniform asymptotic formulas for the eigenvalues and eigenfunctions of the Sturm-Liouville operators L _t (q) with a potential q ∈ L ₁[0,1] and t-periodic boundary conditions, t ∈ (?π, π]. Using these formulas, we find sufficient conditions on the potential q such that the number of spectral singularities in the spectrum of the Hill operator L(q) in L ₂(?∞,∞) is finite. Then we prove that the operator L(q) has no spectral singularities at infinity and it is an asymptotically spectral operator provided that the potential q satisfies sufficient conditions.     

Half-inverse problem for diffusion operators on the finite interval

The potential function q(x) in the regular and singular Sturm-Liouville problem can be uniquely determined from two spectra. Inverse problem for diffusion operator given at the finite interval eigenvalues, normal numbers also on two spectra are solved. Half-inverse spectral problem for a Sturm-Liouville operator consists in reconstruction of this operator by its spectrum and half of the potential. In this study, by using the Hochstadt and Lieberman's method we show that if q(x) is prescribed on , then only one spectrum is sufficient to determine q(x) on the interval for diffusion operator.     

On the completeness of the system of root functions of the Sturm-Liouville operator with degenerate boundary conditions

We consider the spectral problem generated by the Sturm-Liouville equation with arbitrary complex-valued potential q(x) ∈ L ₁(0, π) and with degenerate boundary conditions. We obtain sufficient conditions for the completeness of the system of eigenfunctions and associated functions of this operator.     

Bounds on the Non-real Spectrum of a Singular Indefinite Sturm-Liouville Operator on ℝ

A simple explicit bound on the absolute values of the non-real eigenvalues of a singular indefinite Sturm-Liouville operator on the real line with the weight function sgn(·) and an integrable, continuous potential q is obtained. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the divergence of expansions in systems of root functions of the Sturm-Liouville operator with degenerate boundary conditions

We consider the spectral problem generated by the Sturm-Liouville operator with an arbitrary complex-valued potential q(x) ?? L ₁(0, ??) and with degenerate boundary conditions. We show that, under some additional condition, the system of root functions of that operator is not a basis in the space L ₂(0, ??).     

Inverse spectral problem for integro-differential operators

In this paper, we study the inverse spectral problem on a finite interval for the integro-differential operator ? which is the perturbation of the Sturm-Liouville operator by the Volterra integral operator. The potential q belongs to L ₂[0, π] and the kernel of the integral perturbation is integrable in its domain of definition. We obtain a local solution of the inverse reconstruction problem for the potential q, given the kernel of the integral perturbation, and prove the stability of this solution. For the spectral data we take the spectra of two operators given by the expression for ? and by two pairs of boundary conditions coinciding at one of the finite points.     

Characterization of the spectrum of regular boundary value problems for the Sturm-Liouville operator

On the interval (0, τ), we consider the spectral problem generated by the Sturm-Liouville equation with an arbitrary complex-valued potential q(x) ∈ L ₂(0, τ) and with regular (but not strengthened-regular) boundary conditions. Under certain additional assumptions, we establish necessary and sufficient conditions for a set of complex numbers to be the spectrum of such an operator.     

On a two-point boundary value problem for the Sturm-Liouville operator with a nonclassical asymptotics of the spectrum

We consider a spectral problem generated by a Sturm-Liouville equation on the interval (0, π) with degenerate boundary conditions. We prove the existence of potentials q(x) ∈ L ₂(0, π) such that the multiplicities of the eigenvalues λ _n monotonically tend to infinity as n → ∞.     

Absolute continuity and spectral concentration for slowly decaying potentials

We consider the spectral function ϱ(μ) (μ⩾0) for the Sturm-Liouville equation y″ + (λ −qq)y=0 on [0,∞) with the boundary condition y(0)=0 and where q has slow decay O(x^−a (a > 0) as x → ∞. We develop our previous methods of locating spectral concentration for q with rapid exponential decay (this Journal 81 (1997) 333–348) to deal with the new theoretical and computational complexities which arise for slow decay.     

Relative oscillation theory for Sturm-Liouville operators extended

We extend relative oscillation theory to the case of Sturm-Liouville operators Hu=r⁻¹(−^′(pu^′)+qu) with different p's. We show that the weighted number of zeros of Wronskians of certain solutions equals the value of Krein's spectral shift function inside essential spectral gaps.     

On the solutions of (ry′)′+qy=f

A number of mathematicians have focused attention on conditions involvingq andr which guarantee the boundedness of the solutionsy, and its derivativey′, of the equation (ry′)′+qy=f. Others, because of Sturm-Liouville theory, have focused attention on solutions which are inL ² (0, ∞). This article employs the assumption ofL ²-ness to show the boundedness ofy and its derivativey′.     

On the uniqueness criterion for solutions of the Sturm-Liouville equation

We consider the Sturm-Liouville equation $ - y'' + qy = \lambda ^2 y $ in an annular domain K from ? and obtain necessary and sufficient conditions on the potential q under which all solutions of the equation ?y″(z) + q(z)y(z) = λ ² y(z), z ∈ γ, where γ is a certain curve, are unique in the domain K for all values of the parameter λ ∈ ?.     

Schur Q-polynomials, multiple hypergeometric series and enumeration of marked shifted tableaux

We study Schur Q-polynomials evaluated on a geometric progression, or equivalently q-enumeration of marked shifted tableaux, seeking explicit formulas that remain regular at q=1. We obtain several such expressions as multiple basic hypergeometric series, and as determinants and pfaffians of continuous q-ultraspherical or continuous q-Jacobi polynomials. As special cases, we obtain simple closed formulas for staircase-type partitions.     

Finite difference methods for half inverse Sturm-Liouville problems

We investigate finite difference solution of the Hochstadt-Lieberman problem for a Sturm-Liouville operator defined on (0, π): given the value of the potential q on (c, π), where 0 < c < π, use eigenvalues to estimate q on (0, c). Our methods use an asymptotic correction technique of Paine, de Hoog and Anderssen, and its extension to Numerov’s method for various boundary conditions. In the classical case c = π/2, Numerov’s method is found to be particularly effective. Since eigenvalue data is scarce in applications, we also examine stability problems associated with the use of the extra information on q when c < π/2, and give some suggestions for further research.     

Inverse indefinite Sturm-Liouville problems with three spectra

We prove that the potential q(x) of an indefinite Sturm-Liouville problem on the closed interval [a,b] with the indefinite weight function w(x) can be determined uniquely by three spectra, which are generated by the indefinite problem defined on [a,b] and two right-definite problems defined on [a,0] and [0,b], where point 0 lies in (a,b) and is the turning point of the weight function w(x).     

On the Eigenvalue Accumulation of Sturm-Liouville Problems Depending Nonlinearly on the Spectral Parameter

A nonlinear spectral problem for a Sturm-Liouville equation-(p(x, λ)y'(x, λ))' + q(x, λ) y(x, λ) = 0 on a finite interval [a, b] with λ-dependent boundary conditions is considered. The spectral parameter λ is varying in an interval ∧ and p(x, λ), q(x, A) are real, continuous functions on [a, b] × ∧ Some criteria to the eigenvalue accumulation at the endpoints of A will be established. The results are applied to concrete problems arising in magnetohydrodynamics.     

A curious q-analogue of Hermite polynomials

Two well-known q-Hermite polynomials are the continuous and discrete q-Hermite polynomials. In this paper we consider a new family of q-Hermite polynomials and prove several curious properties about these polynomials. One striking property is the connection with q-Fibonacci and q-Lucas polynomials. The latter relation yields a generalization of the Touchard-Riordan formula.