Stability of eigenvalues of quantum graphs with respect to magnetic perturbation and the nodal count of the eigenfunctions

(a) Quadratic form of H_γ

The quadratic form of the operator H⁰ on the graph Γ is

with the domain The Dirichlet conditions, if any, are also imposed on the domain D. For , the quadratic form of the operator H_γ acting on the tree T is formally with the domain Importantly, the domain D_γ is larger than D since we no longer impose continuity at the cut points c_j on the functions from D_γ. More precisely, we can represent the domain D as Note that the domain D_γ is independent of the actual value of γ.

Remark 4.1

Observe that any function f∈D_γ can be written as

where f₀∈D, and ρ_j∈D_γ. Moreover, we require that ρ_j have a jump at c_j, but be continuous at all other cut points c_k, k≠j (i.e. each ρ_j represents one jump of the function f). In particular, for a given λ, we will use the family of functions ρ_j,λ that satisfy on every edge, the δ-type conditions (2.1) at the vertices of Γ, and the following conditions at the cut points: and Note that condition (4.4) essentially glues these cut points back together.

Existence and uniqueness of the functions satisfying the above conditions is assured (see, for example [6], section 3.5.2) provided λ stays away from the Dirichlet spectrum . Since we are interested in λ close to the eignevalue λ_n(H⁰) of the uncut graph, we check that λ_n(H⁰) does not belong to the Dirichlet spectrum described above. The corresponding ‘Dirichlet graph’ can be viewed as the uncut graph with an extra Dirichlet condition imposed at the vertex c_j (in place of the Neumann condition effectively imposed there by H⁰). However, by a simple extension of the interlacing theorem of [6,17] (see lemma 6.4 for a precise formulation), one can see that the interlacing between Neumann and Dirichlet eigenvaules is strict since λ_n(H⁰) is assumed to be simple and the corresponding eigenfunction ψ is non-zero at the cut points.

(b) Properties of Wronskian on graphs

It will be important to relate the values of the derivatives of the functions ρ_j,λ at the cut points . We will do this using the Wronskian. Therefore, in this subsection, we investigate the properties of the Wronskian on graphs. We will do this for the most general self-adjoint vertex conditions on the graph Γ.

Given any two functions f₁,f₂∈D that satisfy the differential equation −f′′(x)+(q(x)−λ)f(x)=0, we know by Abel’s formula that the Wronskian of f₁ and f₂ is constant on any interval, or in particular, on any edge. Observe that the Wronskian is a one-form, that is, its sign depends on direction. We will now show that the total sum of Wronskians at any vertex with self-adjoint conditions is zero (all Wronskians must be taken in the outgoing direction).

Lemma 4.2

Let Γ be a graph and let be two functions that satisfy the differential equation −f′′(x)+(q(x)−λ)f(x)=0 and real self-adjoint vertex conditions. Then

where E_v denotes the set of all edges attached to the vertex v and each Wronskian W_e(f₁,f₂) is taken outward.

Proof.

We denote the self-adjoint operator acting as −d²/dx²+q(x) by H. Define a smooth compactly supported function ζ on Γ such that ζ≡1 in a neighbourhood of the vertex v and is zero at all other vertices of Γ. For the sake of convenience, we denote ζf_j by g_j. Then using the self-adjointness of H and integrating by parts, we obtain

since g_j=f_j near vertex v and g_j are zero near all other vertices. □

Lemma 4.3

Let a and b be two leaves (i.e. vertices of degree one) of a graph Γ. Let f₁ and f₂ be two solutions of −f′′(x)+(q(x)−λ)f(x)=0 on Γ that satisfy the same self-adjoint vertex conditions at all vertices except a and b. Then W( f₁,f₂)(a)=−W( f₁,f₂)(b).

Proof.

In graph theory, the flow η between two vertices a and b is defined as a non-negative function on the edges of a directed graph Γ that satisfies Kirchhoff’s current conservation condition at every vertex other than a or b: the total current flowing into a vertex must equal the total current flowing out of it (see figure 2 for an example). Given a flow η between a and b, it is a standard result of graph theory that the total current flowing into b is equal to the total current flowing out of a [18].

