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The subject of this work concerns to the classical inverse spectral problem. This inverse problem can be formulated as follows: do the Dirichlet eigenvalues and the derivatives (which order?) of the normalized eigenfunctions at the boundary determine uniquely the coefficients of the corresponding differential operator? For operators of order two this type of theorem is called Borg-Levinson theorem. In the case of the Schrödinger operators the knowledge of the Dirichlet eigenvalues and the normal derivatives of the normalized eigenfunctions at the boundary uniquely determine unknown potential. Borg-Levinson theorem for the Schrödinger operators was proved for the first time by Nachman, Sylvester and Uhlmann [1] for the potentials from the space  Their proof remains however valid if one assumes that potential just from L¥(W). This problem was reduced finally to the fact that the Dirichley-to-Neumann map uniquely determines such potentials. The same result was obtained independently by Novikov [2]. For singular potentials from the space Lp(W); n/2

For the magnetic Schrödinger operator with singular coefficients Borg-Levinson theorem was proved for the first time by Serov [6]. The proof of this result for the magnetic Schrödinger operators uses the same technique as for the Schrödinger operators with singular potentials. It must be mentioned here that the magnetic Schrödinger operator cannot be considered as a "small" perturbation of the Schrödinger (or Laplace) operator.

For the operator of order 4 which is the first order perturbation of the bi-harmonic operator with Navier boundary conditions on a smooth bounded domain in Rn, n³3, it is proved by Krupchyk, Lassas and Uhlmann [7] that the Dirichlet-to-Neumann map uniquely determine this first order perturbation.

The main goal of present work is to show that the knowledge of the discrete Dirichlet spectrum and some special derivatives up to the second order of the normalized eigenfunctions at the boundary uniquely determine the Dirichlet-to-Neumann map that corresponds to an arbitrary perturbation of the bi-harmonic operator with singular coefficient from some Sobolev spaces on the smooth bounded domain in Rn, n³2. The next step might be to prove that the knowledge of the Dirichlet-to-Neumann map uniquely determines the coefficients of the operator H4 which is an arbitrary perturbation of the bi-harmonic operator. The solution of this problem will be given in future publications.

Below we use the following notations. The space Wpt(W), t³0,1£p£¥, denotes the Lp – based Sobolev space in the domain W and the space  denotes the Besov space on the boundary of the domain W. The Besov space  for negative t is defined as the adjoint space to the corresponding Besov space with positive index of smoothness – t. Throughout this article we use the trace type theorem for the functions from the Sobolev spaces at the boundary (see, for example, [8]). More precisely, for any function f from the Sobolev space  with t ³ 1 and p > 1 there is the trace on the boundary of W from the Besov space

Green’s function

Let W be a bounded domain with smooth boundary in Rn, n³2. We consider in this domain the following operator of order four that we will call the "magnetic" operator of 4th order with variable coefficients

         (1)

where Ñ denotes the gradient in Rn, D denotes the Laplacian in Rn and where the coefficients , F(x),  and V(x) are assumed to be real-valued. We assume also that

(2)

and                        (3)

where it is assumed (WLG) that the value of p is the same for all these spaces. It is well-known that under the conditions (2) and (3) for the coefficients the following Gårding’s inequality holds (see, for instance, [9]):

,                    (4)

where 00. This inequality allows us to define symmetric operator (1) by the method of quadratic forms. Then H4 has a self-adjoint Friedrichs extension denoted by (H4)F with domain

where  denotes the closure of the space  by the norm of Sobolev space . It is possible to prove actually that under the conditions (2) and (3)

                            (5)

The spectrum of this extension is purely discrete, of finite multiplicity and has an accumulation point only at the +¥: l1£l2£…£lk£… ®+¥.

The corresponding orthonormal eigenfunctions  form orthonormal basis in L2(W). The Gårding’s inequality (4) allows us also to conclude that there is a positive constant m0 such that the operator (H4)F+m0I is positive.

Therefore for any l ³ m0 the operator (H4)F+lI is positive and its inverse

((H4)F + lI)–1: L2(W) ® L2(W)                          (6)

is compact. It is an integral operator with kernel denoted by G(x, y, l). If we use for this integral operator the symbol  then we have

,  

                                         (7)

Definition 1. The kernel G(x, y, l) of the integral operator  is called the Green’s function of the operator (H4)F + lI.

Our first result is:

Theorem 1. Suppose that (x), F(x), (x) and V(x) satisfy the conditions (2) and (3). Then for any l³m0 the Green’s function of the operator (H4)F+lI exists and satisfies the following estimates:

     (8)

(9)

and                 (10)

where x, yÎW and constants C>0 and d>0 do not depend on x, yÎW and l.

Proof. It is well-known that the fundamental solution of the operator D2+lI, l>0, has exactly these estimates (8)–(10). The conditions (2) and (3) for the coefficients that are in front of the derivatives of order one and higher of the operator H4 allow us to conclude that they belong to the Kato space Kn–1(Rn) (if we extend them by zero outside of the domain W), i.e.

