CC BY
24
WEYL FUNCTION FOR SUM OF OPERATORS TENSOR PRODUCTS
A.A. Boitsev1, H. Neidhardt2, I.Yu. Popov
1 Saint Petersburg National Research University of Information Technologies, Mechanics and Optics, 49 Kronverkskiy, Saint Petersburg, 197101, Russia
2Weierstrass Institute for Applied Analysis and Stochastic, Berlin, Germany
boitsevanton@gmail.com, hagen.neidhardt@wias-berlin.de, popov1955@gmail.com
The boundary triplets approach is applied to the construction of self-adjoint extensions of the operator having the form s = a ® it + i a ® t where the operator a is symmetric and the operator t is bounded and self-adjoint. The formula for the Y-field and the Weyl function corresponding the the boundary triplet n^ is obtained in terms of the 7-field and the Weyl function corresponding to the boundary triplet nA.
Keywords: operator extension, Weyl function, boundary triplet. 1. Introduction
The spectral theory of differential operators is very important for mathematics and has many applications in quantum physics (see, e.g., [1]. The theory of self-adjoint operators and especially of self-adjoint extensions of symmetric operators occupies a special place in the operator theory [2]. In many interesting problems of quantum physics (like the interaction of photons with electrons) the operators take on the form of the sum of tensor products [3], [4]. From general position, the extensions are usually described in terms of so-called boundary triplets [5]. Up to now, there is no boundary triplets method for obtaining all self-adjoint extensions of such an operator.
In particular, we consider a closed densely defined symmetric operator
where A is a closed densely defined symmetric operator on the separable Hilbert space HA and T is a bounded self-adjoint operator acting on the separable infinite dimensional Hilbert space Ht. Notice that the deficiency indices of S are infinite even if A has finite deficiency indices.
Our aim is to describe all self-adjoint extensions of S using the boundary triplet approach. More precisely, assuming that nA = {'HA, rA, TA1} is a boundary triplet for A* we construct a boundary triplet nS = {HS, rf, rf} for S*. In addition, using the 7-field 7a(-) and the Weyl function MA(-) of the boundary triplet nA we express the 7-field 7S(•) and Weyl function Ms(•) of ns.
The present note generalizes results of [6]. In [6] on the Hilbert space H = ¿2(R+, H) the operator
PACS 03.65 Nk
5 = A 0 It + I a 0 T
(1.1)
(Sf)(x) = - dt; f (t)+ Tf(t),
f G dom (S) := [f G W2'2(R+, H) : f (0) = f'(0) = 0}.
(1.2)
was considered where T is a bounded self-adjoint operator. One easily checks that the operator (1.2) has the form (1.1) where A acts on L2(R+) and is given by
(Af)(t) = -f (t), f e dom(A) := {W2'2(R+) : f (0) = f' (0) = 0}.
In [6] it was verified that nS = {'H, r0, ri}
rof := f (0), rf = f '(0), f e dom (S*) = W2'2(R+, H).
defines a boundary triplet for S*. The corresponding Weyl function is given by MS(z) = iyjz — T, z e C±.
Notation. Let H and H be separable Hilbert spaces. The set of bounded linear operators from H1 to H2 is denoted by [H1,H2]; [H] := [H,H]. By Sp(H), p e (0, ro], we denote the Schatten-v.Neumann ideals of compact operators on H; in particular, S^(H) denotes the ideal of compact operators in H.
By dom (T), ran (T) and a(T) we denote the domain, range and spectrum of the operator T, respectively. The symbols ap(-), ac(-) and ar(•) stand for the point, continuous and residual spectrum of a linear operator. Recall that z e ac(H) if ker (H — z) = {0} and ran (H — z) = ran (H — z) = H; z e ar(H) if ker (H — z) = {0} and ran (H — z) = H.
2. Preliminaries
2.1. Linear relations
A linear relation 6 in H is a closed linear subspace of H®H. The set of all linear relations in H is denoted by C(H). Denote also by C(H) the set of all closed linear (not necessarily densely defined) operators in H. Identifying each operator T e C (H) with its graph gr (T) we regard C (H) as a subset of C(H).
The role of the set C(H) in extension theory becomes clear from Proposition 2.3. However, its role in the operator theory is substantially motivated by the following circumstances: in contrast to C(H), the set C(H) is closed with respect to taking inverse and adjoint relations 6_1 and 6*. The latter are given by: 6_1 = {{g, f} : {f,g} e 6} and
6* = { (£) : (h', k) = (h, k') for all (fy e ^ .
