Eigenvalue and nodal properties on quantum graph trees

Waves in Random and Complex Media Vol. 16, No. 3, August 2006, 167–178

Eigenvalue and nodal properties on quantum graph trees∗ PHILIPP SCHAPOTSCHNIKOW Cardiff School of Computer Science, Cardiff CF24 3AA, UK Technische Universit¨at at M¨unchen, Centre for Mathematical Sciences, 85748 Garching, Germany (Received 11 January 2006; in final form 18 March 2006) We consider the Schr¨odinger operator −y

+ q(x)y on a finite tree with linear separate (e.g. Dirichlet or Neumann) conditions at the boundary. The potential q is assumed to be real-valued, positive and finite. At the interior vertices the solution has to be continuous and smooth in a certain way (Kirchhoff’s or δ-type condition). Using oscillation theory we derive first for radial and then for general trees the interlacing property: the eigenvalues of the tree problem lie between the eigenvalues of the problems of subtrees joining at an arbitrary vertex. From this we can derive some eigenvalue asymptotics. Moreover we can generalize Sturm’s Theorem from the interval to a tree: the nth eigenfunction has n nodal domains.

1. Introduction For several reasons quantum graphs have become objects of detailed mathematical investigation in recent years. A special section in Waves in Random Media (2004, Vol. 14) is dedicated to this topic. In his reviews [1, 2] Peter Kuchment shows the role of quantum graphs in models coming from physics and chemistry. Finite graphs are one-dimensional objects although they lie in the plane. They have some features from an interval and some from a closed domain in R2 . One of the important examples is the Schr¨odinger operator. Its nth eigenfunction on a closed interval has by Sturm’s Theorem n nodal domains [3]; and on a closed domain in the plane the number of nodal domains of the nth eigenfunction is less than or equal n [4]. In [5] the nodal statistics on a general finite graph are studied. There it is shown that for a graph with loops (valency at each vertex ≥3) the number of nodal domains becomes strictly smaller than the eigenvalue index. This result can be seen as the counterpart of the generalized Sturm Theorem in our work. In [6] it has been shown that the spectrum of the Laplacian on a star graph interlaces with the eigenvalues of the Laplacian on the edges; in [7, 8] the same result has been proved for the star graph with three edges and a potential. In this work, this result will be extended to ∗ In memoriam Z. E. Schapotschnikow. Corresponding author. E-mail: [email protected]

Waves in Random and Complex Media c 2006 Taylor & Francis ISSN: 1745-5030 (print), 1745-5049 (online)  http://www.tandf.co.uk/journals DOI: 10.1080/17455030600702535

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the finite tree. En passant, the well-known eigenvalue asymptotics [9], and references therein, will be shown using the interlacing property. 1.1 Definitions A finite directed graph consists in a set of N vertices V = {v1 , . . . , v N } and a set of edges E ⊂ V × V . The edge (vi , v j ) starts in vi and ends in v j . Two vertices are neighboured if they are connected with an edge. The number s of vertices neighbouring with vk is called the valence of vk . A boundary vertex is a vertex contained in exactly one edge, which is called boundary edge. An interior vertex belongs to two or more edges. A path is a finite sequence of vertices such that no vertex appears twice and any two consecutive vertices are connected with an edge. A path is called directed if any two consecutive vertices vi , v j are connected with (vi , v j ). This work treats an important type of graph: trees. The graph is a tree if between any two vertices exists a unique path. A tree has exactly N − 1 edges. In the sequel edges will be labelled from time to time, and we will switch between (vk , vl )- and ei -notation. The subtree Tk,l of the tree T contains the vertex vk and each path from vk to the boundary of T leading through vl . If vl1 , . . . , vls are all vertices neighbouring to vk , then the intersection of any two subtrees Tk,li and Tk,l j is vk and the union of the subtrees Tk,l1 , . . . , Tk,ls is T . On an edge the distance is measured from the vertex where the edge starts. Denote the length of the edge ei by L i . The free one-dimensional variable on the edge ei is denoted by xi . A function F defined on a graph is an ensemble of functions ( f i ) defined on the edges (ei ) so that F(xi ) = f i (x). The Schr¨odinger operator on the edge ei is −

d2 + Q(xi ) d xi2

where Q is L 1 . In this work we consider only bounded potentials Q. A function F satisfies interior conditions if 1. it is continuous on the graph and if 2. for every interior vertex vk
f i
(L i ) − ei ends in vk




f i
(0) = βk F(vk )

(1)

ei starts in vk

where βk is a real number. Equation (1) is called a δ-type condition. The special case βk = 0 is called Kirchhoff’s condition. A function F satisfies boundary conditions if for every boundary vertex v j cos α j F(v j ) = sin α j F
(v j ),

0 ≤ α j < π.