Using this fact we can easily prove the existence of the fundamental solution for the operator H4+lI. Moreover, the same estimates (8)–(10) will be fulfilled for this fundamental solution too. In order to obtain the same estimates (8)–(10) for the Green’s function G(x, y, l) we refer to the paper [10] (with some changes that are connected to the operator of order 4). Theorem 1 is proved.

We have three immediate corollaries of Theorem 1 (see again [10]).

Corollary 1. Assume that (x), F(x), (x) and V(x) are as above and , n³2. Then for any function f (x)ÎL2(W) the following inequality holds

,

where l³m0 with m0 as in Theorem 1.

Corollary 2. Assume that , n ³ 2. Then there is a constant C>0 depending only on W, such that the estimate  holds uniformly in xÎW and l³m0.

Corollary 3. Assume that , n³2. Then the following series

                                                        (11)

converges.

Remark 1. It can be mentioned here that the estimates (8)–(10) of the Green’s function of the "magnetic" operator (1) are obtained in Theorem 1 for quite weak conditions of the coefficients of H4. As far as we know they never appeared in the literature.

Dirichlet-to-Neumann map and eigenfunctions

Lemma 1. Under the conditions (2) and (3) for the coefficients of H4 we have that for any two functions u and f from  the following equality holds

where L1 and L2 are defined as

L1u(x)=¶v(Du)(x)+i¶v(×Ñu)(x)+iv×D u(x)–

–F¶vu(x)–iv×u(x), xζW                                    (12)

and L2u(x)=–Du(x)–i×Ñu(x), xζW,                (13)

respectively. Here v is unit outward normal vector at the boundary of the domain W.

Proof. The proof of this lemma is straightforward and is based only on the divergence theorem ("integration by parts"). And the main purpose of this lemma is to define the map L=(L1, L2). It must be mentioned also here that the conditions (2) and (3) for the coefficients of the operator H4 and the conditions for the functions u and f from this lemma guarantee the existence of the traces of the corresponding terms in L1u(x), L1f(x), L2u(x) and L2f(x) on the boundary ¶W from some Besov spaces. This fact justifies the correct application of the integration by parts and proves this lemma.

Let l³m0 with m0 as in Chapter 1. Consider the following Dirichlet problem:

((H4)F+lI)u(x)=0; xÎW, u(x)=f0(x), ¶vu(x)=f1(x),

xζW,                                                                       (14)

where the boundary functions f0(x) and f1(x) satisfy the following conditions:

(15)

where  denotes Besov space on the boundary and p is the same as in (2) and (3).

Using the technique of the multipliers from Sobolev spaces (see, for example, [11]) it can be proved that there exists a unique solution of the Dirichlet boundary value problem (14)–(15) from the spaces

                                   (16)

Thus, we may define the Dirichlet-to-Neumann map Ll.

Definition 2. The Dirichlet-to-Neumann map Ll for Dirichlet boundary problem (14)–(15) is defined as the following two-dimensional vector

      (17)

where v is outward normal vector at the boundary ¶W.

Conditions (2), (3), (15) and (16) imply that the Dirichlet-to-Neumann map (17) acts as (for fixed l)

(18)

with p as in (15).

The following theorem can be considered as one of the main results of this work.

Theorem 2. Assume that  and V1, V2 satisfy the conditions (2), (3) and f0, f1 satisfy the condition (15). In addition we assume that  on the boundary ¶W. Then, for any  

        (19)

where  denotes the corresponding Dirichlet-to-Neumann map for .

Proof. Let w (x): =u1(x)–u2(x), where uj(x), j=1,2, solves the problem (14) with , respectively. We denote the corresponding operators (1) by . Then w (x) solves the boundary value problem  w(x)=0, ¶vw(x)=0, xζW.

This problem can be rewritten in the domain W as

                 (20)

and with Dirichlet boundary conditions

w(x)=0, ¶vw(x)=0, xζW.                                 (21)

Denote by  the integral operator with the kernel which is the Green’s function of the operator D2+lI with Dirichlet boundary conditions (21). Applying  to the left and to the right hand-sides in (20) we obtain the following integral equation

(I + K)w(x)=F(x),                                                    (22)

where the integral operator K and the function F are given by

                  (23)

and

            (24)

We consider this equation (22) in the space of functions from the Sobolev space  which are vanishing with their first normal derivatives at the boundary ¶W.

Due to the assumptions (2) and (3) for the coefficients  and Vj, j=1,2, and embedding (16) we may conclude that F belongs to this space and K is compact there. Since the operator  is positive for l³m0 then the boundary value problem

w(x)=0, ¶vw(x)=0, xζW has only trivial solution wº0. The same is true for the homogeneous equation corresponding to (22). By the Fredholm’s alternative the operator I+K has a bounded inverse in the indicated Sobolev space and therefore the solution w of the equation (22) satisfies the following inequality

                                          (25)

where constant C is independent on l³m0.

