LOCATION AND MULTIPLICITIES OF EIGENVALUES

. LOCATION AND MULTIPLICITIES OF EIGENVALUES FOR A STAR GRAPH OF STIELTJES STRINGS VYACHESLAV PIVOVARCHIK AND CHRISTIANE TRETTER

Abstract. In this paper we consider a star graph of Stieltjes strings with prescribed number of masses on each edge, with or without a mass at the central vertex. At the latter Dirichlet or Neumann conditions are imposed while all pendant vertices are subject to Dirichlet conditions. We establish necessary and sufficient conditions on the location and multiplicities of two (finite) sequences of numbers {ζk } and {λk } to be the corresponding Dirichlet and Neumann eigenvalues. Moreover, we derive necessary and sufficient conditions for one (finite) sequence {λk } to be the Neumann eigenvalues of such a star graph. Here a key role is played by the possible multiplicities; the corresponding conditions are formulated by means of the notion of vector majorization. Our results include, as a special case, some results for star-patterned matrix inverse problems where only multiplicities, not the location of eigenvalues, were prescribed.

1. Introduction Finite dimensional direct and inverse problems appear in many fields of physics, such as vibrations of strings, electrical circuits synthesis etc. Nice reviews may be found e.g. in [3], see also [12], [6], [7], [4]. Recently direct and inverse problems on graphs, or more specifically on trees or star graphs, have attracted a lot of attention, see e.g. [1]. There are different settings for finding necessary and sufficient conditions on one, or two respectively, (finite) sequences to be the Neumann, and Dirichlet respectively, spectra of a problem generated by Stieltjes string equations on a star graph. In this paper we consider the case where Neumann and Dirichlet conditions are imposed at the central vertex, while at the pendant vertices we impose Dirichlet boundary conditions; following [16] these problems will be denoted by (N1) and (D1). In [16] a complete description of the corresponding direct and inverse problems was given in the case when the numbers of point masses on the edges were not part of the given data. An interesting feature of the Neumann eigenvalues proved there is that out of two neighbouring Neumann eigenvalues one must be simple. In this paper we study the problem where, apart from the number of edges and the lengths of all edges, also the number of masses on each edge is prescribed. The latter results in restrictions on the eigenvalue multiplicities that need to be satisfied in the inverse problem. Our main results include necessary (Theorem 2.2 below) and sufficient (Theorem 3.3 below) conditions for two (finite) sequences {±ζk } and {±λk }, counted with multiplicities, to be the Dirichlet and Neumann spectra of problems (D1) and (N1) as well as necessary (Theorem 2.3 below) and 1991 Mathematics Subject Classification. 34A55, 39A70, 70F17, 70J30. Key words and phrases. Star graph, inverse problem, Nevanlinna function, S-function, continued fractions, transversal vibrations, Dirichlet boundary condition, Neumann boundary condition, point mass, eigenvalue. 1

2

deleted nj , was not correct

VYACHESLAV PIVOVARCHIK AND CHRISTIANE TRETTER

sufficient (Theorem 3.4 below) conditions for one (finite) sequence {±λk }, counted with multiplicities, to be the Neumann spectrum of a problem (N1). In both cases, the formulation of the necessary and sufficient conditions requires the notion of vector majorization. For example, suppose that there are q edges and ni masses on the i-th edge enumerated such that n1 ≥ n2 ≥ · · · ≥ nq . Denote by D the rD the number of different positive values in the sequence {ζk }, by (p↓j (D))rj=1 vector of all their multiplicities in decreasing order, and by Nj the number of edges 1 for which the number of masses is ≥ j for j = 1, 2, . . . , n1 . Then the vector (Nj )nj=1 D , majorizes the vector (p↓j (D))rj=1 N1 , N2 , . . . , Nn1 p↓1 (D), p↓2 (D), . . . , p↓rD (D) , which means that n1 ≤ rD and n1 X j=1

do you agree with this formulation?

Nj =

rD X j=1

p↓j (D)

and

κ X j=1

Nj ≥

κ X

p↓j (D)

(κ = 1, 2, . . . , n1 ).

j=1

The necessary and sufficient conditions for one (finite) sequence to be a Neumann spectrum are similar, but slightly more involved; e.g. they include that the number of simple eigenvalues is greater than or equal to the maximal number n1 of masses on one string and that the number of different Neumann eigenvalues is greater or equal than the sum n1 + n2 of the two largest numbers of masses on one string. Our results generalize results for star-patterned matrices, in particular [9, Theorem 9] (see also [11], [15]). There the existence of a tree-patterned matrix with prescribed multiplicities of eigenvalues was proved, while our result shows the existence, and provides a method of construction, when a set of eigenvalues with these prescribed multiplicities, together with the lengths of the strings, is given. It should be mentioned that spectral theory of a star graph of general strings preserves many properties of star of Stieltjes strings, see [17]. 2. Direct problem: necessary conditions Throughout this paper, we consider a plane star graph of q Stieltjes strings, q ∈ N, q ≥ 2, joined at the central vertex called the root where a mass M ≥ 0 is placed and with all q pendant vertices fixed; here, following Gantmakher and Krein (see [5], [18]), a Stieltjes string is a thread (i.e. an elastic string of zero density) bearing a finite number of point masses. In the sequel, we label the edges of the star graph by j = 1, 2, . . . , q such that (j) the j-th edge carries nj > 0 point masses mk in its interior (k = 1, 2, . . . , nj ) and n1 ≥ n2 ≥ · · · ≥ nq ; here we do not count the possible mass at the centre and there (j) are no masses at the pendant vertices. The masses mk subdivide the j-th edge (j) into nj + 1 intervals of length lk (k = 0, 1, . . . , nj ) where we count both, masses and intervals between them, from the exterior towards the centre; the length of the Pnj (j) j-th edge is denoted by lj := k=0 lk . The total number P of masses on the star q graph without the mass M in the centre is denoted by n := j=1 nj . We assume that this web is stretched and study the small transverse vibrations (j) (j) vk (t) of the masses mk in two different cases: (N1) the mass M at the central vertex is free to move in the direction orthogonal to the equilibrium position of the strings (Neumann problem),

