BRUCE A. WATSON University of the Witwatersrand School of Mathematics Private bag 3 P.O. Wits 2050 South Africa [email protected]

Abstract: We consider the Sturm-Liouville operator on a graph and give bounds for the norms of the boundary values of solutions to the non-homogeneous boundary value problem in terms of the norm of the non-homogeneity. In addition the eigenparameter dependence of these bounds is studied. Key–Words: Differential Operators on Graphs, Boundary Estimates

1 Introduction We consider the Sturm-Liouville equation ly := −

d2 y + q(x)y = λy, dx2

(1)

where q is real-valued and essentially bounded (with respect to Lebesgue measure), on a weighted graph G with formally self-adjoint boundary conditions at the nodes. For characterisations of self-adjoint boundary value problems on graphs and associated boundary conditions we refer the reader to [5] and [11]. In [10], it was shown that the geometry of a noncommensurate simple graph is uniquely dependent on the spectrum of the Laplacian on the graph. I.e. for zero potential they reconstructed the boundary conditions (of a specific type) from a single spectrum. In [6] spectral asymptotics were given for l on graphs where all edges are of equal length while in [7] and [8] eigenvalue asymptotics were given for l on general compact graphs via matrix Pr¨ufer angle techniques and Dirichlet-Neumann bracketing respectively. Variational aspects of boundary value problems on graphs were studied in [2], [8] and [20], and on trees in [19]. Sturmian oscillation theory was extended to Sturm-Liouville operators on graphs by Pokornyi and Pryadiev, and Pokornyi, Pryadiev and Al-Obeid, in [15] and [16]. Sturm-Liouville problems on finite graphs arise naturally in quantum mechanics and circuit theory, [3, 9]. In [13] and the bibliography thereof one can find an extensive collection of physical systems that give rise to Sturm-Liouville problems on graphs.

Here we consider solutions of non-homogeneous Sturm-Liouville problems on graphs and their a priori estimates. Particular attention is given to the relationship between the norm of the solutions to the non-homogeneous boundary value problem on the boundary of the graph and the norm of the nonhomogeneous term on the graph, see Theorem 3.2. In addition the eigenparameter dependence of this relationship is explored. To complete the paper an example is provided in Section 4, illustrating Theorem 3.2. The results obtained in this paper rely on an ability to make the transition between local results (on each edge) and global results (on the whole graph). Thus the method employed involves two main steps: establishing a local result on each edge; and the transition from the local results to a global result on the whole graph. It should be noted that for Sturmian systems it is only possible to find boundary estimates, of the form given in this paper, for two specific types of systems. Firstly for non-weighted systems with general, essentially bounded, Hermitian potential (not necessarily diagonal) and secondly for diagonal systems, which are equivalent to considering Sturm-Liouville equations on graphs as is done in this paper, see [7] for the equivalence. In [17, 18], Schauder considers interior estimates and estimates near the boundary for solutions of second order elliptic boundary value problems. His estimates near the boundary are for solutions of the Dirichlet problem. Estimates near the boundary for other than Dirichlet boundary conditions have been obtained by Miranda, [14], for second order elliptic

boundary value problems and by Agmon, Douglis, Nirenberg and Browder, [1, 4], for arbitrary order elliptic operators. In the above references it should be noted that the estimates are given in a region near the boundary whereas our results provide estimates on the boundary.

2 Preliminaries Let G denote a directed graph with a finite number of nodes and edges, with each edge parametrized by path-length and having finite length. Each edge, ei , of length say li can thus be considered as the interval [0, li ]. Having made this identification, it is possible to consider the differential equation (1) on the graph G to be the collection of differential equations d2 yi (2) − 2 + qi (x)yi = λyi , x ∈ [0, li ], dx for i = 1, ..., K, where qi and yi denote the restriction of q and y to ei , respectively. It is now possible, at each node, ν, to specify boundary conditions in terms of the values of y and y ′ at ν on each of the incident edges. In particular if the edges which originate at node ν are ei , i ∈ Λs (ν) and the edges which terminate at node ν are ei , i ∈ Λe (ν) then the boundary conditions at ν are of the form j∈Λs (ν)

h

X

h

i

i

′

γij yj (lj ) + δij y j (lj ) = 0,

j∈Λe (ν)

(3)

Remark It should be noted that by setting αij = 0 = βij for i = 1, ..., N (ν) with j 6∈ Λs (ν) and γij = 0 = δij for i = 1, ..., N (ν) with j 6∈ Λe (ν), after relabelling the conditions (3) may be written as

K h X

i

γij yj (lj ) + δij y ′ j (lj ) = 0,

j=1

(f, g) =

i=1 0

li

f |ei g¯|ei dt,

Theorem 3.1 Let λ = −k2 , k > 0, then for y a solution of the boundary value problem (2)-(3), 1 1 ||y||L2 (G) = √ ||y|∂G ||L2 (∂G) 1 + O k 2k

(5)

as k → ∞, where ∂G denotes the boundary of G. Proof: Consider the second order Sturm-Liouville problem on the interval [0, li ] given by −yi′′ + qyi = λyi

(6)

with non-homogeneous Dirichlet boundary conditions and

yi (li ) = βi (k).

