Patrick J. Browne

Sturm-Liouville problems with eigenparameter dependent boundary conditions

Sturm theory is extended to the equation for 1/ p , q , r ∈ L 1 [0, 1] with p, r > 0, subject to boundary conditions and Oscillation and comparison results are given, and asymptotic estimates are developed. Interlacing of eigenvalues with those of a standard Sturm–Liouville problem where the boundary conditions are a j y ( j ) = c j ( py ′)( j ), j =0, 1, forms a key tool.

Inverse Spectral Problems for Sturm–Liouville Equations with Eigenparameter Dependent Boundary Conditions

Inverse Sturm–Liouville problems with eigenparameter-dependent boundary conditions are considered. Theorems analogous to those of both Hochstadt and Gelfand and Levitan are proved. In particular, let l y = (1/ r )(−( py ′)′+ qy ), l˜ y = (1/ r˜ )(−( p˜ y ′)′+ q˜ y ), formula here where det Δ = δ > 0, c ≠ 0, det [sum ] > 0, t ≠ 0 and ( cs + dr − au − tb ) 2 cr − ta )( ds − ub ). Denote by ( l ; α; Δ) the eigenvalue problem ly = λ y with boundary conditions y (0)cosα+ y ′(0)sinα = 0 and ( a λ+ b ) y (1) = ( c λ+ d )( py ′)(1). Define (l˜; α; Δ) as above but with l replaced by l˜ . Let w n denote the eigenfunction of ( l ; α; Δ) having eigenvalue λ n and initial conditions w n (0) = sin α and pw ′ n (0) = −cos α and let γ n = − aw n (1)+ cpw ′ n (1). Define w˜ n and γ˜ n similarly. As sample results, it is proved that if ( l ; α; Δ) and (l˜; α; Δ) have the same spectrum, and ( l ; α; Σ) and (l˜; α; Σ) have the same spectrum or ∫ 1 0 [mid ] w n [mid ] 2 rdt +([mid ]γ n [mid ] 2 /δ) = ∫ 1 0 [mid ] w˜ n [mid ] 2 r˜ dt +([mid ]γ˜ n [mid ] 2 /δ) for all n , then q / r = q˜/r˜.

Spectral properties of two parameter eigenvalue problems II

We study the self-adjoint eigenvalue problem W (λ) x = 0, (*), in Hilbert space for one equation in two parameters. Here is bounded below with compact resolvent for each λ = (λ 1 , λ 2 ). We give necessary and sufficient conditions for the existence of λ so that (*) holds with W (λ)= ≧0 and we investigate the geometry of the set Z 0 of such λ. We also discuss higher order solution sets Z i where the ith eigenvalue of W (λ) vanishes, deriving various asymptotic results in a unified fashion.

Asymptotics of eigencurves for second order ordinary differential equations, I

Abstract The regular two parameter Sturm-Liouville equation −(py′)′ + qy = (λf − μr)y is studied for L1 coefficients with p, r > 0. For each fixed number n of internal zeros of the eigenfunctions y, μ = μn is analytic in λ. Necessary and sufficient conditions (which are in fact independent of n) are given for lim μ n gl to exist as λ → ∞ (or −∞). Asymptotic expansions for μn are derived in cases of existence and non-existence of lim μ n λ .

OSCILLATION THEORY FOR INDEFINITE STURM-LIOUVILLE PROBLEMS WITH EIGENPARAMETER-DEPENDENT BOUNDARY CONDITIONS

In previous papers we have studied oscillation properties of Sturm–Liouville problems (− Py ′)′ + qy = λry , with λ-dependent boundary conditions, under various ‘definiteness’ conditions. Here we present a new, unified, approach which also covers cases previously untreated, e.g. of semidefinite weight, and also the fully indefinite problem.

A Variational Approach to Multi-Parameter Eigenvalue Problems in Hilbert Space

Let

Abstract multiparameter theory, I

T_r

A uniqueness theorem for inverse eigenparameter dependent Sturm - Liouville problems

and

Comparison Cones for Multiparameter Eigenvalue Problems

V_{rs}

Nonlinear multiparameter Sturm-Liouville problems

be self-adjoint linear operators on Hilbert spaces