(2)

Boundary and interior conditions are seen here as a part of the tree. In particular, to each boundary vertex v j a coefficient α j is attached; to each interior vertex vk corresponds the fixed coefficient βk . One convention must be agreed. The subtree Tk,l inherits the conditions from T and the coefficient αk at vk is 0. Definition 1.1. A real function Y is an eigenfunction of the Schr¨odinger operator on the tree T to the eigenvalue λ if

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1. Y is continuous on the tree; 2. the equation −yi

+ qi yi = λyi

(3)

is satisfied in the interior of each edge ei ; 3. boundary and interior conditions are satisfied everywhere; 4. Y does not vanish along the whole tree. The vector space of eigenfunctions to λ is called the eigenspace. The set of eigenvalues is called the spectrum. The spectrum of the Schr¨odinger operator on a tree is real and discrete. Note that eigenvalues and eigenfunctions do not depend on directions of edges since the interior conditions do not. Definition 1.2. An eigenfunction is called degenerate if it vanishes at an interior vertex with valency ≥3. The corresponding eigenvalue is also called degenerate. LEMMA 1.3. 1. Degenerate eigenfunctions vanish along entire edges unless the eigenvalue is multiple. 2. Non-degenerate eigenvalues are simple. Non-degenerate eigenfunctions belong to nondegenerate eigenvalues. 3. If the eigenvalue is m-multiple, then for every given boundary edge there are at least m − 1 linearly independent eigenfunctions vanishing on it.

Proof 1. Assume that the eigenfunction Y to the eigenvalue λ vanishes at an interior vertex v with valency s ≥ 3. Split the tree at v in s subtrees and consider Y1 , . . . , Ys the parts of Y on those subtrees. Without loss of generality let the vertices containing v end there. Then Kirchhoff’s condition becomes s


Yi
(v) = 0.

(4)

i=1

Assume Y does not vanish along an edge. Hence, Yi
= 0 for all i. Then the linear equation (4) has a solution (U1 , . . . , Us ) linearly independent from (Y1
(v), . . . , Ys
(v)). Define solui tions on subtrees Z i = YU
(v) Yi and Z as their composition. Then Z is an eigenfunction to λ i linearly independent from Y . 2. This statement will be shown implicitly in the proof of Lemma 3.4 and will be not required for this. 3. Consider a boundary edge e with the boundary vertex v. All eigenfunctions satisfy an α-boundary condition. Consider a different linear condition with the angle β. Then the m-dimensional eigenspace has an m − 1 dimensional subspace satisfying the β-boundary condition. Together with α-condition it implies that all functions in that subspace vanish along e.


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1.2 Prufer ¨ transform Consider on the interval [0, L] the differential equation −y

+ qy = λy

(5)

with a bounded integrable potential q and homogeneous boundary conditions cos α1 y(0) = sin α1 y
(0),

cos α2 y(L) = sin α2 y
(L).

(6)

y
. y

(7)

y
= r cos θ.

(8)

The Pr¨ufer transform is r 2 = y 2 + (y
)2

cot θ =

The inverse transform is given by y(x) = r sin θ,

Differentiating (7) we obtain the equations for the Pr¨ufer angle θ and for the amplitude r : θ
= (λ − q) sin2 θ + cos2 θ

(9)

1 (1 + q) r sin 2θ. (10) 2 From the second equation we see that the amplitude r never changes its sign. For this reason the first of these two equations is relevant when investigating qualitative properties of the solution such as zeros, extrema and eigenvalues. The initial condition for θ is θ0 := θ(0) = α1 modulo π . A function y is an eigenfunction if θ (L) = α2 + nπ and the amplitude does not vanish. In the sequel we will consider θ as a function of the free variable x, spectral parameter λ and initial value θ0 . r
=

THEOREM 1.4. (Oscillation theorem).

