Eigenfunction expansions associated with an operator differential equation nonlinearly depending on a spectral parameter. (English) Zbl 1313.34276

The author obtains eigenfunction expansions for operator-differential equations of the form \[ l[y]-\lambda m[y]-n_\lambda [y]=m[f], \] where \(l,m,n_\lambda\) are symmetric differential expressions with bounded operator coefficients on a Hilbert space, \(n_\lambda\) is a Nevanlinna function of the spectral parameter \(\lambda\). The case \(n_\lambda =0\) was considered in the same generality in the author’s earlier paper [Methods Funct. Anal. Topol. 15, No. 2, 137–151 (2009; Zbl 1199.34450)].


34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators
34B07 Linear boundary value problems for ordinary differential equations with nonlinear dependence on the spectral parameter
34G10 Linear differential equations in abstract spaces
47E05 General theory of ordinary differential operators


Zbl 1199.34450
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