A linear relation 6 is called symmetric if 6 C 6* and self-adjoint if 6 = 6*.
2.2. Boundary triplets and proper extensions
Let us briefly recall some basic facts regarding boundary triplets. Let S be a densely defined closed symmetric operator with equal deficiency indices n±(S) := dim(N±i), Nz := ker (S* — z), z e C±, acting on some separable Hilbert space H.
Definition 2.1.
(i) A closed extension S of S is called proper if dom (S) C dom (S) C dom (S*).
(ii) Two proper extensions S', S are called disjoint if dom (S') fl dom (S) = dom (S) and transversal if in addition dom (S') + dom (S) = dom (S*).
We denote by ExtS the set of all proper extensions of S completed by the non-proper extensions S and S* is denoted. Any self-adjoint or maximal dissipative (accumulative) extension is proper.
Definition 2.2 ( [7]). A triplet П = {H, Г0, Г}, where H is an auxiliary Hilbert space and Г0, Г1 : dom (S*) ^ H are linear mappings, is called a boundary triplet for S* if the "abstract Green's identity"
(S*f,g) - (f,S*g) = (Г1 f, Год) - (Го/, Г^), f,g E dom (S*). (2.1)
is satisfied and the mapping Г := (Г0, Г1 )т : dom (S *) ^H®H is surjective, i.e. ran (Г) = H®H. ♦
A boundary triplet П = {H, Г0, Г1} for S* always exists whenever n+(S) = n-(S). Note also that n±(S) = dim(H) and ker (Г0) П ker (Г1) = dom (S).
In general, the linear maps Г : H —> H, j = 0,1, are neither bounded nor closed. However, equipping the domain dom (S*) with the graph norm
\\f := l|S*||2 + \\f II2, f E dom (S*), (2.2)
one gets a Hilbert space, which is denoted by H+(S*), and regarding the maps Г : H —> H, j = 0,1, as acting from H+(S*) into H it turns out that that the operators Г : H+(S*) —> H, j = 0,1, are bounded. In the following work we denote the operator Г : H+(S*) —> H by Г j : H+(S*) —> H, j = 0,1. From surjectivity it follows that ran (Г) = H®H, where Г := (Г 1, Г 1). Notice that the abstract Green's identity (2.1) can be written as
( S* f, g) - (f, S* g) = (Г 1 f, Г og) - (Г о f, Г 1g), f,g E dom(S*). (2.3)
where S* denotes the operator S* regarded as acting from H+(S*) into H.
With any boundary triplet П one associates two canonical self-adjoint extensions Sj := S* \ ker (Г), j E {0,1}. Conversely, for any extension S0 = SO E Exts there exists a (non-unique) boundary triplet П = {H, Г0, Г1} for S* such that S0 := S* \ ker (Г0).
Using the concept of boundary triplets one can parameterize all proper extensions of A in the following way.
Proposition 2.3 ( [8,9]). Let П = {H, Г0, Г1} be a boundary triplet for S*. Then the mapping
Exts Э S ^ Гdom (S) = (Гof, Гlf)т : f E dom (S)} =: в E C(H) (2.4)
establishes a bijective correspondence between the sets Exts and C(H). We write S = S© if S corresponds to в by (2.4). Moreover, the following holds:
(i) S© = S©*, in particular, S© = S© if and only if в* = в.
(ii) S© is symmetric (self-adjoint) if and only if в is symmetric (self-adjoint).
(iii) The extensions S© and S0 are disjoint (transversal) if and only if there is a closed (bounded) operator B such that в = gr (B). In this case (2.4) takes the form
S© := Sgr(S) = S* Г ker (Г1 - ВГо). (2.5)
In particular, Sj := S* \ ker (Г) = S©j, j E {0,1}, where в0 := {0} x H and в1 := Hx {0} = gr (O) where O denotes the zero operator in H. Note also that C(H) contains the trivial linear relations {0} x {0} and HxH parameterizing the extensions S and S*, respectively, for any boundary triplet П.
2.3. 7-fleld and Weyl function
It is well known that Weyl functions are important tools in the direct and inverse spectral theory of Sturm-Liouville operators. In [8,11] the concept of Weyl function was generalized to the case of an arbitrary symmetric operator S with n+(S) = n-(S) ^ x>. Following [8], we briefly recall basic facts on Weyl functions and 7-fields associated with a boundary triplet n.