Let θ := θ (x; λ, θ0 ) be defined as above. Then

1. θ is continuous in all arguments. 2. θ is strictly increasing in the third component; and for any positive integer n θ (x, λ, θ0 + nπ ) = nπ + θ (x, λ, θ0 )

(11)

3. θ > 0 for all x ∈ (0, L]. 4. θ increases strictly with λ and is unbounded. 5. limλ→−∞ θ = 0. This theorem follows straight from the properties of (9) and its dependence on λ. A proof can be found in [3], pages 258–262. 2. Radial trees Consider a tree with N ≥ 2 edges and exactly one interior vertex where they all meet. We call it a radial tree or a star. Without loss of generality we can assume that boundary vertices correspond to 0. The equations at the interior vertex are yi (L i ; λ) = y j (L j ; λ), N
i=1

yi
(L i ; λ) = β y1 (L 1 ).

1 ≤ i, j ≤ N

(12) (13)

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171

Now let yi (L i ; λ) = 0 for one and hence all i. Then we can rewrite Kirchhoff’s condition as N
yi
(L i ; λ) = β. y (L i ; λ) i=1 i

(14)

The Pr¨ufer transform yi (x; λ) = r (x; λ) sin θ (x; λ), yi
(x; λ) = r (x; λ) cos θ (x; λ) yields N


cot θi (L i ; λ) = β.

(15)

i=1

Equation (15) is the interior condition for the Pr¨ufer angle. It motivates a different consideration. A homogeneous boundary condition always yields an initial value problem for the Pr¨ufer phase θ . Thus, functions θi (x, λ) are well defined as functions of λ. As result we can define dθ (λ) :=

N


cot θi (L i ; λ) − β.

(16)

i=1

LEMMA 2.1. The function dθ (λ) decreases strictly between any two of its consecutive poles. Proof By the Oscillation Theorem, for all i, θ (L i ; λ) is an increasing function in λ. The cotangent decreases strictly between two consecutive poles. Summation over i conserves this property.
Let ν ∈ R be a pole of dθ . Then there exists at least one j ∈ 1, . . . , N such that sin (θ j (L j ; ν)) = 0. Hence ν is an eigenvalue of the following boundary value problem on the edge j: −z

+ q j (x)z = λz

with

cos α j z(0) = sin α j z
(0), z(L j ) = 0.

(17)

Let (νk ) be the sequence of poles of dθ arranged in increasing order. Eigenvalues µ of the Schr¨odinger operator on the tree different from every νk are its zeros. Conversely, let dθ (µ) = 0. Then putting yi (L i ; µ) = 1, ∀i ∈ 1, . . . , N , and solving the N resulting BVPs we obtain a non-trivial solution of the homogeneous problem. Hence, µ is an eigenvalue of the tree problem. THEOREM 2.2. Let T be a star with N ≥ 2 edges; −Y

+ Q · Y = λY the Schr¨odinger equation with interior conditions (12)–(13) and homogeneous boundary conditions. Let (µk ) be its eigenvalues arranged in increasing order, including multiplicity. Let (λk ) be the eigenvalues of the Schr¨odinger operator on the intervals corresponding to the N edges arranged in increasing order, including multiplicity. Boundary conditions are the same as in 17. Then the sequences (λk ) and (µk ) interlace exactly, i.e. µ1 < λ1 and for k > 1  · · · ≤ λk−1 < µk < λk ≤ · · · if λk−1 < λk · · · < λk = µk+1 = · · · = µk+m = λk+m < · · · if λk−1 = · · · = λk+m , m = 1, 2, . . . Proof We make a case differentiation. In the first case we consider λk−1 < λk . By Lemma 2.1 and succeeding considerations it follows that between λk−1 and λk lies exactly one zero µ ¯ of dθ , which is an at least simple eigenvalue of the tree problem.

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Now we investigate the multiplicity of µ. ¯ The continuity condition for a corresponding eigenfunction yields r1 (L 1 ; µ) ¯ sin θ1 (L 1 ; µ) ¯ = · · · = r N (L N ; µ) ¯ sin θ N (L N ; µ) ¯ = 0.

