An index theorem for Schrödinger operators on
                    metric graphs

Latushkin, Yuri; Sukhtaiev, Selim

doi:10.1090/conm/741/14922

Cited by 8 publications

(9 citation statements)

References 27 publications

(55 reference statements)

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“…In the generic picture, the various intersection terms in (11) disappear. The authors of [26] study the Morse index of Schrödinger operators on graphs. By reducing the problem to an interval, they provide a Morse index formula as the Maslov index of a curve in a Lagrangian Grassmanian of a sufficiently big dimension.…”

Section: Main Results and Structure Of The Papermentioning

confidence: 99%

Index theorems for graph-parametrized optimal control problems

Agrachev¹,

Baranzini²,

Beschastnyi³

2022

Preprint

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In this paper we prove Morse index theorems for a big class of constrained variational problems on graphs. Such theorems are useful in various physical and geometric applications. Our formulas compute the difference of Morse indices of two Hessians related to two different graphs or two different sets of boundary conditions. Several applications such as the iteration formulas or lower bounds for the index are proved.

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Section: Main Results and Structure Of The Papermentioning

confidence: 99%

Index theorems for graph-parametrized optimal control problems

Agrachev¹,

Baranzini²,

Beschastnyi³

2022

Preprint

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“…In this section, we apply our framework to an operator on the half-line that arises through consideration of nonlinear Schrödinger equations on quantum graphs with n infinite edges extending from a single vertex (i.e., on star graphs). Our direct motivation for considering this example is the recent analysis of Kairzhan and Pelinovsky (see [12]), and we also note that Kostrykin and Schrader have shown how the symplectic framework fits well with such problems (see [13]) and that Latushkin and Sukhtaiev have recently developed this framework in the case of quantum graphs with edges of finite length (see [15]). Finally, we mention that our general approach to quantum graphs is adapted from the reference [1].…”

Section: Application To Quantum Graphsmentioning

confidence: 99%

The Maslov and Morse indices for Sturm-Liouville systems on the half-line

Howard¹,

Sukhtayev²

2020

Discrete &Amp; Continuous Dynamical Systems - A

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We show that for Sturm-Liouville Systems on the half-line [0, ∞), the Morse index can be expressed in terms of the Maslov index and an additional term associated with the boundary conditions at x = 0. Relations are given both for the case in which the target Lagrangian subspace is associated with the space of L 2 ((0, ∞), C n ) solutions to the Sturm-Liouville System, and the case when the target Lagrangian subspace is associated with the space of solutions satisfying the boundary conditions at x = 0. In the former case, a formula of Hörmander's is used to show that the target space can be replaced with the Dirichlet space, along with additional explicit terms. We illustrate our theory by applying it to an eigenvalue problem that arises when the nonlinear Schrödinger equation on a star graph is linearized about a half-soliton solution.(A2) We assume that P , V , and Q all approach well-defined asymptotic endstates at exponential rate. That is, we assume there exist self-adjoint matrices P + , V + , Q + ∈ C n×n , with P + , Q + positive definite, and constants C and η > 0 so thatand similarly for V (x) and Q(x). In addition, we assume |P ′ (x)| ≤ Ce −η|x| for a.e. x ∈ (0, ∞).(A3) For the boundary conditions, we take α 1 , α 2 ∈ C n×n , and for notational convenience, set α = (α 1 α 2 ). We assume rank α = n, αJα * = 0, which is equivalent to self-adjointness in this case. Here, J denotes the standard symplectic matrix J = 0 −I n I n 0 , with I n denoting the usual n × n identity matrix. We can think of (1.1) in terms of the operatorwith which we associate the domainand the inner productWith this choice of domain and inner product, L is densely defined, closed, and self-adjoint, so σ(L) ⊂ R.Our particular interest lies in counting the number of negative eigenvalues of L (i.e., the Morse index). We proceed by relating the Morse index to the Maslov index, which is described in Section 3. We find that the Morse index can be computed in terms of the Maslov index and an additional term associated with the boundary condition at x = 0.As a starting point, we define what we will mean by a Lagrangian subspace of C 2n . For comments about working in C 2n rather than R 2n , the reader is referred to Remark 1.1 of [10], and the references mentioned in that remark.Definition 1.1. We say ℓ ⊂ C 2n is a Lagrangian subspace of C 2n if ℓ has dimension n and (Ju, v) C 2n = 0, (1.3)for all u, v ∈ ℓ. Here, (·, ·) C 2n denotes the standard inner product on C 2n . In addition, we denote by Λ(n) the collection of all Lagrangian subspaces of C 2n , and we will refer to this as the Lagrangian Grassmannian.

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“…where L D has the same differential representation as L and H 2 D ( ) differs from H 2 NK ( ) by the Dirichlet conditions at the boundary vertices in V B instead of the NK conditions. The technique resembles the surgery principle widely used in the spectral analysis of differential operators on graphs [7,20].…”

Section: Remarkmentioning

confidence: 99%

Multi-pulse edge-localized states on quantum graphs

Kairzhan

Pelinovsky

2021

Anal.Math.Phys.

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We construct the edge-localized stationary states of the nonlinear Schrödinger equation on a general quantum graph in the limit of large mass. Compared to the previous works, we include arbitrary multi-pulse positive states which approach asymptotically a composition of N solitons, each sitting on a bounded (pendant, looping, or internal) edge. We give sufficient conditions on the edge lengths of the graph under which such states exist in the limit of large mass. In addition, we compute the precise Morse index (the number of negative eigenvalues in the corresponding linearized operator) for these multi-pulse states. If N solitons of the edge-localized state reside on the pendant and looping edges, we prove that the Morse index is exactly N . The technical novelty of this work is achieved by avoiding elliptic functions (and related exponentially small scalings) and closing the existence arguments in terms of the Dirichlet-to-Neumann maps for relevant parts of the given graph. We illustrate the general results with three examples of the flower, dumbbell, and single-interval graphs.

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An index theorem for Schrödinger operators on metric graphs

Cited by 8 publications

References 27 publications

Index theorems for graph-parametrized optimal control problems

Index theorems for graph-parametrized optimal control problems

The Maslov and Morse indices for Sturm-Liouville systems on the half-line

Multi-pulse edge-localized states on quantum graphs

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