Definition 2.4 ( [8,11]). Let n = {H, r0, rj be a boundary triplet for S* and S0 = S* \ ker(r0). The operator valued functions 7^) : p(S0) ^ [H, H] and M(•) : p(S0) ^ [H] defined by
7(z) := (r0 r Nz)-1 and M(z) := ^7(z), z e PS0), (2.6)
are called the 7-field and the Weyl function, respectively, corresponding to the boundary triplet n.
Clearly, the Weyl function can equivalently be defined by
M(z)rfz = rfz, fz e Nz, z e pS).
The 7-field 7(•) and the Weyl function M(•) in (2.6) are well defined. Moreover, both 7(•) and M(•) are holomorphic on p(S0) and the following relations
7(z) = (I + (z — Z)(S0 — z)-1)7(Z), z, Z e p(S0), (2.7)
and
M(z) — M(Z)* = (z — Z)7(Z)*7(z), z, Z e p(S0), (2.8)
hold. Identity (2.8) yields that M(•) is [H]-valued Nevanlinna function (M(•) e R[H]), i.e. M(•) is [H]-valued holomorphic function on C± satisfying
M(z) = M(z)* and Im(M(Z)) ^ 0, z e C+ U C_.
Im(z)
It also follows from (2.8) that 0 e p(Im(M(z))) for all z e C±.
2.4. Krein-type formula for resolvents
Let n = {H, r0, r1} be a boundary triplet for S*, M(•) and 7(•) the corresponding Weyl function and 7-field, respectively. For any proper (not necessarily self-adjoint) extension S© e ExtS with non-empty resolvent set p(S©) the following Krein-type formula holds
(cf. [8,11,12])
(S© — z)-1 — (S0 — z)-1 = 7(z)(6 — M(z))-17*(z), z e p(S0) f p(S©). (2.9)
Formula (2.9) extends the known Krein formula for canonical resolvents to the case of any S© e ExtS with p(S©) = Moreover, due to relations (2.4), (2.5) and (2.6) formula (2.9) is connected with the boundary triplet n. We emphasize, that this connection makes it possible to apply the Krein-type formula (2.9) to boundary value problems.
2.5. Operator spectral integrals
Let us recall some useful facts regarding operator spectral integrals. We follow in essentially [10, Section I.5.1].
Definition 2.5. Let E(•) be a spectral measure defined on the Borel sets B of the real axis R. Let us assume that the support supp (E) is a bounded set, i.e. supp (E) C [a, b), —<x < a < b < <x. Further, let G(^) : [a, b) —> B(H) be a Borel measurable function. Let Z be a partition of the interval [a, b) of the form [a, b) = [A0, Ai) U [Ai, A2) U ••• U [An_;L, An) where A0 = a and An = b, and put Am := [Am_1, Am), m = 1,...,n. Thus [a, b) = (J nm=\ Am and the intervals Am are pairwise disjoint. Let |Z| := maxm |Am| and let
n
Fz(G) G(xm)E(Am), Xm G Am.
m=1
If there is an operator F0 G B(H) such that lim|3|^0 ||FZ(G) — F0|| =0 independent of Z and {xm}, then F0 is called the operator spectral integral of G(-) with respect to E(•) and is denoted by
Fo = Í G(A)dE(A).
J a
Remark 2.6. Similarly the operator spectral integral J^ dE(A)G(A) can be defined as above.
If f (•) : [a, b) —> H is a Borel measurable function, then the vector spectral integral /a dE(A)f (A) can be defined similarly.
Let us indicate some properties of the operator spectral integral.
(i) If G(A) := g(A)I where #(•) G C([a, b]), then J^ G(A)dE(A) exists and coincides with scalar spectral integral J^ g(A)dE(A).
(ii) If a G(A)dE(A) exists and &(•) G C([a,b]), then also £ h(A)G(A)dE(A) exists and one has
r-b r-b r-b
/ h(A)G(A)dE(A) = G(A)dE(A) h(A)dE(A).