EIGENVALUES OF A STAR GRAPHS OF STIELTJES STRINGS

3

(D1) the mass M at the central vertex is fixed (Dirichlet problem). (j)

Following [5, Chapter III.1] (see also [16, Section 2]), separation of variables vk (t) = (j) uk eiλt leads to the following difference equations for the displacement amplitudes (j) uk (k = 0, 1, 2, . . . , nj , j = 1, 2, . . . , q) for the Neumann and Dirichlet problem: Neumann problem (N1). If the central vertex carrying the mass M ≥ 0 can move freely, we obtain (j)

(2.1)

(j)

(j)

uk −uk+1 (j)

+

(j)

uk −uk−1 (j)

lk

lk−1

(1)

(j)

(j)

− mk λ2 uk = 0

(2)

(k = 1, 2, .., nj , j = 1, 2, .., q),

(q)

(2.2) un1 +1 = un2 +1 = . . . = unq +1 , (2.3)

(j) (j) q X unj +1 − unj (j) lnj

j=1 (j)

(2.4) u0 = 0

(1)

= M λ2 un1 +1 ,

(j = 1, 2, . . . , q).

Dirichlet problem (D1). If all strings are clamped at the central vertex, the problem decouples and consists of the q separate problems on the edges with Dirichlet boundary conditions at both ends, (j)

(2.5)

(j)

uk −uk+1 (j)

(j)

+

(j)

uk −uk−1 (j)

lk

lk−1

(j)

(j)

− mk λ2 uk = 0

(k = 1, 2, .., nj ),

(j)

(2.6)

unj +1 = 0,

(2.7)

u0 = 0

(j)

for all j = 1, 2, . . . , q. Throughout this paper, we use the following notation for the eigenvalues of the spectral problems (N1), (D1) and their multiplicities. Notation 2.1. We denote by {λ±k }n+1 0 k=1 if M > 0, λ (1) ΛN := −k = −λk , λk ≥ λk0 > 0 for k > k > 0, the {λ±k }nk=1 if M = 0, eigenvalues of the Neumann problem (2.1)–(2.4) on the star graph, q S (j) nj (2) ΛD := {ζ±k }nk=1 = ν±κ κ=1 , ζ−k = −ζk , ζk ≥ ζk0 > 0 for k > k 0 > 0, the j=1

eigenvalues of the Dirichlet problem (D1) on the star graph where (j) n

(j)

(j)

(j)

(j)

j , ν−κ = −νκ , νκ > νκ0 > 0 for κ > κ0 > 0, are the eigenvalues (3) {ν±κ }κ=1 of the Dirichlet problem (2.5)–(2.7) on the j-th edge for j = 1, 2, . . . , q, e±k }rN , λ e−k = −λ ek , λ ek > λ ek0 > 0 for k > k 0 > 0 the set of different (4) ΛN = {λ k=1 Neumann eigenvalues, D (5) ΛD = {ζe±k }rk=1 , ζe−k = −ζek , ζek > ζek0 > 0 for k > k 0 > 0 the set of different Dirichlet eigenvalues, N D (6) (pk (N ))rk=1 and (pk (D))rk=1 the vectors of multiplicities of the different ek and ζek for k > 0. positive Neumann and Dirichlet eigenvalues λ

4


Remark 2.2. The number rD of different positive Dirichlet eigenvalues satisfies rD ≥ n1 since n1 is the maximal number of masses on one string, labelled as the (1) (1) (1) first, and ν1 < ν2 < · · · < νn1 . Note that since the equations in (N1) and (D1) only depend on λ2 , the Neumann and Dirichlet eigenvalues lie symmetrically to the origin and the multiplicities of e−k = −λ ek and λ ek as well as of ζe−k = −ζek and ζek coincide. Moreover, by [16, λ Theorem 2.5], 0 is neither a Neumann nor a Dirichlet eigenvalue. In order to derive necessary conditions on the multiplicities of Neumann and Dirichlet eigenvalues, we need the notion of majorization which goes back to Muirhead [14] for the case of integers and was generalized to non-negative numbers by Hardy, Littlewood, and Poly´ a (see [8, 2.18]). Definition 2.3. Let x = (xi )si=1 and y = (yi )ti=1 be two vectors with non-negative entries ordered decreasingly, xs ≥ xs−1 ≥ · · · ≥ x1 ≥ 0, yt ≥ yt−1 ≥ · · · ≥ y1 ≥ 0. If s = t, then x is said to majorize y, (2.8)

xy

:⇐⇒

t X

xi =

i=1

t X

yi ,

i=1

τ X

xi ≥

i=1

τ X

yi

(τ = 1, 2 . . . , t − 1).