(18)

Recall that r (x) = 0 unless the solution is trivial and sin θi = 0 due to the assumption. Since all the sines in (18) are determined by the boundary conditions and are nonzero, all the amplitudes ri (L i ) are determined uniquely up to a constant multiplier (one for all ri for the fixed µ). ¯ Moreover, as solutions of linear initial value problems all the amplitudes are unique up to the common multiplier. Thus, the entire eigenfunction Y (x) is unique up to a constant multiplier and the (geometric) multiplicity of µ ¯ is one. We move on to the second case, when different interval eigenvalues coincide (λk = · · · = λk+m ). Without loss of generality let sin θ1 (L 1 ; λk ) = sin θ2 (L 2 ; λk ) = . . . = sin θm+1 (L m+1 ; λk ) = 0 and sin θi (L i ; λk ) = 0 for all m + 1 < i ≤ N . By the continuity condition we obtain ri (L i ; λk−1 ) = r1 (L 1 ; λk−1 )

sin θ1 (L 1 ; λk−1 ) =0 sin θi (L i ; λk−1 )

∀ m + 1 < i ≤ N.

(19)

Thus, yi (x; λk ) ≡ 0 on all edges where λk is not an eigenvalue of the interval problem. Now the Kirchhoff’s condition reduces to N


yi
(L i ; λk−1 ) =

m+1


i=1

yi
(L i ; λk−1 ) = 0.

(20)

i=1

This linear equation in m +1 variables y1
(L 1 ; λk−1 ), . . . , ym
(L m+1 ; λk−1 ) determines uniquely an m-dimensional linear space of solutions of the Schr¨odinger equation on the whole tree. Hence, λk = · · · = λk+m corresponds to an eigenvalue µ of the tree problem with multiplicity m, and we can write it as · · · < λk = µk+1 = λk+1 = · · · = µk+m = λk+m < · · · It remains to prove that µ0 < λ0 . By the Oscillation Theorem dθ (λ) → ∞ as λ → −∞, and hence there exists a unique simple tree eigenvalue below λ0 .


3. General trees We move on to the general case. We will need several ingredients in order to generalize the results from the previous section. Definition 3.1. We call a tree directed towards a vertex vk if there exists a (unique) directed path from every vertex to vk . Remark

The following properties hold true:

1. One can always redirect a tree towards each of its vertices. 2. A tree can be directed towards at most one vertex. 3. If a tree is directed towards vk , then from every vertex different from vk exactly one edge starts. All edges containing vk end there. Now we need to generalize the Pr¨ufer angle to interior edges. We do it in a recursive way.

Eigenvalues on quantum trees

173

Definition 3.2. Let the tree T be directed towards a vertex. We consider the edge (vk , vl ) with the label j. This is then the only edge starting in vk . Let s be the valence of the vertex vk . Denote by e1 , e2 , . . . , es−1 all vertices ending in vk . The Pr¨ufer angle on the edge e j satisfies the differential equation dθ = (λ − q j ) sin2 θ + cos2 θ. dx j

(21)

The initial condition is given as following:

r If vk is a boundary vertex then in the usual way θ j (0) = αk ,

0 ≤ θ j (0) < π.

(22)

r If vk is an interior vertex, then θ j (0; λ) is given by 1. the interior condition at vk cot(θ j (0) : λ) :=

s−1


cot(θi (L i ; λ)) − βk ;

(23)