J a J a J a
Proposition 2.7 (Proposition I.5.1.2 of [10]). Let G(^) be defined on [a,b) and assume the existence of the derivative G'(A) with respect to the operator norm on [a,b). Further, let G'(^) be Bochner integrable on [a,b) and assume that A(A) = A(a) + £ G'(x)dx. Then ¡¡Ob G(A)dE(A) exists and the estimate
G(X)dE (A)
^ ||G(a)| W ||G'(A)|dA
is valid.
Similar existence theorems can be proven for the other types of spectral integrals. For instance the vector spectral integral exists if f (•) is strongly continuous, strongly differentiate on [a, b] and if f'(^) is also strongly continuous. In particular, the operator and vector spectral integrals exist if the integrands G(^) and f (•) are holomorphic.
b
b
3. Main results
Let A be a closed symmetric operator with equal deficiency indices acting in the separable Hilbert space Ha and let T be a bounded self-adjoint operator acting in the separable Hilbert space . We consider the operator S = A ® IT + IA ® T. To define
the operator S we first consider the operator A 0 IT. The operator A 0 IT is defined as the closure of the operator A © Ii defined by
dom (A © It) := < f = gk 0 hk : gk e dom (A), hk eHt, r e N
I k=1
and
r
(A © It)f = ^2 Agk 0 hk, f e dom (A © It). k=1
One can easily check that A © IT is a densely defined symmetric operator which yields that A 0 IT is a densely defined closed symmetric operator. By H+(A) we denote Hilbert space which is obtained equipping the domain dom (A) with the graph norm of A, cf. (2.2). dom (A 0 IT) = H+(A) 0 HT. By Proposition 7.26 of [2] we have (A 0 IT)* = A* 0 IT. Its domain is given by dom (A* 0 IT) = H+(A*) 0 HT.
Similarly, the operator IA 0 T can be defined. IA 0 T is found to be a bounded self-adjoint operator with norm \\T||. The operator S := A 0 IT + IA 0 T is a well-defined closed symmetric operator with domain dom (A 0 IT). Notice that
5 = A © It + I a © T = A © It + I a 0 T.
Its adjoint is given S * = A* 0 IT + IA 0 T.
Let r j := ?A 0It : H+(A*) 0Ht —► Ha0Ht, j = 0,1. Since ran (fA ) = Ha®Ha we have ran(r) = (HA 0 HT) © (HA 0 HT) where r := (To , Ti ). Let us consider the embedding operator J : H+(A*) 0 HT —> HA 0 HT. We introduce the operator Tj : dom (A* 0 It) —>Ha 0Ht by setting
T,Jf := Tj f, f E H+(A*) 0Ht, j = 0,1. (3.1)
Notice that ran ( J) = dom (A* 0 It). Since ran(f ) = (Ha 0 Ht) © (Ha 0 Ht) we get ran (r) = (HA 0 HT) © (HA 0 HT) where r = (r0, r1). Let us introduce the triplet n = {H, r0, r1} where H := HA 0 HT and r, are given by (3.1).
Proposition 3.1. If nA = {HA, rA, rA} is a boundary triplet for A*, then n = {H, r0, r1} is a boundary triplet for S *
Proof. First, we are going to show that n is a boundary triplet for (A 0 Ii)* = A* 0 IT. The surjectivity of r = (r0, r1) was already shown above. Next, we check that the "Green's
N M
identity" holds Let gk, g'k E H+(A*), hk, h'k E HT so that f = gk 0 hk and f ' = gj 0 hj.
