First-order asymptotic perturbation theory for extensions of symmetric operators

Abstract

This work offers a new prospective on asymptotic perturbation theory for varying self-adjoint extensions of symmetric operators. Employing symplectic formulation of self-adjointness we obtain a new version of Krein formula for resolvent difference which facilitates asymptotic analysis of resolvent operators via first order expansion for the family of Lagrangian planes associated with perturbed operators. Specifically, we derive a Riccati-type differential equation and the first order asymptotic expansion for resolvents of self-adjoint extensions determined by smooth one-parameter families of Lagrangian planes. This asymptotic perturbation theory yields a symplectic version of the abstract Kato selection theorem and Hadamard-Rellich-type variational formula for slopes of multiple eigenvalue curves bifurcating from an eigenvalue of the unperturbed operator. The latter, in turn, gives a general infinitesimal version of the celebrated formula equating the spectral flow of a path of self-adjoint extensions and the Maslov index of the corresponding path of Lagrangian planes. Applications are given to quantum graphs, periodic Kronig-Penney model, elliptic second order partial differential operators with Robin boundary conditions, and physically relevant heat equations with thermal conductivity. ContentsDate: December 2, 2020. Key words and phrases. Hadamard variational formula, Krein's resolvent formula, Lagrangian planes, Kato selection theorem.YL was supported by the NSF grant DMS-1710989 and would like to thank the Courant Institute of Mathematical Sciences and especially Prof. Lai-Sang Young for the opportunity to visit CIMS. We are thankful to M. Ashbaugh, V. Derkach, C. Gal, F. Gesztesy, R. Schnaubelt for pointing out very useful sources in the literature. SS was supported by the Simons travel grant and would like to thank D. Damanik and J. Fillman for discussion of the periodic Kronig-Penney model. 1 2 Y. LATUSHKIN AND S. SUKHTAIEV 5.3. Continuous dependence of solutions to heat equation on thermal conductivity 56 5.4. The Hadamard formula for star-shaped domains 59 5.5. Maslov crossing form for elliptic operators 60 6. Krein-type formulas for dual pairs 62 Appendix A. The Krein-Naimark resolvent formula revisited 66 Appendix B. Dirichlet and Neumann trace operators 70 References 72

Resolvent expansions for self-adjoint operators via boundary triplets

“…Date: October 15, 2021. This work naturally emerged from [19] and [5] as the result numerous stimulating discussions with Prof. G. Berkolaiko. We thank him for generously sharing his ideas as well as his notes on related topics.…”

“…We stress that the abstract setting considered in [19] is significantly more general than the one assumed in the current work but the expansion carried out in [19] is only to the first order. The major difference is in the type of trace maps used in [19] where they are assumed to be merely densely defined and with dense range while in the current work the traces are bounded (in particular, have full domain) and surjective. In short, in [19], (H, Γ 0 , Γ 1 ) is not required to be a classical boundary triplet, [2,13,24], but it is in the current manuscript.…”

“…[6,23], for partial differential operators on varying domains or subject to varying boundary conditions, a subject that recently attracted much attention, cf. [8,9,10,11,19,20] and the bibliography therein. For example, Deng-Jones's [11] foundational approach to the spectral count via Maslov index for Robin Laplacian essentially reduces to finding the sign of the derivative of the eigenvalue curves, see recent papers [8,20].…”

“…This work should be considered as a sequel to [19] where a symplectic form of the Krein resolvent formula was derived in order to establish the first order expansion for resolvents and eigenvalues of self-adjoint extensions. We stress that the abstract setting considered in [19] is significantly more general than the one assumed in the current work but the expansion carried out in [19] is only to the first order.…”

“…This work should be considered as a sequel to [19] where a symplectic form of the Krein resolvent formula was derived in order to establish the first order expansion for resolvents and eigenvalues of self-adjoint extensions. We stress that the abstract setting considered in [19] is significantly more general than the one assumed in the current work but the expansion carried out in [19] is only to the first order. The major difference is in the type of trace maps used in [19] where they are assumed to be merely densely defined and with dense range while in the current work the traces are bounded (in particular, have full domain) and surjective.…”