i=1

2. θ j (0; λ) is continuous (and hence monotonically increasing) in the second component. Continuity in λmeans the following. Let λ increase from −∞. As we have seen in the s−1 previous section, i=1 cot(θ ji (L i ; λ))−βk is a strictly decreasing function of λ on the intervals where this function has no poles. Hence θ j (0; λ) comes from 0 and is strictly increasing until θ j (0; λ) reaches π from below. There cot(θ j (0)) jumps from −∞ to +∞. Denote this pole by λ1 . Define θ j (0; λ1 ) = π . For larger multiples of π proceed in the same way, putting θ j (0; λn ) = nπ . We obtain as a consequence of the Oscillation Theorem the following lemma. LEMMA 3.3. Let T and (vk , vl ) be as in Definition 3.2. The function θ j (x j ; λ) fulfils some properties of the Pr¨ufer angle in the Oscillation Theorem. It is (as a function of λ) 1. continuous and monotonically increasing; 2. unbounded above; 3. it tends to 0 as λ → −∞. The next lemma shows us the correspondence between zeros of the Pr¨ufer angle and eigenvalues of a quantum tree problem. LEMMA 3.4. Let T and (vk , vl ) be as above. Consider the subtree Tk,l of T which contains vl and all paths from the boundary to vl which lead through vk , i.e. all directed paths from boundary vertices to vl . Let θ(L j , µ) = nπ. Then µ is an eigenvalue of the Schr¨odinger operator on Tk,l . Proof We need to define a Pr¨ufer amplitude r which does not vanish over all edges and satisfies, together with the angle, the continuity condition. Put r j (L j ) = 1. Then r j (x) is well defined as the solution of a IVP, and we obtain r j (0) = 0. The continuity condition yields r j (0) sin θ j (0) = ri (L i ) sin θi (L i ) for all vertices i joining in vk . We have to make a case by case differentiation

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1. Both sin θ j (0) and sin θi (li ) are = 0. Then ri (li ) =

r j (0) sin θ j (0) sin θi (L i )

defines the Pr¨ufer phase on the edge j. 2. Both sin θ j (0) and sin θi (L i ) are 0. Then we obtain another degree of freedom. Put ri (L i ) = 1 and so ri is defined on the edge i. 3. sin θ j (0) = 0 and sin θi (L i ) = 0. Then ri (L i ) = 0, and the solution vanishes along all paths from vk to the boundary which contain the edge i. 4. sin θ j (0) = 0 and sin θi (L i ) = 0. This case cannot occur due to the definition of θ j (0). Continuing along the edges we can define the amplitude on the whole tree, and this amplitude does not vanish everywhere.
From the proof one can see immediately that non-degenerate eigenvalues are always simple and how zeros at interior vertices can cause multiplicity. The converse of this lemma is only valid, if some eigenfunctions do not vanish along the edge (vk , vl ). In particular, θ (l j , µ) = 0 if µ is a non-degenerate eigenvalue on Tk,l . Now we have all components for the main result of this section. THEOREM 3.5. Let T be a quantum tree, vk an interior vertex with valence s and vl1 , . . . , vls its neighbouring edges. Then the eigenvalues (µn ) of the T -problem interlace with the eigenvalues (λn ) of the Sch¨odinger operator on the s subtrees Tk,l1 , . . . , Tk,ls , i.e µ1 < λ1 and for n > 1 . . . ≤ λn−1 ≤ µn ≤ λn ≤ µn . . . Moreover, equality occurs only for degenerate eigenvalues. Proof Redirect the tree towards vk and relabel the edges so that (vli , vk ) has the label i, i = 1, . . . , s. If an eigenfunction is not degenerate, the corresponding eigenvalue solves the equation dk (λ) :=

s


cot θ (L i ; λ) − βk = 0.

(24)

i=1

The angles θi are defined by 3.2. The function dk has by Lemma 3.3 the same properties as dθ in Lemma 2.1, and the same arguments as in the proof of Theorem 2.2 can be applied. A pole of dk is by Lemma 3.4 an eigenvalue of the Schr¨odinger operator on a subtree Tk,i . Between any two consecutive poles lies one simple zero; between any two consecutive zeros lies a simple pole. By the same argument as in the proof of Theorem 2.2, the least eigenvalue of the tree problem is smaller than the least eigenvalue of the subtree problems. The case when a pole of dk is a subtree eigenvalue of m different subtree problems leads to a degenerate eigenvalue of the tree problem with multiplicity m − 1 in the same way as in the proof of Theorem 2.2. It remains to treat degenerate eigenvalues of subtree problems. Let ω be an eigenvalue of the Tk,i problem with multiplicity m which is not a pole of dk . Then ω is also an eigenvalue of the tree problem with multiplicity m. The corresponding eigenfunctions can be taken from the subtree Tk,i and completed with 0 on other subtrees. Let ν be an eigenvalue of the Tk,i problem with multiplicity m ≥ 2 which is a pole of dk . Then its eigenspace has an (m − 1)-dimensional subspace of eigenfunctions vanishing along (vli , vk ). Thus, ν is an eigenvalue with multiplicity m − 1 of the tree problem.

