1 Embedded eigenvalues in 1-periodic graphs

Eigenfunctions of Unbounded Support for Embedded Eigenvalues

of Locally Perturbed Periodic Graph Operators

Stephen P. Shipman

Department of Mathematics, Louisiana State University

Baton Rouge, LA 70803, USA

Abstract. It is known that, if a locally perturbed periodic self-adjoint operator on a combinatorial or quantum graph admits an eigenvalue embedded in the continuous spectrum, then the associated eigenfunction is compactly supported—that is, if the Fermi surface is irreducible, which occurs generically in dimension two or higher. This article constructs a class of operators whose Fermi surface is reducible for all energies by coupling several periodic systems. The components of the Fermi surface correspond to decoupled spaces of hybrid states, and in certain frequency bands, some components contribute oscillatory hybrid states (corresponding to spectrum) and other components contribute only exponential ones. This separation allows a localized defect to suppress the oscillatory (radiation) modes and retain the evanescent ones, thereby leading to embedded eigenvalues whose associated eigenfunctions decay exponentially but are not compactly supported.

Key words: quantum graph, graph operator, periodic operator, bound state, embedded eigenvalue, reducible Fermi surface, local perturbation, defect state, coupled graphs, Floquet transform

 

If a periodic self-adjoint difference or differential operator on a combinatorial or quantum graph is perturbed by a localized operator , and if admits an eigenvalue embedded in the continuous spectrum, then the corresponding eigenfunction (bound state) typically has compact support [9]. The obstruction to unbounded support is the algebraic fact that a generic polynomial in several variables cannot be factored. This is reflected in the irreducibility of the Floquet (Fermi) surface of , which is the zero set of a Laurent polynomial that describes the complex vectors for which admits a quasi-periodic solution with quasi-momentum vector , where is the vector of Floquet multipliers.

The Fermi surface is known to be irreducible for all but finitely many energies for the discrete 2D Laplacian plus a periodic potential [4] and for the continuous Laplacian plus a potential that is separable in a specific way in 2D and 3D [2, 8]. In the latter case, the principle of unique continuation of solutions of elliptic equations precludes the emergence of eigenfunctions of compact support under local perturbations. Thus no embedded eigenvalues are possible. But unique continuation fails for periodic combinatorial graph operators and quantum graphs [3, 7] and for higher-order elliptic equations [6]. In these cases, spectrally embedded eigenfunctions with compact support do exist, even for unperturbed periodic operators. In the graph case, they can be created by attaching a finite graph to the periodic one at a vertex of the finite graph where one of its eigenfunctions vanishes.

This article constructs a class of periodic graph operators for which the Fermi surface is reducible for all energies and for which local perturbations create embedded eigenvalues whose eigenfunctions have unbounded support. These operators are constructed by coupling different operators on identical graphs. The resulting operator decouples into invariant subspaces of hybrid states with different spectral bands. A non-embedded eigenvalue for one of these hybrid spaces that lies in a spectral band of another is an embedded eigenvalue for the full system. A simple example is two copies of the integer lattice , placed one atop the other, endowed with the discrete Laplace operator, or the quantum version in which edges connect adjacent vertices. The reducibility of the Fermi surface for all energies is automatic: each of its components corresponds to an invariant subspace of the operator.

Questions on the analytic structure of the Fermi surface, in particular the determination of (ir)reducibility, are not easy (see [5], for example). Reducibility for the class of operators in the present work results intentionally from its explicit construction. Each irreducible component is contained in the Fermi surface for an invariant subspace of the graph operator. If a component corresponding to an invariant subspace fails to intersect at an energy , then is not in the spectrum for that subspace and one can create a defect that supports an eigenfunction (bound state) within that subspace. This evokes the question of whether each irreducible component of the Fermi surface always corresponds to an invariant subspace of the operator, because this would raise the prospect of creating a defect that produces an eigenvalue whenever at least one irreducible component of the Fermi surface does not intersect . This was conjectured for Schrödinger operators in [8, §5, point 3].