i=1 max{s,t}

If s 6= t, we fill up the shorter vector with zeros and set x e = (xi )i=1 and max{s,t} ye = (yi )i=1 with xi = 0 for i = s + 1, . . . , max{s, t}, yi = 0 for i = t + 1, . . . , max{s, t}. Then x is said to majorize y, x y, if x e majorizes ye, x e ye. Remark 2.4. If a vector x = (xi )si=1 majorizes a vector (yi )ti=1 , then the number of non-zero entries of x is less or equal to the number of non-zero entries of y, x y =⇒ #{i ∈ {1, . . . , s} : xi > 0} ≤ #{i ∈ {1, . . . , t} : yi > 0}. In fact, denote the two numbers by s0 , t0 and assume s0 > t0 . Then x y if and 0 0 only if x e ye where x e = (xi )si=1 and ye = (yi )si=1 with yi = 0 for i = t0 + 1, . . . , s0 . By (2.8) this implies s0 X i=1

xi =

s0 X i=1

yi =

t0 X

yi ≤

i=1

t0 X i=1

xi ≤

s0 X

xi ,

i=1

and hence we have equality everywhere. Since all xi are non-negative, this shows that xi = 0 for i = t0 + 1, . . . , s0 , a contradiction to the assumption. Notation 2.5. For a vector x = (xi )si=1 ∈ Rs we denote by x↓ = (x↓i )si=1 ∈ Rs the vector with the same entries but ordered decreasingly, i.e. there exists a permutation π of {1, 2, . . . , s} such that x↓i = xπ(i) , i = 1, 2, . . . , s,

x↓1 ≥ x↓2 ≥ · · · ≥ x↓s .

The following elementary lemma on the inversion of the non-increasing function {1, 2, . . . , q} → N, j 7→ nj , will be used throughout this paper. Lemma 2.6. Let q ∈ N, q ≥ 2, (nj )qj=1 ⊂ N with n1 ≥ n2 ≥ · · · ≥ nq , and set Pq n := j=1 nj , nq+1 := 0. For i = 1, 2, . . . , n1 define Ni := #{j ∈ {1, 2, . . . , q} : nj ≥ i} = max{j ∈ {1, 2, . . . , q} : nj ≥ i}. Then N1 ≥ N2 ≥ · · · ≥ Nn1 and Pn1 i) i=1 Ni = n;


5

ii) Ni > 1 (i = 1, 2, . . . , n2 ), Ni = 1 (i = n2 + 1, n2 +2, . . . , n1 ); iii) #{i ∈ {1, 2, . . . , n1 } : Ni = j} = nj − nj+1 (j = 1, 2, . . . , q); iv) nj = #{i ∈ {1, 2, .., n1 } : Ni ≥ j} = max{i ∈ {1, 2, .., n1 } : Ni ≥ j} (j = 1, 2, .., q). Proof. Claims i), ii), iii) follow from the definition of Ni , and claim iv) from iii).

Theorem 2.7. Let ΛN be the set of all Neumann eigenvalues λ±k of (N1), λk > 0, λ−k = −λk (k > 0), and ΛD the set of all Dirichlet eigenvalues ζ±k of (D1), ζk > 0, ζ−k = −ζk (k > 0) (see Notation 2.1 (1) and (2)), both counted with mulitplicities. Denote by rD the number of different positive Dirichlet eigenvalues, D the vector of their multiplicities in decreasing order, and by p↓ (D) = (p↓i (D))ri=1 by Nj the number of edges for which the number of masses is ≥ j (j = 1, 2, . . . , n1 ), i.e. Ni := #{j ∈ {1, 2, . . . , q} : nj ≥ i}. Then ( 0 < λ21 < ζ12 ≤ . . . ≤ λ2n ≤ ζn2 < λ2n+1 if M > 0, 1) if M = 0; 0 < λ21 < ζ12 ≤ . . . ≤ λ2n ≤ ζn2 2) ζk−1 = λk if and only if λk = ζk ; 3) N1 , N2 , . . . , Nn1 p↓1 (D), p↓2 (D), . . . , p↓rD (D) . Proof. The first two claims 1) and 2) were proved in [16, Theorem 2.5]. In order to prove 3), we first note that all entries in the vectors in 3) are non-zero and n1 ≤ rD by Remark 2.2. The maximal multiplicity of a Dirichlet eigenvalue is equal to the number q of strings, which is in turn equal to N1 (the number of strings carrying at least 1 mass), i.e. p↓1 (D) ≤ q = N1 . If p↓1 (D) = q is maximal, then in order to achieve the next highest multiplicity p↓2 (D) there are only Dirichlet eigenvalues left on edges with at least 2 masses, i.e. p↓2 (D) ≤ N2 so that altogether p↓1 (D)+p↓2 (D) ≤ q+N2 = N1 +N2 ; in the general case p↓1 (D) ≤ q, in order to achieve the multiplicity p↓2 (D) there are only those Dirichlet eigenvalues left on edges with at least 2 masses and egdes with one mass which have not contributed to the highest multiplicity p↓1 (D), i.e. p↓2 (D) ≤ N2 + (N1 − p↓1 (D)) and hence p↓1 (D) + p↓2 (D) ≤ N2 + N1 . Inductively, the same reasoning yields that (2.9)

τ X

p↓i (D) ≤

i=1

τ X

Ni ,

τ = 1, 2, . . . , n1 .

i=1

The total number of Dirichlet eigenvalues counted with multiplicities is n or, equivalently, the sum of all multiplicities is n. Together with Lemma 2.6 i), this implies rD n1 X X (2.10) p↓i (D) = n = Ni . i=1

i=1

Now 3) follows from the inequalities (2.9) and the equality (2.10).

