Spectral Theory And Root Bases Associated With Multiparameter Eigenvalue Problems

Abstract

Consider (1) -yn1+ q1y1 = (λr11 + µr12)y1 on [0, 1] y’1(0) = cot α1 and = y’1(1) = a1λ + b1 y1(0) y1(1) c1λ+d1

(2) - yn2 + q2y2 = (λr21 + µr22)y2 on [0, 1] y’2(0) = cot α2 and = y’2(1) = a2µ + b2 y2(0) y2(1) c2µ + d2

subject to certain definiteness conditions; where qi and rij are continuous real valued functions on [0, 1], the angle αi is in [0, π) and ai, bi, ci, di are real numbers with δi = aidi − bici > 0 and ci = 0 for I, j = 1,2. Under the Uniform Left Definite condition we have proved an asymptotic theorem and an oscillation theorem. Analysis of (1) and (2) subject to the Uniform Ellipticity condition focus on the location of eigenvalues, perturbation theory and the local analysis of eigenvalues. We also gave a bound for the number of nonreal eigenvalues. We also have studied the system T1(x1) = (λA11 + µA12)(x1) and T2(x2) = (λA21 + µA22)(x2) where Aij (j =1, 2) and Ti are linear operators acting on finite dimensional Hilbert spaces Hi (i = 1, 2). For a pair of commutative operators Γ = (Γ0, Γ1) constructed from Aij and Ti on the Hilbert space tensor product H1 ⊗ H2, we can associate a natural Koszul complex namely Dºr-(λ,μ) D1 r-(λ,μ) 0 H H ø H H 0

We have constructed a basis for the Koszul quotient space N(D1Г−(λ,µ))/R(D0Г−( λ,µ)) in terms of the root basis of (Г0, Г1). (For equations pl refer the PDF file)