The Fermi “surface” of a 1-periodic combinatorial or quantum graph operator or ODE is always reducible; its components are simply the roots of the Laurent polynomial of one variable . It is easy to construct embedded eigenvalues with exponentially decaying eigenfunctions because of the explicit decoupling of the Floquet modes , where and is the restriction of the mode to one period (e.g. [1, 11, 12, 13])—one splices an exponentially growing mode to the left of a defect together with an exponentially decaying mode to the right. An examination of some 1D examples that can be computed by hand motivates the constructions in higher dimensions.

1 Embedded eigenvalues in 1-periodic graphs

The purpose of this section is to illustrate the ideas of the paper through three examples of 1D graph operators for which one can straightforwardly compute spectrally embedded bound states of unbounded support. Example 1 shows how embedded eigenvalues are easily created in 1D periodic graphs simply because the Laurent polynomial is generically a product of linear factors, where is the dispersion relation between energy and quasi-momentum (or wavenumber) . The construction does not generalize to higher dimensions, where generically fails to factor. Example 2 for a combinatorial graph does generalize to higher dimensions (sec. 2) because the construction of bound states is devised specifically to be independent of dimension. It relies on an explicit decoupling of a graph operator into two independent subsystems with different continuous spectrum. Example 3 shows how to modify Example 2 to accommodate quantum graphs; it is generalized to higher dimensions in section 3.

1.1 Example 1: Finite-difference operator of order 4

Consider the fourth-order difference operator on given by

The -transform (i.e., the Floquet transform evaluated at ),

converts into a multiplication operator

It is a Hilbert-space isomorphism from to , where is the complex unit circle . This shows that the spectrum of consists of those for which for some . This “dispersion relation” between and ,

is shown in Fig. 3 for real . The spectrum of is the range of this trigonometric polynomial in .

As seen in Fig. 3, for the spectrum is of multiplicity 2—there is exactly one pair of solutions of of the form with . This can be seen algebraically by writing as

(1.1)

Each choice of sign of the square root gives a pair of solutions of the form , which are of unit modulus if and only if . In the -interval , the plus sign yields () and the minus sign yields , with .

This means that there are both oscillatory solutions and exponential solutions of . This is because acts on fields of the form (eigenfunctions of the shift operator not in ) by multiplication by :

In this spectral interval, , the exponential solutions can be used to construct a spectrally embedded eigenfunction (bound state) for a localized perturbation of , by splicing an exponentially growing solution for with an exponentially decaying one for ,

Let the potential be given by a multiplication operator

with for all but finitely many values of . By enforcing the equation , one obtains for and

(1.2)

A typical perturbation of will destroy the bound state and the embedded eigenvalue of , resulting in resonant scattering of the extended eigenstates  [13].

Figure 3: This is the dispersion relation for the fourth-order difference operator in Example 1. It characterizes solutions of the form to the equation . The spectrum of is the range of this graph; it has multiplicity 4 in and multiplicity 2 in .

1.2 Example 2: Decoupling by symmetry in a combinatorial graph

The construction of Example 1 does not extend to -periodic graphs for because the Floquet surface111The complex dispersion relation between energy and quasi-momentum is , and this zero-set of -values is the Bloch variety; the Fermi surface for an energy is , and the Floquet surface for is [9]. for , (more generally, ) is generically irreducible over . Example 2 illustrates a construction, which generalizes to a class of -periodic combinatorial graph operators (sec. 2), for which the Floquet surface is reducible for all and embedded eigenvalues with eigenfunctions of unbounded support can be created by local defects.

The combinatorial graph in Fig. 6 consists of two coupled 1D chains. A function on the vertex set of can be viewed as a -valued function of . The edges of indicate interactions between neighboring vertices realized by a periodic self-adjoint operator on :

(1.3)
(1.4)

Under the -transform, becomes multiplication by a matrix function ,

(1.5)
(1.6)

The equation has a solution of the form222A function is a non- eigenfunction of the shift operator with eigenvalue . The function is a Floquet-Bloch, or quasi-periodic, solution of , is the Floquet multiplier, and is the quasi-momentum. if and only if is a null vector of , and the spectrum of is all such that holds for some . The Floquet surface reduces to

By putting , one obtains two branches of a dispersion relation between energy and wavenumber . The parts of these branches where is real correspond to two -intervals, whose union is the spectrum of ,

The quantity is the magnitude of the splitting of these two energy bands and is akin to the Rabi frequency. When the bias vanishes, the plus-branch has eigenvector , corresponding to symmetric solutions of , and the minus-branch has eigenvector , corresponding to anti-symmetric solutions.