Remark 2.8. The well-known property that the multiplicity of a Dirichlet eigenvalue ζek is at most q is an immediate consequence of the majorization property in Theorem 2.7 3). In fact, the first inequality therein is N1 ≥ p↓1 (D) and, by definition, N1 = q and p↓1 (D) is the largest multiplicity of a ζek .

This definition of Nj has been already given in Lemma 2.6 but maybe we should repeat it

6


The following properties of the Dirichlet and Neumann eigenvalues and their multiplicities, which follow from Theorem 2.7, will be used in the proof of the next theorem. Proposition 2.9. i) If we set (2.11) κN (l) := #{λk ∈ ΛN ∩ (0, ∞) : λk has multiplicity l} κD (l) := #{ζk ∈ ΛD ∩ (0, ∞) : ζk has multiplicity l} This definition of rD has been already given on page 2 but maybe we should repeat it

(l = 1, 2, . . . , q−1), (l = 1, 2, . . . , q),

and rD denotes the number of different positive Dirichlet eigenvalues, then ( κD (2) + rD + 1 if M > 0. κN (1) = κD (2) + rD if M = 0, κN (l) = κD (l + 1) (l = 2, 3, . . . , q − 1). ii) If rN denotes the number of different positive Neumann eigenvalues and reD the number of multiple positive Dirichlet eigenvalues, then ( rD + reD + 1 if M > 0, (2.12) rN = rD + reD if M = 0, and      ↓   ↓  p1 (N )+1 p↓1 (N ) p1 (N ) p↓1 (D)        .    . . .     .. .. ..    ..                 .. ↓  ..  p↓ (N )+1   p (N )           reD .    = (2.13)  .. = re↓D  ↓  .      .  p  p . (N ) (N )  n    .  .   reD +1   reD +1    −1, M > 0,  .     .. .. .. = rrN     ..    M = 0; N, . . .      .      ↓ ↓ ↓  .. prD (D) prD (N )  prD (N )          ..      1  .       ..      . ..    .reD      p↓rD +erD (N )  1 

( iii)

N1 , N2 , . . . , Nn1 p↓1 (N ), p↓2 (N ), . . . , p↓rN −1 (N ) , N1 , N2 , . . . , Nn1 p↓1 (N ), p↓2 (N ), . . . , p↓rN (N ) ,

M > 0, M = 0.

Proof. i) By Theorem 2.7 1) and 2), the only two possibilities for a simple Neumann eigenvalue λk with k ∈ {2, . . . , n} to appear is either strictly between two Dirichlet eigenvalues or coinciding with a double Dirichlet eigenvalue, i.e. ζk−1 < λk < ζk

or

λk−1 < ζk−1 = λk = ζk < λk−1 ;

in any case there appears the simple eigenvalue λ1 and, if M > 0, the simple eigenvalue λn+1 . This shows that κN (1) = rD + κD (2) if M = 0 and κN (1) = rD + κD (2) + 1 if M > 0. If λk is an eigenvalue of multiplicity equal l ≥ 2, then k 6= 1 and, if M > 0, also k 6= n + 1 and there exists a k0 ∈ {2, . . . , n − l} such that k0 ≤ k and λk0 −1 < λk0 = λk0 +1 = . . . = λk0 +l−1 < λk0 +l . Then Theorem 2.7 1) and 2) show that λk0 −1 < ζk0 −1 = λk0 = ζk0 = λk0 +1 = . . . = ζk0 +l−2 = λk0 +l−1 = ζk0 +l−1 < λk0 +l .


7

Hence to each Neumann eigenvalue of multiplicity l ≥ 2 there corresponds a Dirichlet eigenvalue of multiplicity l + 1, and the same holds vice versa. ii) Using the above relations for κN (l) we find, e.g. for M = 0, rN =

q−1 X

κN (l) = κD (2) + rD +

l=1

q−1 X

κD (l + 1) = rD +

l=2

q X

κD (l) = rD + reD ;

l=2

for M > 0, one only has to add 1 from the second equality on. In order to D , prove (2.13), we note that Theorem 2.7 2) yields that for the vectors (p↓j (D))rj=1 ↓ rN (pj (N ))j=1 of Dirichlet and Neumann multiplicities ordered decreasingly, we have (2.14)

p↓j (D) > 1,

p↓j (N ) = p↓j (D) − 1

(j = 1, 2, . . . , reD ),

(2.15)

p↓j (D) = 1,

p↓j (N ) = p↓j (D)

(j = reD + 1, . . . , rD ).

This implies the first and the last equality in (2.13) if we note (2.12). The majorization property claimed in (2.13) is obvious from Definition 2.3 since the number of components where +1 is added on the left hand side is equal to reD and coincides with the number of new components 1 added on the right hand side. iii) The last claim is immediate from Theorem 2.7 3) and from (2.13) since the majorization property is transitive. In the following we consider the properties of the set of Neumann eigenvalues alone. Here we distinguish the case when there is no mass at the central vertex, i.e. M = 0, and when there is a mass, i.e. M > 0. The effect of this central mass is that it adds the simple eigenvalues ±λn+1 and, viewed as functions of M , the Neumann eigenvalues ±λk (M ) tend monotonically to the Dirichlet eigenvalues ±ζk−1 as M grows; more precisely, with λn+1 (0) := ∞, we have λk (M ) & ζk−1 (k = 1, 2, . . . , n + 1) and λ1 (M ) & 0 for M → ∞ (see [16, Prop. 2.8]). Theorem 2.10. Let ΛN be the set of all Neumann eigenvalues λ±k of (N1), λk > 0, λ−k = −λk (k > 0), counted with multiplicities (see Notation 2.1 (1)). Denote by ek , by (pi (N ))rN the rN the number of different positive Neumann eigenvalues λ i=1 ↓ r N vector of their multiplicities, and by p↓ (N ) = (pi (N ))i=1 the corresponding vector of multiplicities in decreasing order. If M = 0, then 1) 0 < λ21 < λ22 ≤ λ23 ≤ . . . ≤ λ2n−2 ≤ λ2n−1 ≤ λ2n ; 2) if pi (N ) > 1, then pi−1 (N ) = pi+1 (N ) = 1 (i = 2, . . . , rN − 1), and if pr (N ) > 1, then prN −1 (N ) = 1; PrNN 3) i=1 pi (N ) = n; 4) rN ≥ n1 + n2 ; 5) {N1 − 1, N2 − 1, . . . , Nn1 − 1} {p↓1 (N ), p↓2 (N ), . . . , p↓rN −n1 (N )}. If M > 0, then 1’) 2’) 3’) 4’)