Eigenvalues on quantum trees

175

The ω-case can occur in combination with another kind of degeneracy. Then the multiplicities can be added.
COROLLARY 3.6. The least eigenvalue of the Schr¨odinger operator on a quantum tree is always non-degenerate.

4. Eigenvalue asymptotics THEOREM 4.1. The number (t) of the eigenvalues of the Schr¨odinger operator below t on a quantum tree behaves as  N √
t Li (x) = + O(1) (25) π i=1 where · denotes the integer part of a real number. Proof We prove the statement by induction over the number of interior vertices. For an interval the statement is well known, see e.g. [3], page 265. Let the statement be true for all trees with up to V interior vertices. Consider a tree T with V +1 vertices. Let vk be an interior vertex with valence s and neighbouring vertices vl1 , . . . , vls . Denote by i the eigenvalue counting function on the subtree Tk,li . Then by Theorem 3.5 and the induction assumption   √    s s V +1  √


t Lj t Lj 
(t) = O(1) + i (t) = O(1) + + O(1) = + O(1).  π π j, i=1 i=1 j=1 e j ∈Tk,l

i

(26)


5. Nodal counting The goal of this section is to count zeros, or equivalently domains between zeros of nondegenerate eigenfunctions. For the star the main result is a straight consequence of Sturm’s Theorem for intervals. THEOREM 5.1. Consider a star graph T with N ≥ 2 edges. Then the kth eigenfunction of the Schr¨odinger operator has k − 1 zeros in the interior or is degenerate. The number of nodal domains is k. The eigenvalues are counted as µ1 , µ2 , . . . Proof Assume the kth eigenfunction Y (k) is not degenerate, i.e. does not vanish at the interior vertex. Then λk−1 < µk < λk as in Theorem 2.2. Put λ0 = −∞. Hence there are exactly k interval eigenvalues below µk . Let ki be the number of eigenvalues below µk of the Schr¨odinger operator on the ith edge. Then N
i=1

ki = k − 1.

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Since µk lies between the ki th and (ki + 1)th eigenvalue of the ith interval problem, Sturm’s Theorem implies that Y (k) has ki zeros in the interior of the ith edge. Summing up over i we obtain k − 1 zeros and k nodal domains.
For the general tree we need an interlacing property for different types of boundary conditions. LEMMA 5.2. Consider a tree T with a boundary vertex vk and two sets of homogeneous boundary conditions differing at vk only: (P1 ) (P2 )

αk = 0, i.e. Y (vk ) = 0; αk > 0, i.e.

Y
(vk ) = cot(αk ). Y (vk )

Then the sequences (λn ) and (νn ) of eigenvalues of the Schr¨odinger operator with boundary conditions P1 and P2 respectively interlace exactly ν 1 < λ 1 ≤ ν2 ≤ λ2 ≤ . . . with equality only for degenerate eigenvalues. Proof As first we redirect the tree towards vk in order to perform the Pr¨ufer transform later. Let k be the label of the edge ending in vk . As in Theorem 3.5 we first consider the non-degenerate eigenvalues and then fit the degenerate ones into the inequality chain. The non-degenerate eigenvalues satisfy θk (L k ; λn ) = mπ

(27)

θk (L k ; νn ) = (m − 1)π + αk

(28)

where m is positive integer. Recall that θk increases strictly in the second component. Since the least eigenvalues are not degenerate, they correspond to m = 0, and hence ν1 < λ1 We can also state that the solutions of (27) interlace exactly with the solutions of (28). This is the initial inequality chain. Eigenvalues µ which do not satisfy (27) or (28) have only eigenfunctions vanishing along ek . Such eigenvalues obviously have the same multiplicity for P1 and P2 . They fit in the inequality chain with equality signs without destroying the interlacing order, e.g. . . . ≤ λn < νn+1 = µ = λn+1 < νn+2 ≤ . . .