In the -intervals of multiplicity 2, where the two bands do not overlap, one can create exponentially decaying eigenfunctions for embedded eigenvalues of a locally perturbed operator similarly to Example 1; this is carried out in [11]. Generic perturbations of destroy the embedded eigenvalue. In [11], it is shown that the resulting scattering resonance is detuned from the bound-state energy because of asymmetry created by the bias .

Figure 2: This is the dispersion relation for the fourth-order difference operator in Example 1. It characterizes solutions of the form to the equation . The spectrum of is the range of this graph; it has multiplicity 4 in and multiplicity 2 in .
Figure 6: The combinatorial graph of Example 2. The edges indicate interactions between neighboring vertices realized by the operator (1.3) that acts on .

1.3 Example 3: Decoupling by symmetry in a quantum graph

Figure 7 depicts a simple 1-periodic metric graph (e.g. [3, §1.3]). The group acts as a translational symmetry group with a fundamental domain consisting of two vertices and , two horizontal edges and coordinatized by , and a vertical edge coordinatized by . Define an operator by

(1.7)

acts on functions , such that the restriction of to each edge is in the Sobolev space ,  is continuous at each vertex, and the sum of the derivatives of at each vertex directed away from must vanish (-flux, or Neumann, condition [3, p. 14]). The additional requirement that be integrable over makes a self-adjoint operator in , thus creating a quantum graph .

In analogy to Examples 1 and 2, one seeks solutions of (not in ) that satisfy the quasi-periodic condition

On each edge, has the form , where , as depicted in Fig. 7. Observe that is invariant on the symmetric and anti-symmetric spaces of functions with respect to the horizontal line of reflectional symmetry of . By requiring that be anti-symmetric, it has the form on the vertical edge , and the continuity and flux conditions at the vertex impose three homogeneous linear conditions on and the coefficients and for the edge :

(1.8)

For symmetric functions , one just changes to and, in the first column of the matrix, to and to . Setting the determinants of these matrices to yields conditions for nonzero quasi-periodic solutions to ,

(1.9)

By setting , one obtains symmetric and antisymmetric branches of the dispersion relation, which are depicted in Fig. 10. Each branch exhibits spectral bands, indicated by the solid lines, separated by gaps, but these bands overlap so that the spectrum of consists of all .

Figure 7: Left. The metric graph of Example 3. A fundamental domain for its translational symmetry group consists of the bold edges , , and the vertices , . Edges and are identified with the -interval , and edge with in the direction of the arrows, and the vertices carry no mass. The functions on the edges are Floquet solutions of () with Floquet multiplier . Symmetric and anti-symmetric solutions with respect to a horizontal line are depicted; the () sign for corresponds to () on . Right. One vertical edge is made defective by adding a potential so that and an antisymmetric solution is with . For spectral values of multiplicity around a multiple of (see Fig. 10) and appropriate choice of , one can construct an -eigenfunction that is symmetric about the defective edge and anti-symmetric about the central horizontal line.

Figure 5: The combinatorial graph of Example 2. The edges indicate interactions between neighboring vertices realized by the operator (1.3) that acts on .
Figure 10: Branches of the dispersion relation (energy ) for the operator of Example 3 on the metric graph in Fig. 7 (one period in is shown). The upper branch corresponds to fields that are anti-symmetric about the central horizontal line of ; is real on the spectral band indicated by the upper solid line, where . The lower branch corresponds to symmetric fields and the spectral band indicated by the lower solid lines.

Let the operator be perturbed locally by adding to it a potential that vanishes everywhere except on the edge of the fundamental domain in Fig. 7. This means that the action of is

in which is a real number. Thus solutions of have the form on each edge except , where it has the form , where . The continuous spectra of and are identical because is a relatively compact perturbation of .

