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  • Spectral Theory and Spectral Gaps

    for Periodic Schrödinger Operators

    on Product Graphs

    Robert Carlson

    University of Colorado at Colorado Springs Colorado Springs, CO 80933


    Floquet theory and its applications to spectral theory are devel-

    oped for periodic Schrödinger operators on product graphs � × � ,

    where �

    is a finite graph. The resolvent and the spectrum have de-

    tailed descriptions which involve the eigenvalues and singularities of

    the meromorphic Floquet matrix function. Existence and size esti-

    mates for sequences of spectral gaps are established.

    Keywords: Floquet theory, quantum graphs, spectral gaps.



    1 Introduction

    Among the interesting problems posed by nanoscale technology is the anal- ysis of propagation in periodic networks. Such networks might be used to describe paths for quantum mechanical electron propagation in ‘wires’ which are macroscopic in length, but have atomic scale width and thickness. These geometries are seen for instance in single wall carbon nanotubes [17]. In addition to electron propagation, there are analogous problems for acoustic or thermal propagation on such networks. Motivated in part by these appli- cations, this work considers spectral problems for differential equations on certain periodic graphs.

    Two methods of spectral analysis are available for periodic differential equations. The direct integral analysis typically used for partial differential equations [13, 16] offers broad applicability, but makes sacrifices in detail. The Floquet matrix analysis most common for ordinary differential equations is not as flexible, but often provides more detailed information and sharper results. This work uses the Floquet matrix approach to analyze products of finite graphs

    � with the integer graph � . The vertices of � are the integers,

    and vertices m,n ∈ � are connected by an edge if and only if n = m ± 1. If V � and V � denote the vertex sets of � and � respectively, then the graph product G = � × � has vertices (v, n) for v ∈ V � and n ∈ V � . Vertices (v1, n1) and (v2, n2) are connected in G if and only if n1 = n2 and v1, v2 are connected in

    � , or v1 = v2 and n1, n2 are connected in � . The edges of

    are assumed to connect distinct vertices. Our differential operators on the graph G are actually operators L =

    −D2 + q on the Hilbert space ⊕e∈GL2(e). That is, the operators act on functions defined on a set of intervals indexed by the abstract edges of the graph. The edges of � will have length 1 and may be identified with the intervals [m,m+1]; the resulting topological graph � may be identified with the real line, with global coordinate x. The N(E) edges of

    � will have lengths

    ri, for i = 1, . . . , N(E), and local coordinate ti, with 0 < ti < ri. Edge lengths for G are induced in the obvious way, and points on an edge of G may be described by pairs (v, x) for v ∈ V � , or (ti, m) for m ∈ V � . The group of integers acts on G by n : (v, x) → (v, x+n) and n : (ti, m) → (ti, m+n). An example of a graph G is illustrated in Figure 1.

    The eigenvalue equation

    (−D2 + q)y = λy, λ ∈ � . (1)


    is basic to this study. The function y is complex valued on each edge, and the potential q is assumed to be real valued, measurable, bounded, and in- variant under the action by the group of integers. (The requirement that q be bounded can certainly be relaxed, [7, p. 97], [10, pp. 343–346].)

    Figure 1 A type of boundary value problem is created by using the structure of G

    to define vertex condition constraints on admissible solutions to (1). Denote by ye the function y restricted to the edge e, and write e ∼ v if the vertex v is an endpoint of the edge e. The conditions are

    lim t→v

    ye(1)(t) = lim t→v

    ye(2)(t), e(1), e(2) ∼ v, (2) ∑

    e∼v lim t→v

    ∂νye(t) = 0.

    The first condition requires continuity of y at the vertices of G. In the second condition ∂νye(t) is the derivative in outward pointing local coordi- nates, so in a fixed set of local coordinates ∂νye(t) = y

    ′ e(t) if t is decreasing as

    t approaches v along e, otherwise ∂νye(t) = −y′e(t). These vertex conditions may be used to define a domain on which L = −D2 + q is self adjoint on the Hilbert space ⊕e∈GL2(e). Details are provided in [4, 5].

    In overview the development unfolds as follows. On the edges of G where x is not an integer, (1) is a system of ordinary differential equations on intervals m < x < m + 1. Solutions satisfying (2) are linked by transition matrices associated to

    � , and satisfy an existence and uniqueness theorem

    like that for differential equations on � , except for λ in a set ΣD comprising

    the Dirichlet eigenvalues from the edges of �

    . Simple examples show that


    points in ΣD may be eigenvalues of L on G; this feature of quantum graphs is not present in classical Schrödinger equations with periodic coefficients.