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We interpret the Wronskian as a flow by assigning directions to the edges of Γ so that the Wronskian is always positive. The current conservation condition is then equivalent to lemma 4.2. Therefore, the flow into b equals the flow out of a so W( f₁,f₂)(a)=−W( f₁,f₂)(b). □

(c) Morse index with Lagrange multipliers

In the proof of theorem 3.3, we will need to find the Morse index of the nth eigenpair (λ_n,ψ) of H⁰. The lemma below will help us to do just that.

Lemma 4.4

Let A be a bounded from below self-adjoint operator acting on a real Hilbert space . Assume that A has only discrete spectrum below a certain Λ and its eigenvalues are ordered in increasing order. Let h[f] be the quadratic form corresponding to A. If the nth eigenvalue λ_n<Λ is simple and ψ is the corresponding eigenfunction, then the Lagrange functional

has a non-degenerate critical point at (λ_n,ψ) whose Morse index is n.

Proof.

We split the Hilbert space into the orthogonal sum . Here, the space is the span of the first n−1 eigenfunctions of A, the space is the span of the nth eigenfunction ψ, and is their orthogonal complement. The quadratic form h is reduced by the decomposition , namely,

On , the quadratic form h is bounded from above, Similarly, on the form h is bounded from below, Finally, on , we have where f₀=sψ, .

To show that (λ_n,ψ) is a critical point and to calculate its index, we evaluate

Expanding δf=δf₋+sψ+δf₊ according to our decomposition of , we see that Simplifying and completing squares, we obtain where the two middle terms are representing 2sδλ. We observe that all terms are quadratic or higher order and hence (λ_n,ψ) is a critical point as claimed. The first two terms represent the negative part of the Hessian. Their dimension is the dimension of plus one. Thus, the Morse index is (n−1)+1=n. □

We remark that in the finite-dimensional case the Hessian of δL at the critical point is known as the ‘bordered Hessian’ (see [19] for a brief history of the term).

(d) Restriction to a critical manifold

We will also use the following simple result from [11].

Lemma 4.5

Let be a direct decomposition of a Banach space. Let also be a smooth functional such that (0,0)∈X is its critical point of Morse index ind( f).

If for any y in a neighbourhood of zero in Y, the point (y,0) is a critical point of f over the affine subspace {y}×Y ′, then the Hessian of f at the origin, as a quadratic form in X, is reduced by the decomposition X=Y ⊕Y ′.

In particular,

where ind(f|_W) is the Morse index of 0 as the critical point of the function f restricted to the space W. Moreover, if (0,0) is a non-degenerate critical point of f on X, then 0 is non-degenerate as a critical point of f|_Y.

The subspace Y , which is the locus of the critical points of f over the affine subspaces (y,⋅), is called the critical manifold. In applications, the locus of the critical points with respect to a chosen direction is usually not a linear subspace. Then a simple change of variables is applied to reduce the situation to that of lemma 4.5, while the Morse index remains unchanged.

(e) Proof of theorem 3.3

In this subsection, λ is used as both an independent variable and a function (eigenvalue as a function of parameters). To reduce the confusion, we denote ξ=λ_n(H⁰). Recall that ψ is the nth eigenfunction of H⁰ and ϕ denotes the number of internal zeros of ψ on Γ.

Proof of part 1 of theorem 3.3.

By design, ψ is an eigenfunction of H_γ when ; the vertex conditions at the new vertices were specifically chosen to fit ψ. We conclude that .

Since ψ is non-zero on vertices, the corresponding eigenvalue of is simple [17,20]. Eigenfunctions on a tree are Courant-sharp [17,20,21]; in other words, the eigenfunction number n has n−1 internal zeros. We use this property in reverse, concluding that ψ is the eigenfunction number ϕ+1 of the tree operator . □

Proof of part 2 of theorem 3.3.

Here, we prove that is a critical point of λ_ϕ+1(H_γ).