An embedded eigenvalue for the defective quantum graph can be created at spectral values of multiplicity 2. This occurs, say, if is near a multiple of in Fig. 10, where the symmetric states are propagating () and the anti-symmetric states are exponential (). An anti-symmetric eigenfunction (bound state) is created by splicing an exponentially decaying quasi-periodic solution of to the right of the defective edge with an exponentially growing solution to the left (Fig. 7, right). Specifically, the second equation of (1.9) gives two solutions , with the smaller one equal to

(1.10)

The bound state has the form

(1.11)

in which and satisfy for and , respectively, subject to (1.10). In fact, and are reflections of one another about because of the corresponding reflection symmetry of and . By setting , the continuity and -flux condition at the vertex in Fig. 7 result in a relation between and ,

(1.12)

Remember that depends on through (1.10) and that . As long as one can solve for in terms of in (1.12), the potential can be determined so that (1.11) satisfies , thus completing the construction of an embedded eigenvalue whose eigenfunction has unbounded support. This is possible because the left-hand side of (1.12) takes on all real values as ranges over .

2 Embedded eigenvalues in coupled -periodic graphs

This section generalizes the 1D Example 2 to higher dimension. The first step (sec. 2.1) is to couple two identical combinatorial graphs, with possibly different operators, in such a way that the resulting system decouples into two spaces of hybrid states with different continuous spectrum. Next (sec. 2.2), a non-embedded eigenvalue is constructed for one of the hybrid systems with energy in the spectral band of the other. The construction is generalized to coupled graphs in section 2.3.

The mathematical development of this coupling-decoupling construction in sections 2.1 and 2.3 is valid in a general Hilbert-space setting, although it is presented in the language of combinatorial graphs. In particular, it can be applied to the coupling of two identical quantum graphs. However, since they are coupled by interactions “at a distance”, the coupled system is not a quantum graph. Section 3 presents a modification of the construction for quantum graphs.

2.1 Decoupling of hybrid states in coupled graphs

Let be a combinatorial or metric graph that is -periodic, meaning that admits a group of symmetries isomorphic to . Assume also that a fundamental domain of the action on is pre-compact. Let be a periodic operator on , whose domain is a dense sub-vector-space of the Hilbert space of square-integrable functions on , and let be self-adjoint in . The periodicity of means that commutes with the action of .

Consider two copies of the same graph , one endowed with the operator and the other with the operator , where the bias is bounded, periodic, and self-adjoint. The two systems and are then coupled through a bounded periodic operator to create a periodic self-adjoint operator on the disjoint union . The domain of is , and its block-matrix representation with respect to this decomposition is

(2.13)

It turns out that, if and are linearly dependent operators, then is unitarily block-diagonalizable. Thus, let and be multiples of a given bounded, periodic, self-adjoint operator on :

(2.14)

Here, is an arbitrary phase, and measures the relative strengths of the bias and the coupling. The operator is decoupled into two operators and by the unitary operator

(2.15)

Indeed, a calculation yields

(2.16)

on the domain , in which

If is a graph operator, then

so that the Floquet surface is reducible for all energies .

The conjugacy (2.16) effects a decomposition of into two orthogonal -invariant spaces and of hybrid states

(2.17)
(2.18)

The action of on is given by applying to each component,

(2.19)

and on the action is by .

Notice that the splitting of into and , and therefore also the spectra of and , depend only on ; they do not depend on , which measures the relative strengths of the bias  and coupling . What changes with are the relative amplitudes of the components of the hybrid fields, as seen from the definitions of and . The energy (square norm) of a hybrid state is divided between the two graphs:

(2.20)
(2.21)
(2.22)

Figure 13 illustrates the relation between the relative strengths of the bias and coupling and the relative amplitudes of the components of the hybrid states. When the two graphs are coupled but no bias is imposed (), the energy of a hybrid state is equally partitioned between the two graphs. If, in addition, , and consist of the symmetric and antisymmetric states:

On the other extreme, corresponds to no coupling, so and .

The operator is reminiscent of the Rabi frequency; if , it is determined through the bias and the coupling by . If , the number is the width of the spectral splitting of the two decoupled spaces and of hybrid states.