    The existence and uniqueness result enables us to construct a Floquet matrix type represention for the action of translation by 1 on the space of solutions to (1) satisfying (2) if λ /∈ ΣD. When the potential q is even on the edges of

    � , the Floquet matrix is symplectic for λ ∈ � and has a

    meromorphic continuation for λ ∈ � with poles in ΣD. This combination of structural features is quite helpful in establishing the existence of sequences of spectral gaps.

    The spectral theoretic consequences of Floquet theory are developed when q is even on the edges of

    � . The resolvent R(λ) = (L− λI)−1 is constructed

    in two parts, with a particularly simple representation when R(λ) acts on functions which vanish on the subgraphs

    � ×m. As one expects, the spec- trum of L consists of the set Σ1 where the Floquet matrix has an eigenvalue of absolute value 1, together with a subset of ΣD. Having connected the spectrum of L with the multipliers of the Floquet matrix, the existence of spectral gaps is considered. A rich class of examples with an infinite sequence of gaps is constructed under the unexpected hypothesis that

    � is not a bi-

    partite graph. These examples exhibit a robust structure which allows us to establish gap size estimates with minimal symmetry assumptions on the potential.

    The recent survey [14] may be consulted for an extensive list of references on the history, physical interpretations, and results in partial differential equations related to this work. The applicability of PDE techniques for the spectral analysis of a broad class of periodic difference operators on graphs was noted in [12]. The use of geometric scatterers to open spectral gaps has been previously explored in other contexts. A periodic problem with wide gaps was discussed in [3]. Recent work [18] on decorated graphs does not require periodicity, and also makes use of meromorphic functions in opening spectral gaps. The paper [19], which treats perturbation theory for eigenval- ues of infinite multiplicity inside gaps for a different type of graph, provided some of the motivation for this work. The author thanks Peter Kuchment for helpful comments.


    2 The Floquet matrix

    2.1 Symplectic matrices and the Wronskian

    Some basic material on symplectic matrices [2, p. 219], [11, p. 11] will be helpful. Let

    J =

    ( 0N IN −IN 0N

    ) .

    A real 2N × 2N matrix T is symplectic if

    T tJT = J.

    Extending this definition, a complex 2N × 2N matrix T is J-unitary if

    T ∗JT = J.

    The characteristic polynomial p(ξ) = det(T − ξI) of a symplectic matrix satisfies [2, p. 226]

    p(ξ) = ξ2Np(1/ξ),

    or equivalently, the coefficients ak of p(ξ) satisfy aj = a2N−j . It follows that the eigenvalues of a symplectic matrix are symmetric with respect to the unit circle and the real axis.

    For complex vectors W,Z with 2N components wi, zi we let

    〈W,Z〉 = 2N∑



    In terms of this form T is J-unitary if and only if

    〈JTW, TZ〉 = 〈JW,Z〉, W, Z ∈ � 2N .

    For notational convenience define

    [W,Z] = 〈JW,Z〉, W, Z ∈ � 2N .

    Suppose thatQ(x) is anN×N matrix valued function defined on an inter- val with Q(x) = Q∗(x). The conventional Wronski identity has an extension to equations

    −Y ′′ +Q(x)Y = λY. (3)


    Taking adjoints gives

    −(Y ∗)(x, λ)′′ + Y ∗(x, λ)Q(x) = λY ∗(x, λ).

    If U(x, λ) and V (x, λ) are N ×N matrix valued functions satisfying (3) then simple algebra leads to

    V ∗(x, λ)U ′′(x, λ)− (V ∗)′′(x, λ)U(x, λ) = 0,

    so the matrix Wronskian

    W(U(x, λ), V (x, λ)) = V ∗(x, λ)U ′(x, λ)− (V ∗)′(x, λ)U(x, λ)

    is a constant. For notational convenience define

    L](λ) = L∗(λ)

    for matrix functions L(λ) of λ ∈ � ; then

    W(U(x, λ), V (x, λ)) = V ]U ′ − (V ])′U.

    Suppose the columns of U(x, λ) and V (x, λ) are Ui and Vi. The matrix Wronskian W(U, V ) has entries

    W(U, V )ij = 〈 ( Uj U ′j

    ) (λ), J

    ( Vi V ′i

    ) (λ)〉 = −[

    ( Uj U ′j

    ) (λ),

    ( Vi V ′i

    ) (λ)].

    This gives the following result.

    Lemma 2.1.

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