0 < λ21 < λ22 ≤ λ23 ≤ . . . ≤ λ2n−1 ≤ λ2n < λ2n+1 ; if pi (N ) > 1, then pi−1 (N ) = pi+1 (N ) = 1 (i = 2, . . . , rN − 1); PrN i=1 pi (N ) = n + 1; rN ≥ n1 + n2 + 1;

5’) {N1 − 1, N2 − 1, . . . , Nn1 − 1} {p↓1 (N ), p↓2 (N ), . . . , p↓rN −n1 −1 (N )}.

This definition of rN has been already given in Proposition 2.9 but maybe we should repeat it

8


Note that 4) is a hidden consequence of the majorization property 5) by Remark 2.4; in fact, the number of non-zero entries of the majorizing vector is n2 , all rN − n1 entries of the majorized vector are non-zero, and hence rN − n1 ≥ n2 . The same holds for 4’) and 5’). Furthermore, the following are necessary conditions for assumptions 1), 2), and 3). Remark 2.11. For M = 0, assumptions 1) and 2) imply 6) p↓h (N ) = p↓h+1 (N ) = . . . = p↓rN (N ) = 1 for h := [ rN2+1 ] (:= s if rN = 2s or 2s−1); 7) the number of simple Neumann eigenvalues is at least n1 ; for M > 0, assumptions 1’) and 2’) imply 6’) p↓h (N ) = p↓h+1 (N ) = . . . = p↓rN (N ) = 1 for h := [ rN2+1 ]; 7’) the number of simple Neumann eigenvalues is at least n1 + 1. Proof. Let M = 0. Claim 6) follows immediately from 2). In Proposition 2.9 i) we showed, by means of 1) and 2), that the number of simple Neumann eigenvalues satisfies κN (1) ≥ rD ; on the other hand, rD ≥ n1 by Remark 2.2). Thus κN (1) ≥ n1 which is Claim 7). The proofs for M > 0 are analogous. Proof of Theorem 2.10. Claims 1) and 1’) are immediate from Theorem 2.7 1). Claims 2) and 2’) follow from Theorem 2.7 2) (see [16, Cor. 2.6]). Claims 3) and 3’) follow from the fact that the sum of all multiplicities of different eigenvalues coincides with the number of all eigenvalues λk counted with multiplicites, which was proved to be n if M = 0 and n + 1 if M > 0 in [16, Thm. 2.5]. Let us prove 5). Theorem 2.7 3) implies (N1 − 1, N2 − 1, ..., Nn1 − 1) (P1↓ (D) − 1, P2↓ (D) − 1, ..., Pr˜↓D − 1, Pr˜↓D +1 , ..., Pr↓D −(n1 −˜rD ) ) = (P1↓ (N ), P2↓ (N ), ..., Pr↓D −(n1 −˜rD ) (N )). We have seen (see (2.13)) that rD + r˜D = rN , thus 5) is proved. Let us show that 5) inplies 4). First of all 5) implies that the number of nonzero elements in (N1 − 1, N2 − 1, ..., Nn1 ) is ≤ rN − n1 . That means that the number of {k : Nk > 1} ≤ rN − n1 . This means that n1 − {nubmer of k withNk = 1} ≤ rN − n1 . According to Lemma 2.6. iii) {number of k withNk = 1} = n1 − n2 . Combining these two we obtain n1 − (n1 − n2 ) ≤ rN − n1 .

ek Remark 2.12. The property that the multiplicity of a Neumann eigenvalue λ is at most q − 1 (see [16, Theorem 2.5 3)]) is an immediate consequence of the majorization properties in Theorem 2.10 5) and 5’). In fact, the first inequality therein is N1 − 1 ≥ p↓1 (N ) and, by definition, N1 = q and p↓1 (N ) is the largest ek . multiplicity of a λ


9

3. Inverse problem (j) n

j In this section we consider the problem of recovering the sequences {mk }k=1 (j) nj of masses on each edge, the central mass M ≥ 0, and the lengths {lk }k=0 of subintervals between the masses on each edge, from the Neumann and Dirichlet spectra {λk } and {ζk } on the whole star graph and from the total lengths lj of the strings if the number of masses nj on the j-th string is given as well. The inverse problem where not just the set of all Dirichlet eigenvalues {ζk }, but (j) the subsets of the Dirichlet eigenvalues {νk } for all edges j = 1, 2, . . . , q, are given was considered in [10] for trees with M = 0 at the root. In the case when the tree is a star, the following result is an immediate corollary of [10, Theorem 3.3].