   

Figure 9: Branches of the dispersion relation (energy ) for the operator of Example 3 on the metric graph in Fig. 7 (one period in is shown). The upper branch corresponds to fields that are anti-symmetric about the central horizontal line of ; is real on the spectral band indicated by the upper solid line, where . The lower branch corresponds to symmetric fields and the spectral band indicated by the lower solid lines.
Figure 13: The relative strengths of the bias and coupling in the operator , defined in (2.13) and (2.14), is represented by an angle . There are two -invariant spaces and of hybrid states given by (2.17,2.18). The relative amplitudes of the components of states in (resp. ) are given by the angle (resp. ), as indicated by the dots.

2.2 Spectrally embedded eigenfunctions

The coupling-decoupling technique of sec. 2.1 can be used to create locally perturbed periodic graph operators with an embedded eigenvalue whose eigenfunction has unbounded support. By specializing the operator to a multiple of the identity, the spectra of the hybrid systems can be shifted at will. To create an embedded eigenvalue for the operator for some , it suffices that possess an eigenvalue, embedded or not.

Theorem 1.

Let be a periodic self-adjoint operator on a combinatorial or metric graph and a compactly supported self-adjoint perturbation of , and suppose that satisfies . Let be such that , where denotes the continuous spectrum of . For each and , is an embedded eigenvalue of each the self-adjoint operators

(2.23)

and

(2.24)

in . In each case, the eigenfunction corresponding to is

(2.25)

that is, .

In particular, if is a combinatorial graph and has unbounded support, then is an eigenfunction (bound state) of unbounded support for an embedded eigenvalue of the self-adjoint operators on a combinatorial graph whose vertex set is , where is the vertex set of . Note that, in this case, .

Remark. The perturbation vanishes (acts as the zero operator) on the subspace , and therefore does not affect the extended states associated with . The bound state is in the subspace , on which acts as a local perturbation of . The perturbation , on the other hand, affects the action of in both subspaces . Since is a local perturbation, it does not modify the continuous spectrum of , and thus neither does modify the continuous spectrum of . Thus the eigenvalue , which is by construction within the continuum of , is also within the continuum of .

Proof.

Assume that, for some , and , and put . Then and . Since is a local graph operator, , and thus . This means that is an embedded eigenvalue of the operators

and

in with eigenfunction . Therefore is an embedded eigenvalue of , where is the unitary operator defined by (2.15). One computes using (2.16) that is equal to in the theorem, and a corresponding eigenfunction is . ∎

This theorem allows one to use any (typically non-embedded) eigenvalue of a locally perturbed periodic operator on a combinatorial graph to construct an embedded eigenvalue for an operator on another graph, namely the union of two disjoint copies of . Non-embedded eigenvalues whose eigenfunctions have unbounded support and exponential decay are commonplace for locally defective periodic structures; a construction for graphs is given below. This observation, together with Theorem 1 yields

Corollary 2.

There exist self-adjoint -periodic () finite-degree combinatorial graph operators that admit localized self-adjoint perturbations possessing an embedded eigenvalue whose eigenfunction has unbounded support and exponential decay.

The following discussion shows how to construct, for a simple class of graph operators, a non-embedded eigenvalue whose eigenfunction has unbounded support and exponential decay. Let be a degree- () -periodic difference operator on a graph whose fundamental domain consists of a single vertex. The graph can be identified with the integer lattice . The perturbation will be a multiplication operator with support at a single vertex.

First consider the forced equation

(2.26)

in which and for all nonzero . Application of the -transform gives the scalar equation

in which is a Laurent polynomial in . Assuming that , the number is nonzero for all and the function

is bounded on . By the Fourier inversion theorem, is the -transform of a function in , which satisfies . The solution has bounded support if and only if is a Laurent polynomial in . Assuming that is not a multiplication operator (there are interactions between vertices), is non-constant. Moreover, since is self-adjoint, the coefficients of satisfy , and thus has at least two nonzero terms. It follows that vanishes at some so that cannot be a Laurent polynomial. Thus has unbounded support. Since is analytic in a complex neighborhood of , is an exponentially decaying function of .

The Floquet inversion theorem gives as the average of over the -torus:

The value is real and nonzero. The reason is the identity coming from the self-adjointness of , which makes real valued on . Since is real, non-vanishing, and continuous at each , the integrand is of one sign.

Define the multiplication operator on by

By this definition, , and by (2.26), one obtains

so that is an eigenfunction of with unbounded support and exponential decay.