k=1 j=1
We have
N M N M
(A* 0 It )jJ2gk 0 hk, Jj^gj 0 K) - J^gk 0 hk, J(A* 0 It ) ^ j 0 hj k=1 j=1 k=1 j=1
N M
= Y,Y.(hk ,h'j ) [(A*Ja* gk, Ja* gjj ) - (Ja* gk ,A*Ja* gj )] k=1 j=1
N M
= Y,Y,(hk,h'j ) [(fAJa* gk, rA Ja* gj ) - (rAJA* gk, tAJa* gj )] k=1 j=1
where JA* : H+(A*) —> HA is the embeding operator. Similarly we get
N M N M
(rj^gu 0 hk, To J 9'j 0 K) - (ro Jj29k 0 hk, rJ S 9'j 0 j k=1 j=1 k=l j=1
N M
= E5>k ,hj) [(J gk, J gj) - (rA J a* gu, rj gj )] • k=i j=i
Hence we get
N M N M
' (A* 0 It )Jy,gk 0 hk, J^gj 0 hj) - J^gk 0 hk, J (A* 0 It ) ^ gj 0 hj k=1 j=1 k=1 j=1
N M N M
iJ 2^gk 0 hk,LoJ^gj 0 hj) - {LoJ^gk 0 hk,r ij ¿^g^-i
k=1 j=1 k=1 j=1
N M N M
^J^gk 0 hk, ToJ^^gj 0 hj) - (ToJ gk 0 hk, r J^gj 0 hj
which yields
'(A* 0 It)jY,gk 0 hk, J^gj 0 hj) - J^gk 0 hk, (A* 0 It) J^gj 0 hj
k=1 j=1 k=1 j=1
N M N M
r^ gk 0 hk, f0^2 gj 0 hj) - ( f o J] gk 0 hk, f 1 J] gj 0 hj k=1 j=1 k=1 j=1
NM
jj
Since elements of the form f = gk 0 hk and f' = gj 0 hj are dense in H+ (A*) the
k=1 j=1 equality can be closed which gives
((A* 0 It )Jf, Jf) - Jf, (A* 0 It )Jf) = (f1 f, fo f) - ( fo f, f1 f)
for f, f' E H+(A*) 0HT which immediately yields the abstract Green's identity for A* 0IT. Hence n is a boundary triplet A* 0 IT. Since TA 0 T is a bounded self-adjoint operator one proves that n is a boundary for S*. Indeed, since dom (A* 0 IT) = dom (S*) one immediately verifies the abstract Green's identity and rdom (A* 0 IT) = rdom (S*) shows the surjectivity. □
Let us also mention that S0 := S* \ ker (rf) admits the representation
So = Ao 0 Iht + IHa 0 T. (3.2)
Let ET(A), A e R, be the spectral measure of the self-adjoint operator T. Obviously,
Et (A) := I A 0 ET (A), A E R,
defines a spectral measure on HA 0 HT.
Proposition 3.2. Let nA be a boundary triplet for A* with y-field jA(z). If ns is the boundary triplet of Proposition 3.1 of S*, then the y-field ys(•) of ns admits the representation
r-b r-b
Ys(z)= dET(A) ya(z - A) 0 H = Ya(z - A) 0 IHt dET(A) (3.3)
J a J a
z E C± where a(T) C [a, b).
Proof. We set G(A) := jA(z - A) 0 IT, A e [a, b). From (2.7) we get
G'(A) = (Ao - C)(Ao - z + A)-2ya(Z) 0 It, A e R. Since £ \\G'(A)\ dA < ro the operator spectral integral
D(z) := / ya(z - A) 0 IHt dET(A) (3.4)
jr
exists by Proposition 2.7. We will show that ran (D(z)) C H+(S* - z). Let Z be a partition of [a, b) and let us consider the Riemann sum
n
D3(z):=J] Ya(z - Ak) 0 It Et(Ak), Ak e Ak. (3.5)
k=1
For every z e C± one has lim|3|^0 \\D3(z) - D(z)\\ = 0. Obviously, for each Z we have Dzf e H+(S*), f e H. Let us estimate the operator norm of (ya(z - Ak) 0 IT) ET(Ak) with respect to the Hilbert space H+(S* - z). Obviously we have
(S* - z)(ya(z - Ak) 0 It) Et(Ak)
= (A* - z)ya(z - Ak) 0 Et (Ak) + ya(z - Ak) 0 TEt (Ak).
which yields
(S* - Z)(Ya(Z - Ak) 0 It ) Et (Ak) = ya(z - Ak) 0 (TEt (Ak) - Ak Et (Ak)) Hence we find
\\(S * - z )(ya(z - Ak) 0 It ) Et (Ak )\\ ^ \\ya(z - Ak )\\ \\TEt (Ak) - Ak Et (Ak )\\.
Since \\TET(Ak) - AkET(Ak)\ ^ |Ak|, where |-| is the Lebesgue measure of the set Ak, we find
\\(S* - z )(ya(z - Ak) 0 It ) Et (Ak )\\ ^ \\ya (z - Ak )\\ |Ak |. Using that CYA(z) := supAe[ab) \\ya(z - A)\ < we immediately get the estimate
\\(S* - z)Dz(z)\ ^ C7a(z)(b - a), z e C±. (3.6)
In particular we get \\(S* - z)D(z)\ ^ CYA(z)(b - a), z e C±. Let us show that the integral D(z) also exists in the strong sense in H+(S* - z).