Proposition 3.1 ([10]). Let q ∈ N, q ≥ 2, (lj )qj=1 ⊂ (0, ∞), n ∈ N. Suppose a (j)

(j) n

j sequence ΛN := {λ±k }nk=1 and q sequences ΛD = {ν±k }k=1 with (nj )qj=1 ⊂ N, n1 ≥ n2 ≥ . . . ≥ nq , are given such that λk > 0, λ−k = −λk (k = 1, 2, . . . , n), (j) (j) (j) (j) ν−k = −νk , 0 < νk < νk0 for k < k 0 (k, k 0 = 1, 2, . . . , nj , j = 1, 2, . . . , q), and Pq Sq (j) n j=1 nj = n. Assume further that the sets ΛN and ΛD := {ζ±k }k=1 := j=1 ΛD satisfy the conditions 1) 0 < λ21 ≤ ζ12 ≤ · · · ≤ λ2n ≤ ζn2 ; 2) λk = ζk−1 if and only if λk = ζk ;

(j) n

(j) n

j j Then there exist a collection of masses {mk }k=1 and of lengths {lk }k=0 (j = 1, Pnj (j) 2, . . . , q) with k=0 lk = lj such that the corresponding spectral problems (N1) and Sq (j) (D1) with M = 0 have the sets ΛN = {λ±k }nk=1 and ΛD = j=1 ΛD as Neumann and Dirichlet eigenvalues.

Proof. The claim follows from [10, Theorem 3.3] in the special case when the subtrees Tj consist only of edge number j with corresponding length Li,j = lj for j = 1, 2, . . . , q. Remark It was shown in [2] that if all λk are simple then the solution of this inverse problem is unique. Theorem 3.2. Let q ∈ N, q ≥ 2, (lj )qj=1 ⊂ (0, ∞), n ∈ N. Suppose that sequences n ΛN := {λ±k }n+1 k=1 , ΛD := {ζ±k }k=1 are given such that λk , ζk > 0, λ−k = −λk , ζ−k = −ζk for k > 0, and let (nj )qj=1 ⊂ N, n1 ≥ n2 ≥ · · · ≥ nq , be given with Pq j=1 nj = n. Define Ni := #{j ∈ {1, 2, . . . , q} : nj ≥ i} for i = 1, 2, . . . , n1 , denote by rD the number of different positive elements in ΛD , by pk (D) their multiplicities D (k = 1, 2, . . . , rD ), and let (p↓k (D))rk=1 be the corresponding vector of multiplicities in decreasing order. Then the conditions 1) 0 < λ21 < ζ12 ≤ . . . ≤ λ2n ≤ ζn2 < λ2n+1 , 2) ζk−1 = λk if and only if λk = ζk , 3) N1 , N2 , . . . , Nn1 p↓1 (D), p↓2 (D), . . . , p↓rD (D) , are necessary and sufficient such that there exist a collection of (positive) masses Pnj (j) (j) nj (j) nj {mk }k=1 , a mass M > 0, and lengths {lk }k=0 (j = 1, 2, . . . , q) with k=0 lk = lj such that the spectral problems (N1) and (D1) on the corresponding star graph have the sets ΛN and ΛD as Neumann and Dirichlet eigenvalues. If the sequence ΛN is of the form ΛN = {λ±k }nk=1 and 1) is replaced by

10


1’) 0 < λ21 < ζ12 ≤ . . . ≤ λ2n ≤ ζn2 , then the above continues to hold with M = 0. Proof. First we show that it is possible to divide the elements of the set {ζk }nk=1 (j) nj into q groups {νk }k=1 for j = 1, 2, . . . , q such that (3.1)

ΛD =

{ζ±k }nk=1

:=

q [

(j) n

j , {ν±k }k=1

j=1 (j) ν−k

(j) −νk

(j) νk

(j) νk 0

0

= and 6= for k 6= k . For the first group of n1 elements we note that all entries in the vectors in 3) are non-zero and hence the majorization property 3) implies that n1 ≤ rD . Thus we can choose n1 elements such that the first element has multiplicity p↓1 (D) in ΛD , the second element has multiplicity p↓2 (D) up to the reD -th element having multiplicity p↓reD (D) in ΛD , and n1 − reD elements of multiplicity 1 from the tail of the vector p↓1 (D), p↓2 (D), . . . , p↓rD (D) . (1) 1 We denote the n1 selected elements, arranged in increasing order, by {νk }nk=1 . Again by assumption 3), we conclude that N1 − 1, N2 − 1, . . . , Nn1 − 1 (3.2) p↓1 (D) − 1, . . . , p↓reD (D) − 1, p↓reD +1 (D), . . . , p↓rD −n1 +erD (D) = p↓1 (D) − 1, . . . , p↓reD (D) − 1, 1, . . . , 1 ,

where reD is the number of multiple positive elements in ΛD . For the second group of n2 elements we note that the number of non-zero entries of the majorizing vector in (3.2) is n2 by Lemma 2.6 ii) and hence the majorization property (3.2) implies that n2 ≤ rD −n1 +e rD (see Remark 2.4). Thus we can choose 1 n2 elements such that the first element has multiplicity p↓1 (D) − 1 in ΛD \{ν (1) }nk=1 , ↓ the second element has multiple multiplicity p2 (D) − 1 etc. as for the first group. (2) 2 We denote the n2 selected elements, arranged in increasing order, by {νk }nk=1 . Due to the majorization assumption 3), together with the property that n j = Pq #{i ∈ {1, 2, . . . , n1 } : Ni ≥ j} by Lemma 2.6 iv) and with j=1 nj = n, we may (j) n

j continue like this to obtain q groups of elements {νk }k=1 (j = 1, 2, . . . , q) such that (3.1) holds. In the case where ΛN = {λ±k }nk=1 , we can now apply Proposition 3.1 to finish the proof in the case M = 0. In the case where ΛN = {λ±k }n+1 k=1 , we can use the chosen decomposition (3.1) as [16, (2.25)] in the proof of [16, Theorem 2.5] to prove the claim in the case M > 0.