(29)

If λn = · · · = λn+m is a multiple eigenvalue to P1 conditions satisfying 27, then its (m + 1)dimensional eigenspace has an m dimensional subspace of eigenfunctions satisfying P1 and P2 simultaneously (i.e. vanishing along ek ), and hence λn is an eigenvalue with multiplicity m with respect to P2 conditions. So they can also be fitted in the inequality chain preserving the interlacing property: . . . < νn < λn = νn+1 = · · · = νn+m = λn+m < . . . The case of a multiple P2 eigenvalue works in the same way.

(30)


We can finally prove the generalization of Sturm’s Theorem. THEOREM 5.3. The eigenfunction Yn corresponding to the nth eigenvalue µn of the Schr¨odinger operator on a tree has n − 1 zeros in the interior and correspondingly n nodal domains or is degenerate.

Eigenvalues on quantum trees

177

Proof Here we can perform an induction over the number V of interior vertices. In the interval case (V = 0) the statement holds true by Sturm’s Theorem. Assume the statement of the theorem holds true for for all trees with ≤ V interior vertices. Consider a tree T with V +1 interior vertices. Assume that the eigenvalue µn is not degenerate. Redirect the tree towards an interior vertex vk with valence s and neighbouring vertices vl1 , . . . , vls . Relabel the edges so that (vli , vk ) obtains the label i. Consider the subtree Tk,li and let n i be the number of Tk,li eigenvalues below µn . We show that Yn has exactly n i zeros on that subtree. Since Yn is not degenerate, Yn (vk ) = 0. Define 0 < αk,i := arccot

Y
n (vk ) Yn (vk )

with the derivative along (vli , vk ). Yn restricted to Tk,li is obviously an eigenfunction to the Schr¨odinger operator with the original conditions at the boundary of T and cos(αk,i )Z (vk ) = sin(αk,i )Z
(vk ) at vk . The corresponding eigenvalue is µn . By Lemma 5.2 and Theorem 3.5 this is the (n i + 1) eigenvalue of the subtree problem, and it is not degenerate because Yn does not vanish at any interior vertex. By the induction assumption, Yn restricted to Tk,li has n i zeros in the interior. Summing up over i and applying Theorem 3.5 we obtain n − 1 zeros and, correspondingly, n nodal domains.


6. Outline Interior conditions for Pr¨ufer angles provide a numerically stable ansatz for computing eigenvalues and eigenfunctions of the tree problem. Indeed, zero finding for monotonically decreasing functions as dk (λ) is a well-posed and generally well-conditioned task. All results in this work hold true for the Sturm–Liouville operator L Y = −(PY
)
+ QY with a positive diagonal parameter P(x) bounded above and below away from 0. Using generalized Pr¨ufer transforms one could extend the results to trees with infinite edges or unbounded potentials. The methods of this article do not seem to work for graphs with cycles.

Acknowledgements I would like to thank Prof. B.M. Brown and Dennis Wassel from Cardiff University and Prof. Rupert Lasser from Technische Universit¨at Munich for their support and helpful advice. The work was partly supported by the Leonhard–Lorenz–Stiftung, grant 673/05. References [1] Kuchment, P., 2004, Quantum graphs I. Some basic structures. Waves in Random Media, 14, 107–128, in Special Section on quantum graphs.

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[2] Kuchment, P., 2005, Quantum graphs II. Some spectral properties of quantum graphs and combinatorial graphs. Journal of Physics A, 38, 4887–4900. [3] Birkhoff, G. and Rota, G.-C., 1962, Ordinary Differential Equations (Boston: Ginn and Company). [4] Courant, R. and Hilbert, D., 1953, Methods of Mathematical Physics, Vol. 1 (New York: Interscience), pp. 451– 65. [5] Gnutzmann, S., Smilansky, U., and Weber, J., 2004, Nodal counting on quantum graphs. Waves in Random Media, 14, 61–73, in Special Section on quantum graphs. [6] Berkolaiko, J., Keating, J. P. and Winn, B., 2003, Physical Review Letters, 91, 134103. [7] Pivovarchik, V., 2000, Inverse problem for the Sturm–Liouville equation on a simple graph. SIAM Journal on Mathematical Analysis, 32, 801–819. [8] Schapotschnikow, P., 2004, Inverse Sturm–Liouville problems on trees. A variational approach. MPhil thesis, Cardiff University, UK. [9] Kottos, T. and Smilansky, U., 1999, Annals of Physics, 274, 76–124.