Analogs of generalized resolvents and eigenfunction expansions of relations generated by pair of differential operator expressions one of which depends on spectral parameter in nonlinear manner

Abstract: For the relations generated by pair of differential operator expressions one of which depends on the spectral parameter in the Nevanlinna manner we construct analogs of the generalized resolvents which are integro-differential operators. The expansions in eigenfunctions of these relations are obtained.

• The paper extends the theory of generalized resolvents by constructing operator-valued integrals for differential operator relations with nonlinear Nevanlinna-type spectral dependence.
• It develops eigenfunction expansions using transformation and Parseval-type equalities while addressing challenges from degenerate weight operators and complex boundary conditions.
• The study reveals notable spectral effects, such as the emergence of additional bands and gaps, which broaden applications in quantum mechanics, elasticity, and control systems.

Analogs of Generalized Resolvents and Eigenfunction Expansions for Differential Relations with Nonlinear Spectral Dependence

Problem Context and Objectives

The paper investigates relations in separable Hilbert spaces generated by pairs of differential operator expressions, focusing on those where one expression manifests nonlinear dependence on the spectral parameter via Nevanlinna-type dependence. The research addresses the construction of analogs to generalized resolvents for such operator relations—especially where previous linear or semi-linear frameworks prove insufficient—and provides associated eigenfunction expansions. Among the generalizations, two significant complications are present: the possibility of degenerate weight operators (where the leading coefficient may be singular or even vanish in certain regions), and spectral parameter dependence embedded nonlinearly in one of the operators.

Operator Framework and Main Constructions

The primary equation under study is

$l[y] = m[f]$

where $l[\cdot]$ and $m[\cdot]$ are differential operator expressions of orders $r$ and $s$ ($r \geq s$, $s$ even), acting on vector-valued functions in a Hilbert space $H$, with $m$ only assumed nonnegative in a suitable quadratic form sense, rather than coercive. The operator $n_\lambda[\cdot]$, representing the nonlinear dependence on the spectral parameter, enters so that $$l[\cdot] = l^*[\cdot] - A m[\cdot] - n_\lambda[\cdot]$$ with $n_\lambda[y]$ being analytic in $\lambda$ and Nevanlinna in nature. In contrast to standard spectral theory, the weight operator can be degenerate ($m$ does not need to be invertible everywhere), and boundary conditions may explicitly depend on the spectral parameter, potentially involving derivatives of the unknown up to prescribed orders.

To handle these complexities, the equation is systematically reduced to first-order canonical systems (with operator weights) via the introduction of quasi-derivatives and operator matrices. Associated Green's formulae and bilinear relationships are established. In the non-symmetric and non-coercive context, the notion of the generalized resolvent is rigorously defined in analogy to abstract operator theory, leading to an operator family $R(\lambda)$ realized via strongly convergent operator integrals involving characteristic operator functions (c.o.) associated with the canonical system.

Generalized Resolvent and Spectral Expansion

A central achievement is the construction of $R(\lambda)$ as an integro-differential operator satisfying

$R(\lambda) = \int dE_\lambda$

for a generalized spectral family $E_\lambda$ with $E_\lambda \leq I$ (the identity), realized in the Hilbert space $L^2_m(I)$ with metric induced by the potentially degenerate weight $m$. The characteristic operator $M(\lambda)$, analytic and self-adjoint on the non-real axis, governs the boundary behaviour and uniquely corresponds to maximally nonnegative subspaces in indefinite inner product geometry. Sufficient and necessary conditions for the existence of such characteristic operators and projections, as well as for the uniqueness of solutions, are provided and shown to coincide with geometric multiplicity conditions (equality of positive/negative indices of the underlying quadratic form for the 'energy' operator $G = RQ(0)$).

The boundary value problem is formulated with general (including dissipative) boundary operators that may depend on both the unknown function and its derivatives, as well as on the spectral parameter. It is demonstrated that for an extended class of such boundary conditions (not necessarily separated or self-adjoint), the constructed resolvents produce weak solutions corresponding to these parametrized boundary settings.

The paper introduces, proves, and leverages transformation formulae and Parseval (respectively, Bessel-type) equalities, relating the expansion of vectors in terms of the eigenfunctions to the spectral data encoded in the Nevanlinna matrix measure; explicit inversion and expansion formulae are obtained in both the regular and singular (infinite interval or degenerate weight) cases, unifying the analysis across a broad class of operator relations.

Exemplification and Spectral Effects

The theory is authenticated by explicit construction with periodic differential operator coefficients on the real line. A distinct spectral behaviour is observed in comparison to the classical (linear) case. For instance, it is shown that, under nonlinear perturbations in the spectral parameter, the spectral measure may possess jumps when such phenomena are precluded in the purely linear setup. Three illustrative examples demonstrate: nonlinear perturbation modifying spectral band edges; the creation of additional spectral bands due to nonlinearities; and the novel occurrence of a spectral gap with an embedded singular spectral point (i.e., "eigenvalue inside the gap"), effects that do not arise in standard scenarios.

Summary of Strong Results and Assertions

• Analogs of generalized resolvents $R(\lambda)$ are constructed for operator relations with nonlinear Nevanlinna-type dependence on the spectral parameter.
• These resolvents are generally non-injective and are realized as strongly continuous operator-valued integrals; they generate eigenfunction expansions in $L^2_m(I)$.
• Transformation and Parseval-type equalities for the spectral expansion are established, together with inequalities of the Bessel type.
• All generalized resolvents in the regular case are exhausted by the constructed operators, and their correspondence with boundary problems (including spectral parameter dependent conditions) is clarified.
• The resolution structure and spectral measures exhibit qualitative changes under nonlinear spectral parameter dependence: new types of spectral bands and gaps can emerge, and the structure of the spectral family can present jumps not present classically.

Theoretical and Practical Implications

The research provides an advanced and precise extension of the theory of generalized resolvents and spectral expansions to a setting where standard tools based on linear dependence, invertible weights, or symmetric operator frameworks are inapplicable. The implications are substantial: it supplies foundational analytic machinery for spectral and evolution problems associated with operator differential equations where either physical modelling or underlying geometry necessitate degenerate weights and nonlinear structures (as in certain quantum, elasticity, or control systems). The method offers a unifying perspective accommodating both strong degeneracy (e.g., vanishing leading coefficients) and strong spectral parameter nonlinearity.

Practically, results herein could govern the spectral theory of difference equations, indefinite and singular Sturm-Liouville problems, and operator pencils with parameter in the boundary conditions, including the derivation of expansion theorems in these complex settings.

Future Directions

Future research may extend the framework to systems with more complicated (possibly operator-valued or distributional) coefficients, further analysis of the impact of these nonlinearities on spectral multiplicity and resonance phenomena, and numerical realization of the expansion theorems for applications to PDEs and mathematical physics. Moreover, a refined description of the jumps and singularities in the spectral measure for nonlinear perturbations remains of significant interest, especially in the analysis of stability and control problems in applied settings.

Conclusion

The paper develops a rigorous, operator-theoretically grounded framework for generalized resolvents and eigenfunction expansions in settings involving nonlinear spectral parameter dependence and degenerate weights. It establishes all central analytic tools—reduction to canonical form, construction of characteristic operators, Green's formula, boundary value correspondence, and spectral expansions—valid in this high generality, thereby significantly expanding the applicability of operator spectral theory beyond the conventional linear and non-degenerate scope.