Consider the Lagrange functional

where h_γ[f] is the quadratic form of the operator H_γ, given by equation (4.2). Observe that λ_ϕ+1(H_γ)=:λ(γ) is a restriction of F₃ onto a submanifold, namely, where f(γ) is the normalized (ϕ+1)^th eigenfunction of H_γ. We will now show that is a critical point of F₃; then criticality of the function λ_ϕ+1(H_γ) will follow immediately.

We know from lemma 4.4 that the eigenpair (ξ,ψ) is a critical point of the Lagrange functional and therefore

Additionally, we calculate from equation (4.2) that since is continuous at all cut points c_j. This proves that is a critical point of λ_ϕ+1(H_γ). The non-degeneracy of the critical point will follow from the proof of part 3. □

Proof of part 3 of theorem 3.3.

We will calculate the index of the critical point of λ_ϕ+1(H_γ) in two steps. We will first establish that the index of as a critical point of F₃ is equal to n+β. Then we will apply lemma 4.5 to the restriction introduced in (4.8) in order to deduce the final result. In fact the second step is simpler and we start with it to illustrate our technique.

Index of the critical point of λ(H_γ). Assume we have already shown that is a non-degenerate critical point of F₃ of index n+β. Define the following change of variables:

where λ(γ) is the (ϕ+1)^th eigenvalue of the operator H_γ and f(γ) is the corresponding normalized eigenfunction. The eigenvalue λ(γ) is simple when (see the proof of part 1 above) and this property is preserved locally.

The critical point corresponds, in the new variables, to (0,0,0). The change of variables is obviously non-degenerate and therefore the signature of a critical point remains unchanged.

For every fixed γ, the function F₃ is the Lagrange functional of the operator H_γ and by lemma 4.4 we conclude that (λ(γ),f(γ)) is its non-degenerate critical point of index ϕ+1. In the new variables this translates to being a critical point with respect to the first two variables for any value of the third variable. Now, we can apply lemma 4.5 to conclude that is a non-degenerate critical point of with index (n+β)−(ϕ+1).

Since , we obtain the desired conclusion. It remains to verify the assumption that is a non-degenerate critical point of F₃ of index n+β.

Index of critical point of F₃. By remark 4.1, any f∈D_γ can be written as

where f₀∈D and each ρ_j,λ satisfies g′′(x)+(q(x)−λ)g(x)=0. Therefore, the Lagrange functional F₃ can be re-parametrized as follows: where we understand the integral over the graph T as the sum of integrals over all edges of T. We let and investigate F₄ as a function of s and γ while λ and f₀ are held fixed. It turns out that (s,γ)=(0,R) is a critical point. Indeed, we calculate explicitly that is equal to zero when γ_j=R_j(λ). Note that we assumed χ_v=0 at every vertex of the graph Γ. If this is not the case, the corresponding terms cancel out when the integration by parts is performed.

The partial derivatives with respect to γ_j also vanish,

since the function f₀∈D is continuous across the cut vertices c_j. Here, we used the short-hand f=f₀+s⋅ρ_λ.

We can also calculate the Morse index of the critical point (s,γ)=(0,R). The Hessian is block-diagonal with β blocks of the form:

where the value of the second derivative with respect to s_j is irrelevant. Each block has a negative determinant and therefore contributes one negative and one positive eigenvalue. The total index is therefore β and the critical point is obviously non-degenerate.

Finally, we observe that the critical manifold (λ,f₀,0,R(λ)) passes through the critical point . To show this, we need to verify that when f₀=ψ. Applying lemma 4.3 to the Wronskian of ψ and ρ_j, we obtain

Substituting the boundary values of ρ_j (see remark 4.1), we arrive at

By using the non-degenerate change of variables

we can again apply lemma 4.5 with . On the subspace Y the function F₃ is equal to h[f₀]−λ(∥f₀∥²−1), which is precisely the Lagrange functional for the operator H⁰ with the correct domain. By lemma 4.4, it has index n at the point (λ,f₀)=(ξ,ψ). Adding the two indices together, we obtain index n+β for the critical point of F₄, which corresponds to the critical point of F₃. This concludes our proof. □

(a) Points of symmetry

Theorem 5.1

Let σ(α) denote the spectrum of H^α where . Then all points in the set

are points of symmetry of σ(α), i.e. for all and for all ς∈Σ, together with multiplicity.

