The Morse and Maslov indices for Schrödinger operators

Latushkin, Yuri; Sukhtaiev, Selim; Sukhtayev, Alim

doi:10.1007/s11854-018-0043-x

Cited by 9 publications

(7 citation statements)

References 38 publications

(35 reference statements)

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“…We answer this question in the positive by reformulating it in terms of Lagrangian geometry. It is well known that self-adjoint extensions of a symmetric operator with equal deficiency indices are in one-to-one correspondence with the Lagrangian planes in some symplectic Hilbert space [AS80,BF,KS99,Ha00,LS,LSS,McS]. The question of restricting self-adjoint vertex conditions from the original graph to its reduced versionwith some edges shrunk to zero -is reframed in terms of the so-called linear symplectic reduction (see, for example, [McS]) allowing us to show that (1.1)-(1.2) indeed define a valid self-adjoint limiting operator.…”

Section: Introductionmentioning

confidence: 99%

Limits of quantum graph operators with shrinking edges

Berkolaiko

Latushkin

Sukhtaiev

2019

Advances in Mathematics

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We address the question of convergence of Schrödinger operators on metric graphs with general self-adjoint vertex conditions as lengths of some of graph's edges shrink to zero. We determine the limiting operator and study convergence in a suitable norm resolvent sense. It is noteworthy that, as edge lengths tend to zero, standard Sobolev-type estimates break down, making convergence fail for some graphs. We use a combination of functional-analytic bounds on the edges of the graph and Lagrangian geometry considerations for the vertex conditions to establish a sufficient condition for convergence. This condition encodes an intricate balance between the topology of the graph and its vertex data. In particular, it does not depend on the potential, on the differences in the rates of convergence of the shrinking edges, or on the lengths of the unaffected edges.

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Section: Introductionmentioning

confidence: 99%

Limits of quantum graph operators with shrinking edges

Berkolaiko

Latushkin

Sukhtaiev

2019

Advances in Mathematics

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“…The relevant technical properties of the Maslov index are summarized in Appendix B of [10]; a more complete presentation can be found in [4] or [14]. Some applications of the Maslov index to boundary value problems for PDE can be found in [10,11,12,17,22,27].…”

Section: Introductionmentioning

confidence: 99%

“…β(s, t) = {z + J ⊞ A(s, t)z : z ∈ β(0, 0)} , where J ⊞ (x, φ, y, ψ) = (R −1 φ, −Rx, −R −1 ψ, Ry)andR : H 1/2 (Σ) → H −1/2 (Σ) ∼ = (H 1/2 (Σ)) *is the Riesz duality isomorphism (cf. equation(17) in[10]). It suffices to choose A(s, t)(x, 0, 0, ψ) = − R −1 (tψ + sJ x), 0, 0, tRx , which is selfadjoint because the composition R −1 • J :H 1/2 (Σ) → H 1/2 (Σ) is selfadjoint.To prove the selfadjointness of R −1 • J we use the identity R −1 = R * to computeR −1 J f, g H 1/2 (Σ) = J f, Rg H −1/2 (Σ) = (J f )g = Σ f g for any f, g ∈ H 1/2 (Σ),where (J f )g denotes the action of the functional J f ∈ H −1/2 (Σ) on g ∈ H 1/2 (Σ).…”

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confidence: 99%

Manifold decompositions and indices of Schr\\\"{o}dinger operators

Cox

Jones²,

Marzuola³

2017

Indiana Univ. Math. J.

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Abstract. The Maslov index is used to compute the spectra of different boundary value problems for Schrödinger operators on compact manifolds. The main result is a spectral decomposition formula for a manifold M divided into components Ω 1 and Ω 2 by a separating hypersurface Σ. A homotopy argument relates the spectrum of a second-order elliptic operator on M to its Dirichlet and Neumann spectra on Ω 1 and Ω 2 , with the difference given by the Maslov index of a path of Lagrangian subspaces. This Maslov index can be expressed in terms of the Morse indices of the Dirichlet-to-Neumann maps on Σ. Applications are given to doubling constructions, periodic boundary conditions and the counting of nodal domains. In particular, a new proof of Courant's nodal domain theorem is given, with an explicit formula for the nodal deficiency.

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“…In this section, we discuss an infinitesimal version of the formula equating the Maslov index and the spectral flow for the family of operators H t = A t + V t satisfying Hypothesis 4.3, which is assumed throughout this section. Formulas relating these two quantities are quite classical, and we refer the reader to the papers [19,20,21,22,26,35,36,48,71,72,73,88] and the literature therein. Employing the abstract Hadamard-type formula obtained in Theorem 3.23, we prove in Theorem 4.19 that the signature of the Maslov crossing form defined in (4.34) at an eigenvalue λ of the operator H t0 is equal to the difference between the number of monotonically decreasing and the number of monotonically increasing eigenvalue curves for H t bifurcating from λ.…”

Section: 4mentioning

confidence: 99%

First-order asymptotic perturbation theory for extensions of symmetric operators

Latushkin¹,

Sukhtaiev²

2020

Preprint

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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

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The Morse and Maslov indices for Schrödinger operators

Cited by 9 publications

References 38 publications

Limits of quantum graph operators with shrinking edges

Limits of quantum graph operators with shrinking edges

Manifold decompositions and indices of Schr\\\"{o}dinger operators

First-order asymptotic perturbation theory for extensions of symmetric operators

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