(S* - z)Dz(z)g 0 h = ((A* - z) 0 It)D3(z)g 0 h + (Ia 0 T)D3(z)g 0 h
((A* - z) 0 It Ya(z - Ak )g 0 Et (Ak )h + (Ia 0 T ^ ya(z - Ak )g 0 Et (Ak )h k=1 k=1
nn
-Ak YA(z - Ak )g 0 Et (Ak )h + Ya(z - Ak)g 0 TEt (Ak )h k=1 k=1
n
Y,Ya(z - Ak )g 0 (TEt (Ak) - Ak Et (Ak ))h k=1
n
Y,Ya(z - Ak )g 0 (TEt (Ak) - Ak Et (Ak ))h
k=1 Hence
\\ (S * - z)Dz(z )g 0 h\\ = Y, \\YA(z - Ak )g 0 (TEt (Ak) - Ak Et (Ak ))h\\,
k=1
we have
n
\\(S * - z)Dz (z)g 0 h\\ ^ \\ya(z - Ak )g\\ \\(TEt (Ak) - Ak Et (Ak ))h\\. (3.7)
k=1
Finally we obtain
n
\\(S* - z)Dz(z)g 0 h\\ ^ \\h\\ £ \\ya(z - Ak)g\\ |Ak| . (3.8)
k=i
Let Z' be a refinement of Z, that Z' = {A'k,}n^=1 where for each k' there is always a k such that A'k, C Ak. This yields the estimate
n
\\(S* - z)(Dz(z) - Dz>(z))g 0 h\\ ^ Hh^^ ^^ \\(ya(z - A'«) - Ya(z - Ak))g\\ A| . (3.9)
k' = 1
where A'k' e Ak' C Ak 3 Ak. Hence ^^ - Ak| ^ |Ak| ^ |Z|. Using (2.7) we find
So - z
\\ya{z - X'k,)g - ya{z - Xk^ -¡-^ sup
S0 — z + X
\\ya(z )\\ 131
which yields the estimate
1 S — Z
\\(S* - z)(Dz(z) - D3(z))g 0 h\\ ^ (b - a)\\h\\\\g\\ —- sup 0 Z
Im(z) X€[a,b)
S0 — z + X
\YA(Z )\\ 131
(3.10)
Hence the Riemann sums Dz(z) converge strongly in H+(S* — z) as |3| ^ 0. Since the Hilbert spaces H+(S*) and H+(S* — z), z E C±, are isomorph the Riemann sums converge strongly in H+(S*).
It remains to show that (S* — z)D(z) = 0. Recall that
n
(S * - z)Dz(z )g 0 h = J]((A* - z)Ya(z - Xk )g 0 Et (Ak )h + ya(z - Xk)g 0 TEt (Ak )h)
k=i
n
= 5] Ya(z - Xk)g 0 (TEt (Ak ) - Xk Et (Ak ))h.
k=i
For instance,
\\ (S * - z)Dz(z )g 0 h\\ = £ \\ya(z - Xk )g 0 (TEt (Ak ) - Xk Et (Ak ))h\\
k=i
£>a(z - Xk)g\\\\(T - Xk)Et(Ak)h\\
k=i
To the degree that \\ya(z — Xk)\\ is bounded, we have ||(S* — z)Dz(z)g 0 h\\ ^ 0 as |3| ^ 0
n
for any g 0 h. For the element of the form f = gk 0 hk obviously the same result holds.
k=1
n
Then, we use that the set of f = Y1 gk 0 hk is dense in H+(S*) □
k=1
Proposition 3.3. Let nA be a boundary triplet for A* with Weyl function MA(-). If nS is the boundary triplet of Proposition 3.1 of S*, then the Weyl function MS(•) of nS admits the representation
Ms (Z )= i dET (A) Ma(Z - A) 0 IHt
Jab (3.11)
= Ma(Z - A) 0 IHT dET (A),
a
z E C± where a(T) C [a, b). In particular, if n±(A) = 1, then MA(-) is scalar, HS = HT and
Ms (Z ) = Ma(Z - T), Z E C±. (3.12)
Proof. We set G(A) := Ma(z - A) 0 It, A E [a, b). From (2.7), (2.8) we get
G'(A) = -Ya(Z)*Ya(Z - A) + (z - A - Z)ya(z)*(A - Z)(Ao - z + a)"27a(z) 0 It, A E R.