Consequently, if λ_n(α) is the nth eigenvalue of H^α that is simple at α=ς∈Σ, then ς is a critical point of the function λ_n(α).

Proof.

We will show that if f(x) is an eigenfunction of H^ς−α, then is an eigenfunction of H^ς+α. Since the operator H^α is self-adjoint, we know that the eigenvalues are real. Taking the complex conjugate of the eigenvalue equation for f, we see that satisfies the same equation,

Similarly, all vertex conditions at the vertices of the tree T inherited from Γ have real coefficients and therefore satisfies them too. The only change occurs at the vertices .

Note that for every σ∈Σ, σ_j is equal to either 0 or π so e^2iσ_j=1 for all j=1,…,β. Therefore,

Conjugating the vertex conditions of H^ς−α at , we obtain and same for the derivative. Thus, satisfies the vertex conditions of the operator H^ς+α and vice versa. The spectra of these two operators are therefore identical. □

(b) A non-self-adjoint continuation

We now consider the same operator −d²/dx²+q(x) on the tree T with different vertex conditions at :

i.e. the function has a jump in magnitude across the cut. It is easy to see that these conditions are obtained from (3.1) by changing α to −iα. We will denote the operator with vertex conditions (5.3) at the vertices in T\Γ by H^iα.

Remark 5.2

The operator of H^iα is not self-adjoint for . A simple example is the interval [0,π] with q(x)=0 and conditions

which has complex eigenvalues when α≠0. Indeed, the eigenvalues are easily calculated to be

Lemma 5.3

If λ_n(H⁰) is simple, then locally around α=(0,…,0) the eigenvalue λ_n(H^iα) is real. The corresponding eigenfunction is real too.

Proof.

By standard perturbation theory [22] (see also [17] for results specifically on graphs), we know that λ_n(H^iα) is an analytic function of α and since λ_n(H⁰) is simple, λ_n(H^iα) remains simple in a neighbourhood of α=(0,…,0). Since the operator H^iα has real coefficients, its complex eigenvalues must come in conjugate pairs. For this to happen, the real eigenvalue must first become double. Since λ_n(H^iα) is simple near (0,…,0), the eigenvalue is real there. □

We note that since we impose no restrictions on the eigenvalues below λ_n(H⁰), some of them might turn complex as soon as α≠0. In this case, the ‘nth’ eigenvalue λ_n(H^iα) refers to the unique continuation of λ_n(H⁰). Locally, of course, it is the same as having the eigenvalues ordered by their real part.

(c) Connection between H_γ and H^iα

Locally around we introduce a mapping R:γ↦α so that λ_ϕ+1(H_γ)=λ_n(H^iα) when R(γ)=α.

For a given γ, we find the (ϕ+1)^th eigenfunction of H_γ, denoting it by g. We then let

and, since g satisfies equation (3.3), it is now easy to check that g is indeed an eigenfunction of H^iα.

Lemma 5.4

The function R is a non-degenerate diffeomorphism. Therefore, the point α=(0,…,0) is a critical point of the function λ_n(H^iα) of index n−1+β−ϕ.

Proof.

The function R is an analytic function in a neighbourhood of since the eigenfunctions are analytic functions of the parameters and therefore R is a composition of analytic functions. We can define R⁻¹ by reversing the process, i.e. for a given α find the (real) nth eigenfunction φ of H^iα and let γ_j=φ′(c⁺_j)/φ(c⁺_j). By the same arguments, R⁻¹ is also an analytic function in a neighbourhood of (0,…,0). Therefore, R is a non-degenerate diffeomorphism.

A diffeomorphism preserves the index and therefore the index of (0,…,0) of the function λ_n(H^iα) is the same as the index of of the function λ_ϕ+1(H_γ), which was computed in theorem 3.3. □

(d) From H^iα to H^α

Proof of theorem 2.1.