In the next theorem we derive sufficient conditions for one sequence of numbers to be the Neumann eigenvalues of a star graph of Stieltjes strings. We can delete 4) since it follows from 5)

Theorem 3.3. Let q ∈ N, q ≥ 2, (lj )qj=1 ⊂ (0, ∞), n ∈ N. Suppose that a sequence ΛN := {λ±k }n+1 k=1 , is given such that λk > 0, λ−k Pq= −λk for k > 0, and let (nj )qj=1 ⊂ N, n1 ≥ n2 ≥ · · · ≥ nq , be given with j=1 nj = n. Define Ni := #{j ∈ {1, 2, . . . , q} : nj ≥ i} for i = 1, 2, . . . , n1 . Denote by rN the number ˜ k in the sequence {λk }n+1 , by (pi (N ))rN the vecof different positive elements λ k=1

i=1

N tor of their multiplicities, and by p↓ (N ) = (p↓i (N ))ri=1 the corresponding vector of multiplicities in decreasing order. Then the conditions


11

1) 0 < λ21 < λ22 ≤ λ23 ≤ . . . ≤ λ2n−2 ≤ λ2n−1 ≤ λ2n ; 2) if pi (N ) > 1 then pi−1 (N ) = pi+1 (N ) = 1 for i = 2, . . . , rN − 1, and if prN (N ) > 1 then prN −1 (N ) = 1; P rN 3) i=1 pi (N ) = n; 4) rN ≥ n1 + n2 ; 5) {N1 − 1, N2 − 1, . . . , Nn1 − 1} {p↓1 (N ), p↓2 (N ), . . . , p↓rN −n1 (N )}. are necessary and sufficient such that there exist a collection of (positive) masses (j) nj (j) nj {mk }k=1 , central mass M = 0, and lengths {lk }k=0 (j = 1, 2, . . . , q) with Pnj (j) k=0 lk = lj such that the spectral problem (N1) on the corresponding star graph has the set ΛN as Neumann eigenvalues. If the sequence ΛN is of the form ΛN = {λ±k }n+1 k=1 and the conditions 1)–5) are replaced by the conditions 1’)–5’) in Theorem 2.10, then the above continues to hold with central mass M > 0. Note that 4) is a necessary condition for the majorization property 5) by Remark 2.4; in fact, the number of non-zero entries of the majorizing vector is n2 , all rN − n1 entries of the majorized vector are non-zero, and hence rN − n1 ≥ n2 . The same holds for 4’) and 5’). Remark 3.4. Let us compare Theorem 3.3 with Theorem 9 in [9]. Our set {λ2k }n+1 1 in case of M > 0 is the spectrum of the matrix M −1/2 LM −1/2 from [16] while our set {λ2k }n1 in case M = 0 is the spectrum of the linear operator λM − L. Remark 3.5. For M > 0, necessary conditions for 1) and 2) are 6) p↓h (N ) = p↓h+1 (N ) = . . . = p↓rN (N ) = 1 for h := [ rN2+1 ] (:= s if rN = 2s or 2s−1); 7) the number of simple elements λk > 0 is at least n1 ; for M = 0, necessary conditions for 1’) and 2’) are 6’) and 7’) in Remark 2.11. Proof. The necessity of 6), 7) and of 6’), 7’), respectively, follows in the same way as in the proof of Remark 2.11. Proof of Theorem 3.3. In the following we will show that the assumptions above ensure the existence of a sequence ΛD := {ζ±k }nk=1 , ζk > 0, ζ−k := −ζk for k > 0, such that the two sequences ΛN and ΛD satisfy the assumptions of Theorem 3.2. To this end, in the case M = 0, we set λn+1 := ∞ for convenience. Following condition 2), we choose {ζk }nk=1 , as follows: If k ∈ {1, 2, . . . , n} and λk < λk+1 are two positive elements of ΛN of multiplicity 1, then we choose ζk strictly in between, i.e. ζk ∈ (λk , λk+1 ), with multiplicity 1; if one of λk , λk+1 is multiple, say λk+1 = · · · = λk+pj0 (N ) with multiplicity pj0 (N ) ≥ 2 for some j0 ∈ {1, 2, . . . , rN } (whence k < n), then we choose ζk = ζk+1 = · · · = ζk+pj0 (N ) = λk+1 with multiplicity pj0 (N ) + 1 ≥ 3. Note that, for this particular choice, there are no elements in {ζ±k }nk=1 with multiplicity 2. If we define the numbers κN (l), κD (l) as in (2.11), then κD (2) = 0 for the above choice of ΛD = {ζ±k }nk=1 . Hence, if reN denotes the number of multiple positive elements in ΛN , then ( rD if M = 0, (3.3) rN − reN = κN (1) = rD + 1 if M > 0;

I will rewrite this remark but I have problems with these matrices in tex. I would like to copy them from our previous paper.