(3.13)
Since £ ||G'(A)| dA < œ the operator spectral integral
D(Z) := i Ya(Z - A) 0 H dET (A) (3.14)
Jr.
exists by Proposition 2.7.
Analogously to Proposition 3.2, we can prove that the integral exists in the strong sense in H+(S* - z) and in H+(S*), as the spaces are isomorph.
n ^
Let us note D^(z) = ya(z - Ak) 0 ITET(Ak). Then,
k=1
rf Dz(z) = rf£ ya(Z - Ak) 0 It Et (Ak) = k=1
n n
£ tAya(z - Ak) 0 Et (Ak) = J] Ma (z - Ak) 0 Et (Ak) = L3(z) k=1 k=1
As far as L^(z) and D$(z) converge in a strong sense in H+(S*) and rf is bounded in H+(S*), we get the estimate. □
Note: In case T has pure point spectrum, the formula (3.11) becomes simpler
Ms(z) = J] Ma(Z - A) 0 Zx, (3.15)
A
where is an eigenvector of T, corresponding to A 4. Example 1
In this section we will describe a simple example. Let's consider the symmetric operator A = - jx2 with the domain
dom (A) = {f E W22(0; +») : f (0) = f'(0) = 0}
in the Hilbert space L2(R). Notice that n±(A) = 1. Let's consider the following bounded self-adjoint operator
1 0
T
1 ^0 -1
acting on HT = C2. We introduce the operator S = A 0 IT + IA 0 T defined in HA 0 HT. Our goal is to get the Y-field and the Weyl function corresponding to H in terms of Y-field and the Weyl function, corresponding to A, using the results described above.
Obviously, operator A* has the deficiency indices (1;1), so it's deficiency subspace is one-dimensional. Let us calculate the boundary form of the operator A*. Integrating by parts, we get:
(A*f,g) - (f, A*g) = - i f''gdx = -f'g\+- + fg'\+™ - i fg"dx.
So
(A*f,g) - (f, A*g) = -f'g\+™ + fg'\+™.
Recall that an element f from the domain of the adjoint operator also satisfies the condition f (+ro), f'(+x>) = 0. Hence, we have:
(A*f, g) - (f, A*g) = -f (0)^(0) + f'(0)g(0).
Now we can obtain the boundary operators, corresponding to A*:
rAf = f (0), rAf = f '(0)
Recalling the result of Proposition 3.1, we introduce the boundary operators for H*:
rf f = f (0) 01, rf f = f '(0) 0 I.
Let us calculate the y-field, corresponding to A*. The deficiency element of the operator A*, corresponding to the point z, has the form: ei^zx (we choose the branch of the square root in such a way that ^^fz > 0). Applying rA, we have:
VAei^~zx = 1
so that
Ya(Z ) = 1.
Let us describe the y-field, corresponding to S*. As far as T is self-adjoint, the spectral decomposition holds:
T = P1 - P2,
where P1 and P2 are the projectors onto the invariant subspaces of the operator T, corresponding to the eigenvalues 1 and -1, respectively. The projectors have the following forms:
P1=(00), P2=(01
Using the result of Proposition (3.2), we have:
Ys (z) = Ya(z - 1) 0 P1 + Ya(z + 1) 0 P2 =(J 1 ).
The corresponding Weyl function is obviously as follows:
Ma(z ) = TAya(z) = Using the result of Proposition (3.3), we have:
Ms (z) = Ma(z - 1) 0 P1 + Ma(z + 1) 0 P2 = ( iJ0ti ).