The function λ_n(H^α)−ξ is analytic and, locally around (0,…,0), quadratic in {α_j} because (0,…,0) is a critical point so the linear term (the first derivative) is zero. Substituting α→−iα into the quadratic term results in an overall minus, that is,

Therefore, the index of (0,…,0) as a critical point of λ_n(H^α) is the dimension of the space of variables minus the index of λ_n(H^iα). Thus, it is equal to  □

(a) Partitions with few zeros

For eigenfunctions corresponding to low eigenvalues, the nodal set might not break all the cycles of the graph, see figure 3a. In this case, the parametrization of the nearby equipartitions is done via a modification of the operator H_γ. In this section, we describe this parameterization and point out the changes in the proofs of the analogue of theorem 3.3 that the new parameterization necessitates. An outline of the procedure has already appeared in [9,10]; however, some essential details have been omitted there.

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As mentioned previously, the eigenfunctions we are interested in here do not have a zero on every cycle. Hence, unlike the previous case for large eigenvalues where the corresponding eigenfunctions have at least one zero on every cycle, we must carefully pick our cut points to avoid cutting cycles that do not contain any zeros of the eigenfunction. To do this, we look at the zeros of our eigenfunction ψ one at a time. If cutting the edge that contains the zero will disconnect the graph, we do nothing and remove this zero from consideration (see figure 3b). If cutting the edge at the zero will not disconnect the graph, then we cut that edge at a nearby point c_j at which ψ is non-zero, calling the new vertices and as before (see figure 3c). Note that the manner in which we order and analyse the zeros does not matter; while the cut positions and the resulting graph may vary, we will make the same number of cuts.

Let us consider the number of cuts η more explicitly. Denote by the zero set of ψ and remove from Γ to get the (disconnected) graph . Let ν be the number of connected components {Γ_j} after the cutting (the components Γ_j are the nodal domains of Γ with respect to ψ). Denote

where β_κ is the Betti number of the graph κ. It is easy to see that and furthermore For further details, see [6], Lemma 5.2.1. Now, we continue with an alternative statement of theorem 3.3.

Theorem 6.2

Let ψ be the eigenfunction of H⁰ that corresponds to a simple eigenvalue λ_n(H⁰). We assume that ψ is non-zero on internal vertices of the graph. We denote by ϕ the number of internal zeros and ν the number of nodal domains of ψ on Γ. Let be the cut points created by following the procedure above, where η=1+ϕ−ν.

Let H_γ,γ=(γ₁,…,γ_η), be the operator obtained from H⁰ by imposing the additional conditions:

at the cut points.

Define

and let . Consider the eigenvalues of H_γ as functions λ_n(H_γ) of γ. Then

1. where ϕ is the number of zeros of ψ on Γ,

2. is a non-degenerate critical point of the function λ_ϕ+1(H_γ), and

3. the Morse index of the critical point is equal to n−ν.

We will map out the proof of the theorem in §6b, after explaining its significance to the question of equipartitions.

Theorem 6.3

Suppose the nth eigenvalue of Γ is simple and its eigenfunction ψ is non-zero on vertices. Denote by ϕ the number of internal zeros of ψ and by ν the number of its nodal domains. Then the ϕ-equipartitions in the vicinity of the nodal partition of ψ are parametrized by the variables γ=(γ₁,…,γ_η).

The nodal partition of ψ corresponds to the point and is a non-degenerate critical point of the functional Λ (equation (6.1)) on the set of equipartitions. The Morse index of the critical point is equal to n−ν.

The mapping from (γ₁,…,γ_η) to the equipartitions is constructed as follows (see [9] for more detail): the partition in question is generated by the zeros of the (ϕ+1)^th eigenfunction of the operator H_γ placed upon the original graph Γ. Indeed, the groundstates of the nodal domains can be obtained by cutting the eigenfunction at zeros and gluing the cut points together (conditions (6.5) ensure the gluing is possible). To verify that all equipartitions are obtainable in this way we simply reverse the process and construct an eigenfunction of H_γ from the groundstates of the nodal domains. The gluing is now done at zeros and it can be done recursively (since all cycles with zeros on them have been cut).

