2018/04/19 by Casey Bartels, Sonja Currie, Bartels, Casey A. +5
Mathematics · #Differential Equations and Boundary Problems #FOS: Mathematics #Numerical methods in inverse problems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1804.07149
openalex publication_date 2018/04/19 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28
We consider a Sturm-Liouville equation \ℓ y:=-y'' + qy = \λ y on the\nintervals (-a,0) and (0,b) with a,b>0 and q \∈ L2(-a,b). We impose\nboundary conditions y(-a)\cos\α = y'(-a)\sin\α, y(b)\cos\β =\ny'(b)\sin\β, where \α \∈ [0,\π) and \β \∈ (0,\π], together\nwith transmission conditions rationally-dependent on the eigenparameter via\n\-y(0+)
left(
lambda
eta -
xi-
sum
limitsi=1N\n
fracbi2
lambda -ci
right) amp;= y'(0+) - y'(0-),
y'(0-)
left(
lambda\n
kappa +
zeta-
sum
limitsj=1M
fracaj2
lambda -dj
right) amp;= y(0+)\n- y(0-), with bi, aj>0 for i=1,\…,N, and j=1,\…,M.\nHere we take \η, \κ \≥ 0 and N,M\∈ N0. The geometric multiplicity\nof the eigenvalues is considered and the cases in which the multiplicity can be\n2 are characterized. An example is given to illustrate the cases. A Hilbert\nspace formulation of the above eigenvalue problem as a self-adjoint operator\neigenvalue problem in L2(-a,b) bigoplus CN^* bigoplus CM^*, for\nsuitable N^*,M^*, is given. The Green's function and the resolvent of the\nrelated Hilbert space operator are expressed explicitly.\n