5. Example 2
In this example we consider an operator S = A 0 IT + IA 0 B defined in HA 0 HT, HA = L2(a,b), HT = C2. Let us take the symmetric operator A as negative Laplacian A = -¿2 with the domain dom (A) = {0 e W2,2[a; b]|0(a) = 0(b) = 0'(a) = 0'(b) = 0} and a self-adjoint operator T be the same as in previous example. Let us obtain the Y-field for S. The boundary operators for A* are:
ro' =1 №) )' rf =
Then, the boundary operators for the operator S* are:
f(b) f '(a)
Fof=(01 rf= (fjW)01
(5.1)
Due to the fact that the deficiency elements of A corresponding to the point z are
ej^fzx e-iyfzx
we obtain the Y-field Ya(z) for A* in the form
Ya(z) =
—i
2^/z cos(v/i(6 — a)) \—e
e-i^za i fZe-i^~zb
iyfze
i^fzb
So, using the result of Proposition 3.3., we have:
YS(z) = YA(Z - 1) 0 (0 0 ) + ya(z + 1) 0 0 0
Direct calculation of the Weyl function for A* gives us
0 1
MA(Z )
_1__/sin^v/i(b - a)) yfz
yfz cos(^i(b - a)) V yfi y/i si^v/i(b - a)
Then,
Ms(z) = Ma(Z - 1) 0( 0 01+ Ma(Z + 1) 0 0
0 1
(5.2)
(5.3)
(5.4)
(5.5)
(5.6)
6. Concluding remarks
In this paper we considered the Y-field and the Weyl function corresponding to the boundary triplet n^ for the operator S = A 0IT + IA 0 T where the operator A is symmetric and the operator T is bounded and self-adjoint. We obtained the formulas in terms of the y-field and the Weyl function corresponding to the boundary triplet nA. The result can be immediately applied to the scattering theory due to the relation between the Weyl function and the scattering matrix (see, e.g., [13]). There is an interesting question about the case when the operator T is unbounded (it is well known that this case has many specific features [14]). We will present the corresponding result in the next paper.
Acknowledgments
Supported by Federal Targeted Program "Scientific and Educational Human Resources for Innovation-Driven Russia" (contract 16.740.11.0030), grant 11-08-00267 of Russian Foundation for Basic Researches and grants of the President of Russia (state contract 14.124.13.2045-MK and grant MK-1493.2013.1), by Program of Development of Leading Russian Universities. A part of this work was made during visits of IYP and AAB to Berlin. We thank WIAS Berlin for kind hospitality.
References
[1 [2
M.S.Birman, M.Z.Solomyak. Spectral Theory of Self adjoint Operators in Hilbert Space. Kluwer, Dordrecht (1987).
Konrad Schmudgen. Unbounded self-adjoint operators on Hilbert space, volume 265 of Graduate Texts in Mathematics. Springer, Dordrecht, 2012.
[3] B.S.Pavlov. Operator extensions theory and explicitly solvable models. Uspekhi Mat. Nauk. 42 (6). P. 99-131 (1987); English transl in: Russ. Math. Surv. 42, P. 127-168 (1987).
[4] N.Bagraev, G.Martin, B.S.Pavlov, A.Yafyasov. Landau-Zener effect for a quasi-2D periodic sandwich. Nanosystems: Phys. Chem. Math. 2 (4), P. 32-50 (2011).
[5] V.Ryzhov. A general boundary value problem and its Weyl function. Opuscula Math. 27 (2), P. 305-331.
[6] M. Malamud, H. Neidhardt. Sturm-liouville boundary value problems with operator potentials and unitary equivalence. J. Differ. Equations, 252 (11), P. 5875-5922 (2012).
[7] V. I. Gorbachuk, M. L. Gorbachuk. Boundary value problems for operator differential equations, volume 48 of Mathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht (1991).
[8] V. A. Derkach and M. M. Malamud. Generalized resolvents and the boundary value problems for Hermitian operators with gaps. J. Funct. Anal. 95 (1), P. 1-95 (1991).
[9] M. M. Malamud. Some classes of extensions of a Hermitian operator with lacunae. Ukrum. Mat. Zh. 44 (2), P. 215-233 (1992).
[10] H. Baumgartel, M. Wollenberg. Mathematical scattering theory, volume 59 of Mathematical Textbooks and Monographs, Part II: Mathematical Monographs. Akademie-Verlag, Berlin (1983).
[11] V. A. Derkach, M. M. Malamud. On the Weyl function and Hermite operators with lacunae. Dokl. Akad. Nauk SSSR, 293 (5), P. 1041-1046 (1987).
[12] V. A. Derkach, M. M. Malamud. The extension theory of Hermitian operators and the moment problem. J. Math. Sci. 73 (2), P. 141-242 (1995).
[13] J. Berndt, M. M. Malamud, H. Ntidhardt. Scattering matrices and Weyl functions. Proc. London Math. Soc. 97 (3), P. 568-598 (2008).
[14] A. N. Kochubei. On extensions of symmetric operators and symmetric binary relations. Mat. Zametki. 17, P. 41-48 (1975).