Once the parameterization of the equipartitions is accomplished, the Morse index result follows immediately from theorem 6.2.

(b) Proof of theorem 6.2

The proof of theorem 6.2 is identical to the proof of theorem 3.3 once we collect some preliminary results. The following lemma can be found in [6] (Theorem 3.1.8 with a slight modification).

Lemma 6.4

Let Γ_α′ be the graph obtained from the graph Γ_α by changing the coefficient of the δ-type condition at a vertex v from α to α′ (conditions at all other vertices are fixed). If (where corresponds to the Dirichlet condition at vertex v), then

If the eigenvalue λ_n(Γ_α′) is simple and its eigenfunction f is such that either f(v) or is non-zero, then the above inequalities can be made strict. If, in addition, the inequalities become

The following theorem is a generalization of [6], Corollary 3.1.9.

Theorem 6.5

Let Γ be a graph with δ-type conditions at every internal vertex and extended δ-type conditions on all leaves. Suppose an eigenvalue λ of Γ has an eigenfunction f which is non-zero on internal vertices of Γ. Further, assume that no zeros of f lie on the cycles of Γ. Then the eigenvalue λ is simple and f is eigenfunction number ϕ+1, where ϕ is the number of internal zeros of f.

Remark 6.6

The condition that no zeros lie on the cycles of the graph Γ is equivalent to η=0 (see equation (6.3)) or to the number of nodal domains of f being equal to ϕ+1.

Proof.

We use induction on the number of internal zeros of f to show that the eigenvalue is simple. If f has no internal zeros, then we know f corresponds to the groundstate eigenvalue, which is simple.

Now suppose f has ϕ>0 internal zeros. By way of contradiction, assume that λ is not simple. Choose an arbitrary zero ζ of f and another eigenfunction g. Cut Γ at ζ; making this cut will disconnect the graph into two subgraphs since ζ cannot lie on a cycle of Γ. On at least one of these subgraphs, g is non-zero and not a multiple of f (otherwise, it cannot be a different eigenfunction). We will now analyse the eigenfunctions on this subgraph Γ′.

On the graph Γ′, f and g satisfy the same δ-type conditions at all vertices except possibly the new leaf ζ. We denote by Γ′_τ as the graph Γ′ with the conditions Φ′(ζ)=τΦ(ζ). We know that (λ,f) is an eigenpair on and similarly, there exists α such that (λ,g) is an eigenpair on Γ′_α. However, since contains fewer internal zeros of f than Γ does, by the inductive hypothesis λ is simple on so .

Observe that f′(ζ) is non-zero; if it was zero, the function f would be identically zero on the whole edge containing ζ and, therefore, at the end-vertices of the edge. Thus, the inequalities in (6.6) with become strict and Γ′_α and cannot have the same eigenvalue λ.

Now, we show that f is eigenfunction number n=ϕ+1. By remark 6.6, there are ν=ϕ+1 nodal domains. As λ is simple, we know from [6], Theorem 5.2.8 that

where β is the Betti number of Γ. However, since ν=ϕ+1, both inequalities hold only if ν=n=ϕ+1; so, f is indeed the eigenfunction number ϕ+1. □

Below, we only include the parts of the proof that differ from theorem 3.3.

Proof of theorem 6.2.

In the proof of theorem 3.3 (§4e), the fact that H_γ is an operator on a tree was used to show that its eigenvalue is simple and to find the sequence number of λ in the spectrum. Theorem 6.5 allows us to do the same in the graph with fewer cuts.

Indeed, on the cut graph is non-zero on all cycles and internal vertices and therefore by theorem 6.5, the eigenvalue is simple and has number ϕ+1 in the spectrum of . As the eigenvalue is simple, we can still apply lemma 4.4. The rest of the proof goes through, with the amendment that the index of the critical point of F₃ is n+η, since we now have η cuts instead of β cuts. Using equation (6.4), we finally get that the Morse index of the critical point is

 □