(7)

Let λi0 denote the least eigenvalue of (6) on (0, li ) with Dirichlet boundary conditions, yi (0) = 0 = yi (li ). Taking λ < Λ := min λi0 we have that (6)-(7) has a unique solution for each αi (k), βi (k) and each i = 1, . . . , K. From [12, Appendix A1], the fundamental solutions of (6) obeying the boundary conditions u1 (0) = 1 = u′2 (0), u′1 (0) = 0 = u2 (0)

u1 (t) = cosh kt + O (4)

for i = 1, ..., N, where N is the total number of linearly independent boundary conditions. Define L2 (G) to be the set of all f : G → C with fi ∈ L2 (0, li ) and inner product K Z X

3 Boundary Estimates

(8) (9)

are given asymptotically for large k > 0, by

i

αij yj (0) + βij y ′ j (0) +

j=1

with domain D(L) = {f | f, f ′ ∈ AC, l(f ) ∈ L2 (G), f obeying (3) }. In this setting, the formal self-adjointness of (2)(3) ensures that the operator L on L2 (G) is a selfadjoint operator, see [21, p. 77-78].

i=1,...,K

for i = 1, ..., N (ν) where N (ν) is the number of linearly independent boundary conditions at node ν.

K h X

Lf = −f ′′ + qf

yi (0) = αi (k)

αij yj (0) + βij y ′ j (0) +

X

then L2 (G) becomes a Hilbert space. The above boundary value problem on G can be reformulated as an operator eigenvalue problem, [5], by setting

u2 (t) =

1 sinh kt + O k

ekt k

!

ekt k2

,

(10)

!

(11)

with corresponding derivatives u′1 (t) = k sinh kt + O(ekt ), u′2 (t) = cosh kt + O

ekt k

!

(12) ,

(13)

uniformly with respect to t. Note that the Wronskian of u1 (t) and u2 (t) is equal to 1 for all t, i.e. u1 (t)u′2 (t) − u2 (t)u′1 (t) = 1,

u1 (li )αi (k) − βi (k) u2 (t) u2 (li ) [−αi (k)(−u1 (t)u2 (li ) + u1 (li )u2 (t))] u2 (li ) βi (k)u2 (t) . (14) u2 (li )

yi2 (t)

= +

w(t) := −u1 (t)u2 (li ) + u1 (li )u2 (t), then w is the solution of (6) with

+

R li

Thus from [12, Appendix A1], for large k > 0, !

,

u22 (li )

1 1+O k

(15)

.

(16)

=

αi (k) sinh k(li − t) + O

e2kli 8k3 u22 (li )

=

e2kli − 1 2k

1+O

1 k

×

βi2 (k) + α2i (k) + 4αi (k)βi (k)kli e−kli

1 e2kli (α2 + βi2 ) 1 + O k 8k3 u22 (li ) i

ekt k

!!

i

ek(li −t) k

α2i (k) + βi2 (k) ≥ |2αi (k)βi (k)|. +

!!#

Using (16) gives .

Z

0

li

yi2 (t) dt = ||yi ||2L2 (0,li )

Squaring this gives

=

yi2 (t) =

1 k

since by Schwartz’ inequality

1 [βi (k)u2 (t) − αi (k)w(t)] u2 (li ) 1 βi (k) sinh kt + O ku2 (li )

1+O

1 2li αi (k)βi (k)ekli 1+O 4 k # 2kli 2 αi (k) 1 e −1 1+O 4 k 2k

+

=

yi (t)

"

2αi (k)βi (k)ekli 1 1+O 4 k # 2k(l −t) 2 i 1 αi (k)e 1+O . 4 k

"

Substituting (11) and (15) in (14) we obtain

=

"

1 βi2 (k)e2kt 2 2 4 k u2 (li )

βi2 (k) 1 1 1+O 2 2 4 k k u2 (li )

+ e2kli = 4k2

!!2 .

yi2 (t) dt

=

uniformly in t. Now

!!

Integrating from 0 to li gives 0

ek(li −t) k2

×

Hence for large k > 0, yi2 (t) is bounded on (0, li ) so Lebesgues dominated convergence theorem can be used, and only the pointwise limit of yi2 (t) needs to be considered for t ∈ (0, li ). For t ∈ (0, li ) and k → ∞,

Let

w(li ) = 0 w′ (li ) = −u′1 (t)u2 (li ) + u1 (li )u′2 (t) = 1.