12


here, for the case M > 0, we have to observe that the additional element λn+1 is always simple due to the last strict inequality in 1’). If we use assumption 7) and add +1 in the n1 components of the majorizing vector and, on the right hand side, add +1 in the first reN components and n1 − reN new components 1, we conclude that, for M = 0, (3.4) N1 , . . . , Nn1 p↓1 (N )+1, . . . , p↓reN (N )+1, p↓reN +1 (N ), . . . , p↓rN −n1 (N ), 1, . . . , 1 ; | {z } n1 −e rN

note that this is possible since rN − reN ≥ n1 , i.e. rN − n1 ≥ reN , by assumption 6). For M > 0, according to (7’), the group of unchanged elements in the middle of the vector on the right hand side of (3.4) needs to be replaced by p↓reN +1 (N ), . . . , p↓reN −n1 −1 (N ). By (3.3), the number of components on the right hand side is equal to rN − n1 + (n1 − reN ) = rN − reN = rD for M = 0 and equal to rN − n1 − 1 + (n1 − reN ) = rN − 1 − reN = rD for M > 0. Moreover, by the definition of reN as the number of multiple positive elements of ΛN , we have p↓k (N ) > 1, p↓k (D) = p↓k (N ) + 1

(k = 1, 2, . . . , reN ),

p↓k (N ) p↓k (D)

(k = reN + 1, . . . , rN − n1 ),

= 1, =1

p↓k (D)

=

p↓k (N )

=1

(k = rN − n1 + 1, . . . , rD ).

This shows that the majorized vector on the right hand side of (3.4) is, in fact, equal to (p1 (D), . . . , prD (D)) and hence N1 , . . . , Nn1 p1 (D), . . . , prD (D) . Thus we have shown that the sequences ΛN and ΛD satisfy all assumptions of Theorem 3.2 which yields the claim. Acknowledgements. The financial support for this work by the Swiss National Science Foundation, SNF, within the Swiss-Ukrainian SCOPES programme (grant no. IZ73Z0-128135) is gratefully acknowledged. The authors also thank Prof. Dr. Harald Woracek, Vienna University of Technology (Austria), for his kind hospitality. References [1] G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs. AMS, Vol. 186, Providence, RI, 2013, xiv+270 pp. [2] O. Boyko and V. Pivovarchik, Inverse spectral problem for a star graph of Stieltjes strings. Methods Funct. Anal. Topology 14 (2008), no. 2, 159–167. [3] S.J. Cox, M. Embree, J.M. Hokanson, One Can Hear the Composition of a String: Experiments with an Inverse Eigenvalue Problem. SIAM Rev. 54:1 (2012), 157–178. [4] W. Cauer, Die Verwirklichung von Wechselstromwiderst¨ anden vorgeschriebener Frequenzabh¨ angigkeit. Archiv f¨ ur Elektrotechnik 17:4 (1926), 355–388. [5] F.R. Gantmakher, M.G. Krein, Oscillating matrices and kernels and vibrations of mechanical systems (in Russian). GITTL, Moscow-Leningrad, (1950), German transl. Akademie Verlag, Berlin, 1960. [6] G. Gladwell, Inverse problems in vibration. Kluwer Academic Publishers, Dordrecht, 2004. [7] G. Gladwell, Matrix inverse eigenvalue problems. In: G. Gladwell, A. Morassi, eds., Dynamical Inverse Problems: Theory and Applications. CISM Courses and Lectures 529 (2011), 1–29. [8] G.H. Hardy, J.E. Littlewood, G. P´ olya, Inequalities. Second Edition. Cambridge, at the University Press, 1952. xii+324 pp.


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[9] C.R. Johnson, A. Leal Duarte, On the possible multiplicities of the eigenvalues of a Hermitian matrix whose graph is a tree. Linear Alg. Appl. 348 (2002), 7–21. [10] C.-K. Law, V. Pivovarchik, W.C. Wang A polynomial identity and its applications to inverse spectral problems in Stieltjes strings. Operators Matrices 7:3 (2013), 603–617. [11] A. Leal Duarte, Construction of acyclic matrices from spectral data. Linear Algebra Appl. 113 (1989), 173–182. [12] V.A. Marchenko, Introduction to the theory of inverse problems of spectral analysis (in Russian). Acta, Kharkov, 2005. [13] A. Marshall, I. Olkin, B. Arnold, Inequalities: Theory of Majorization and Its Applications. Second Edition. Springer, New York, 2011, xxviii+909 pp. [14] R.F. Muirhead, Some methods applicable to identities and inqualities of symmetric algebraic functions of n letters. Proc. Edinburgh Math. Soc. 21 (1903), 144–157. [15] P. Nylen, F. Uhlig, Realization of interlacing by tree patterned matrices. Linear Multilinear Algebra 38 (1994), 13–37. [16] V. Pivovarchik, N. Rozhenko, C. Tretter, Dirichlet-Neumann inverse spectral problem for a star graph of Stieltjes strings. Linear Algebra Appl. 439:8 (2013), 2263-2292 (open access). [17] V. Pivovarchik, H. Woracek, Sums of Nevanlinna functions and differential equations on star-shaped graphs. Operators Matrices 3:4 (2009), 451–501. [18] T.-L. Stieltjes, Recherches sur les fractions continues. Ann. Fac. Sci. Toulouse 8 (1894), 1–122, 9 (1895), 1–47. South Ukrainian National Pedagogical University, Staroportofrankovskaya str. 26, Odessa 65020, Ukraine E-mail address: [email protected] ¨ t Bern, Sidlerstr. 5, 3012 Bern, Switzerland Mathematisches Institut, Universita E-mail address: [email protected]