2.3 Generalization to several coupled graphs

The construction of two spaces of decoupled hybrid states can be generalized to spaces, leading to embedded eigenvalues in systems of several coupled graph operators.

One does this by generalizing the matrix

of biases and couplings to any Hermitian matrix whose entries will serve as coupling coefficients among identical graphs. The hybrid states are defined through the columns of a unitary matrix that diagonalizes , thus generalizing the matrix

from the two-system case. The operators on the hybrid state spaces are of the form , where the real numbers are the eigenvalues of . All this is made precise below.

Let a self-adjoint operator in a Hilbert space be given, and consider identical copies of the system coupled through multiples of a single bounded self-adjoint operator . This results in an operator in the direct sum :

The matrix of coupling coefficients is Hermitian, which makes self-adjoint. The “self-couplings” provided by the diagonal entries of can be thought of as modifications of the operator on each copy of , generalizing the bias from before. By identifying with a tensor product,

the block-matrix form of is written conveniently as

in which is the identity matrix.

The operator can be block-diagonalized. Let be the unitary matrix that conjugates into a diagonal matrix of real eigenvalues of ,

The tensor product , where is the identity operator on , is an block matrix whose blocks are the multiples of the identity. It is a unitary operator on that decomposes into subsystems:

or, more concisely,

(2.27)

Here, is a block-diagonal matrix whose diagonal blocks are modifications of :

(2.28)

Now let and be periodic difference operators of finite degree on a combinatorial graph with translational symmetry, and let , , etc., be the spectral representations of the corresponding operators under the Floquet transform. From (2.27), one obtains

in which is the identity operator on . The operator has a block-diagonal form obtained by replacing and with their Floquet transforms in (2.28). Thus the Floquet surface of is reducible for each energy :

One can then construct embedded eigenvalues for the operator in the combinatorial graph whose vertex set is the union of disjoint copies of by generalizing the procedure in sec. 2.2.

3 Embedded eigenvalues in quantum graphs

If is a quantum graph and , the system constructed in section 2.1 does not define a quantum graph because of the direct coupling between vertices of the two copies of . The construction can be modified by realizing the coupling through additional edges connecting the two copies of . The main result of this section is that there exist self-adjoint -periodic finite-degree quantum graphs that admit localized self-adjoint perturbations that possess an embedded eigenvalue whose eigenfunction has unbounded support.

First, a general procedure for constructing embedded eigenvalues is developed in section 3.1. It involves “decorating” a given graph by periodically attaching dangling edges, which creates gaps in the spectrum that depend on the condition at the free vertex; see [10] for a proof of this phenomenon for combinatorial graphs. When two identical copies of the decorated graph are connected at the free vertices, the resulting graph decouples into even and odd states whose spectra are equal to those for the free-endpoint (Neumann) and clamped-endpoint (Dirichlet) conditions imposed on the free vertices of the decorated graph. One then tries to construct an eigenvalue in a spectral gap of the even (odd) states that lies in a band of the odd (even) states to produce an eigenvalue that is embedded in the spectrum of the full system. A full proof for a specific 2D graph is presented in section 3.2 (Theorem 3).

3.1 Coupling two quantum graphs by edges and (anti)symmetric states

Let be an -periodic quantum graph, and let be the graph obtained by connecting two identical copies and of by edges that connect vertices in to the corresponding ones in , to obtain a periodic metric graph . Endow with a periodic operator given by on the edges of and and by on the connecting edges. A fundamental domain of the quantum graph consists of two copies of a fundamental domain of connected by, say, just one edge for simplicity. Fig. 16 depicts the case that is the hexagonal graph of graphene. Let be identified with the -interval and be symmetric.

A local perturbation of analogous to that in Example 3 consists of a constant potential applied only to the edge in the fundamental domain , but not to any of the translates of ; call this potential :

The operator is reduced by the decomposition , where () is the space of functions symmetric (anti-symmetric) with respect to reflection about the center of and its translates and switching of and . Thus the spectrum of is the union of the spectra of restricted to the spaces . The restriction is identified with the quantum graph , where is “half” of —its fundamental domain consists of plus half of dangling from one vertex of (Fig. 16). Call this edge