Since p satisfies the inequality  then the following embeddings hold

This fact and the conditions (2) and (3) allow us to obtain from (24) and (25) that

                                         (26)

We apply now the result from [12] and obtain that

                         (27)

By combining the inequalities (26) and (27) we get the following inequality

                                         (28)

The interpolation of (26) and (28) leads us to the inequality

                                    (29)

where 0£s£4. Using the definition (17) and the equalities for the terms   on the boundary ¶W we have

Using trace theorem and (29) we can estimate the latter terms as follows:

                 (30)

Since  taking into account the boundedness of the norm of u2 in l we may conclude from (30) that Theorem 2 is completely proved.

We are in the position now to estimate the normalized eigenfunctions of the magnetic Schrödinger operator.

Lemma 2. Under the assumptions (2) and (3) for the coefficients  and V the orthonormal eigenfunctions jk(x) satisfy the estimate

                                  (31)

where 0£s£4,  and m0 is as in Theorem 1.

Proof. Let lk be an eigenvalue and jk(x) corresponding orthonormal eigenfunction. Then Corollary 2 from Theorem 1 allows us to obtain quite easily that  and

                                      (32)

where 1£p£¥ and constant C>0 depends only on n, p and Vol(W).

Rewriting the equation for the eigenfunctions jk(x) in the form (D2+m0I+Q(x, ¶))jk(x)=(lk+m0)jk(x), xÎW, jk(x)=0, ¶vjk(x)=0, xζW, where Q(x, ¶) is the rest of the operator H4, and applying the inequality (25), we obtain for any  that

  (33)

Now by interpolation of (32) and (33) we obtain (31). Thus, Lemma 2 is proved.

The next lemma shows us the representation for the kernel of the operator Ll.

Lemma 3. For  (here [a] denotes the entire part of a) and f0 and f1 as in (15) we have for the vector  the following representations

                    (34)

where the kernels  are defined by

      (35)

and where for l³m0 the right-hand sides of (35) are converging in Lp(¶W´¶W).

Proof. Solution u of the problem (14)–(15) definitely depends on l and we will denote it from now on as u(x, l) with l³m0. Integration by parts for the problem (14) with the boundary conditions f0 and f1 from (15) leads to

                                   (36)

where G(x, y, l) is the Green’s function of (H4)F+lI defined in (6)–(10) and vy denotes the outward normal vector in y. In our case the Green’s function is given by

                                      (37)

Since u solves the problem (14)–(15) then using J. von Neumann spectral theorem it can be easily proved by induction that

      (38)

The operator ((H4)F + lI)–1 is well-defined by the spectral theorem and it is the integral operator with kernel denoted by Gl(x, y, l)

This fact allows us to represent ((H4)F+lI)–1u(x, l) as follows

    (39)

where uk(l) is given by

Integration by parts in the last equality gives us

                                    (40)

Combining (39) and (40) we obtain the following equality ((H4)F+lI)–1u(x, l)=

                             (41)

which coincides for l=0 with (36).

Since u solves the boundary value problem (15)–(16) using (17) we can obtain

                                           (42)

Thus, the equalities (38), (41) and (42) give us that formally we have for the vector  the following relations

and

Here we have used the following equalities for the eigenfunctions jk(x) at the boundary (see (15) and (31)): ¶vjk=0, ¶v(Djk)=D(¶vjk), xζW.

The latter equalities show that this lemma will be proved if we show the convergence of all series (35) in Lp(¶W´¶W). To this end, the inequality (31) from Lemma 2, the conditions (2) and (3) for the coefficients and Sobolev imbedding theorem allow us to conclude that

By using these estimates and taking now , we have for k, m=1,2 and j=0,1

where  for j=0 and  for j=1. Thus, due to Corollary 3 of Theorem 1 (see estimate (11)) the latter series converges since  and therefore Lemma 3 is completely proved.

From spectral data to Dirichlet-to-Neumann map

Now we are in the position to formulate and to prove the second main result of this work. By symbol  we denote the discrete Dirichlet spectrum of the operator H4 defined in (1) with the coefficients  and by  the corresponding eigenfunctions accounting their multiplicities.

Theorem 3. Assume that    for j=1, 2. Assume in addition that  and  at the boundary ¶W. Assume also that for each k=1, 2, …

                  (43)

and

         (44)

Then for all l³m0

                                     (45)

for any  and .

Proof. The conditions (43) and (44), Lemma 3 (see formulas (34), (35)) imply that for all l³m0 and for  we have   for any  and . This equality can be read as

(46)

where two-dimensional vector-valued operators  are bounded from  to Lp(¶W)´Lp(¶W). But Theorem 2 (see (19)) shows us that the polynomial in the right-hand side of (46) is zero. Hence, the equality (45) holds. It means that Theorem 3 is proved.

Remark 2. This theorem means that the inverse boundary spectral problem for the operator H4 is reduced to the problem of the reconstruction of the unknown coefficients of this operator by the knowledge of Dirichlet-to-Neumann map.

Acknowledgments. This work was supported by the Academy of Finland (Application No. 250215, Finnish Programme for Centres of Excellence in Research 2012–2017).

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