!!

ek(li −t) k

α2i (k) sinh k(li − t) + O

+

1 w(t) = sinh k(t − li ) + O k

ek(li −t) k

sinh k(li − t) + O

yi (t) = αi u1 (t) −

+

2αi (k)βi (k) sinh kt + O

for all t.

From (7), (8) and (9)

=

+

ekt k

1 α2i + βi2 1+O 2k k

.

Therefore 1 k2 u22 (li )

β 2 (k) sinh kt + O i

ekt k

!!2

||yi ||2L2 (0,li )

||yi |∂(0,li ) ||2L2 (∂(0,li )) 1 1+O . = 2k k

Summing over i = 1, . . . , K proves the theorem. The following corollary gives bounds for the boundary norm of solutions to the non-homogeneous boundary value problem in terms of the nonhomogeneous term. Corollary 3.2 There exists a constant C > 0 such that for k > 0 large C k

3 2

||f ||L2 (G) ≥ ||y|∂G ||L2 (∂G)

(17)

for all f ∈ L2 (G) and y the solution of −y ′′ + qy = λy + f,

(18)

where λ = −k2 , obeying the boundary conditions (3). Proof: Let Gλ denote the Green’s operator of the boundary value problem (2)-(3) and let GD λ denote the Green’s operator of the boundary value problem (2) but with Dirichlet boundary conditions at every node (i.e. y at all nodes is zero). We note that (l − λ)(Gλ − GD λ )f = f − f = 0 for f ∈ L2 (G) and where l is as given in (1). Thus (Gλ −GD λ )f is a solution of (2) and from Theorem 3.1 obeys (5) hence we obtain that, since (1 + O( k1 )) ≥ √1 for large k, 2 ||(Gλ − GD λ )f ||L2 (G) ≥ 1 √ ||[(Gλ − GD λ )f ]|∂G ||L2 (∂G) , 2 k

since GD λ vanishes on the boundary of G. Taking y = Gλ f gives (17). Remark For the system −Y ′′ + QY = λW Y + F,

(21)

with general self-adjoint boundary conditions of the form AY (0) + BY ′ (0) + CY (1) + DY ′ (1) = 0, (22) for A, B, C and D constant matrices, where either (i) Q ∈ L∞ (0, 1) is Hermitian (not necessarily diagonal), W = I and F ∈ L2 (0, 1) is Hermitian, i.e. a non-weighted system with general, Hermitian, L∞ potential or (ii) Q ∈ L∞ (0, 1) is real valued and diagonal, W is constant, real valued and diagonal and F ∈ L2 (0, 1) is diagonal, i.e. a Sturm-Liouville boundary value problem on a graph, see [7], the following result, corresponding to the above corollary, is obtained. There exists a constant C > 0 such that for k > 0 large, C 3

k2

||F ||L2 (0,1) ≥ ||Y|∂(0,1) ||L2 (∂(0,1))

(23)

for Y the solution of (21), with λ = −k2 , obeying the boundary conditions (22).

(19)

for all f ∈ L2 (G). But [(Gλ − GD λ )f ]|∂G = [Gλ f ]|∂G giving

4 Example In this section we provide an example to illustrate Theorem 3.2. We also show that (17) is the best possible estimate that can be obtained. Consider the second order differential equation

1 ||(Gλ − GD λ )f ||L2 (G) ≥ √ ||[Gλ f ]|∂G ||L2 (∂G) . 2 k

−y ′′ − λy = eαt

Now as Gλ and GD λ are both resolvent operators we have

on [0, π] where α is a constant, i.e. a graph with a single edge of length π, with the boundary conditions

||f ||L2 (G) ||f ||L2 (G) =C |λ| k2 (20) for λ → −∞, where C > 0 is a constant. Hence, combining (19) and (20), we obtain that

y(0) = 0, y ′ (π) = 0.

||(Gλ − GD λ )f ||L2 (G) ≤ C

2C

||f ||L2 (G) k2

≥ =

1 √ ||[(Gλ − GD λ )f ]|∂G ||L2 (∂G) k 1 √ ||[Gλ f ]|∂G ||L2 (∂G) k

(24)

(25)

Then the solutions of (24) on the interval [0, π] are of the form y=

eαt + aekt + be−kt , − k2

α2

where a and b are constants, and λ = −k2 .

(26)

From (26) and the boundary conditions (25), the constants a and b are given as follows αeπα + ke−πk , k(α2 − k2 )(ekπ + e−kπ ) αeπα − ke−πk . k(α2 − k2 )(ekπ + e−kπ )

a = − b =

Substituting the constants back into (26) and evaluating at 0 and π gives y(0) = 0, (k − α)eπ(k+α) + (k + α)eπ(α−k) − 2k . k(α2 − k2 )(ekπ + e−kπ )

y(π) =

We now look at the case of α = 2k. Then y(π) =

−e2πk −ke3πk + 3keπk − 2k ≃ 3k3 (ekπ + e−kπ ) 3k2

giving

||f ||L2 (G) =

s

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