2003 Colloquium on Differential Equations and Applications, Maracaibo, Venezuela. Electronic Journal of Differential Equations, Conference 13, 2005, pp. 101–112. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

THE DIVISION METHOD FOR SUBSPECTRA OF SELF-ADJOINT DIFFERENTIAL VECTOR-OPERATORS

MAKSIM SOKOLOV

Abstract. We discuss the division method for subspectra which appears to be one of the key approaches in the study of spectral properties of self-adjoint differential vector-operators, that is operators generated as a direct sum of self-adjoint extensions on an Everitt-Markus-Zettl multi-interval system. In the current work we show how the division method may be applied to obtain the ordered spectral representation and Fourier-like decompositions for self-adjoint differential vector-operators, after which we obtain the analytical decompositions for the measurable (relative to a spectral parameter) generalized eigenfunctions of a self-adjoint differential vector-operator.

1. Introduction

Problem Overview. We begin with a physical example of a Schrödinger vectoroperator. Gesztesy and Kirsch [6] in particular considered a Schrödinger operator generated by the Hamiltonian

H = −\frac{d^{2}}{dx^{2}} + (s^{2} − \frac{1}{4})\frac{1}{cos^{2} x} , s > 0 .  (1.1)

Since the potential in Hamiltonian has a countable number of singularities on a discreet set X in ℝ, leading to spoiling of the local integrability, it is impossible to apply the standard methods of the theory of ordinary differential operators. In order to proceed and build a self-adjoint extension of a minimal operator generated by (1.1) on ℝ \ X, one may take self-adjoint extensions T_{i}, generated by the same Hamiltonian (1.1) in the coordinate spaces L^{2}(−\frac{π}{2} + iπ, \frac{π}{2} + iπ), i ∈ ℤ, and then consider the direct sum operator ⊕_{i∈ℤ}T_{i} in the space

⊕_{i∈ℤ}L^{2}( − \frac{π}{2} + iπ, \frac{π}{2} + iπ) .

This direct sum operator appears to be one of the possible self-adjoint extensions of the minimal operator considered on ℝ \ X. Moreover in the case s ⩾ 1 the minimal

2000 Mathematics Subject Classification. 34L05, 47B25, 47B37, 47A16. Key words and phrases. Vector-operator; cyclic vector; spectral representation; ordered representation; multiplicity; unitary transformation. \begin{matrix}c \\ ◯ \end{matrix}2005 Texas State University - San Marcos. Published May 30, 2005. Supported by the grants AC-84 and MISC-03/14 from the Abdus Salam International Center for Theoretical Physics.

on ℝ \ X operator appears to be essentially self-adjoint and its only self-adjoint extension is a direct sum operator.

This physical example gave birth to the theory of general differential direct sum operators, or in the text below vector-operators. Beginning from 1992, the theory of differential vector-operators has been investigated in connection with their nonspectral properties in a Hilbert space ([1], [2, 3] and in complete locally convex spaces [4, 5]. The interest in such a theory is explained by its numerous applications in theoretical physics and pure mathematics. Thus, physical applications may be found in a single or a multi-particle quantum mechanics, especially in problems where a quantum system is split into a number of disconnected subsystems under the influence of a potential. For applications in quantum mechanics see also the respective references in [3].

As it was shown in the fundamental works [2] and [3], a differential vectoroperator is an object which resembles an ordinary differential operator by its general properties, but in fact it has much more complicated structure.

Although the bigger part of studies concerned only non-spectral properties of differential vector-operators, there has been some development of their spectral theory recently. Some results describing position of spectra of Schrödinger vectoroperators were presented in 1985 in [6] and the most recent results for general quasi-differential vector-operators belong to Sobhy El-Sayed Ibrahim [7, 8].

The internal spectral structure of abstract self-adjoint vector-operators was first investigated in [9], for which see also [10, 11]. The structure of coordinate operators as differential operators played the key role in [12] where the unitary transformation making the ordered representation was described in terms of generalized eigenfunctions of a differential vector-operator. These generalized eigenfunctions appear to be only measurable relative to the spectral parameter, therefore it is an essential problem to obtain their decomposition over some set of analytical kernels. This problem is positively solved by Theorem 2.4 of the current work.

Mathematical background. Basic concepts of quasi-differential operators are well described in [2, 3]. A good reference for operators with real coefficients is the book of Naimark [13].

Let Ω be a finite or a countable set of indices. On Ω, we have a multi-interval differential Everitt-Markus-Zettl system {I_{i}, τ_{i}}_{i∈Ω}, where I_{i} are arbitrary intervals of the real line and τ_{i} are formally self-adjoint differential expressions of a finite order. This EMZ system generates a family of Hilbert spaces {L^{2}(I_{i}) = L_{i}^{2}}_{i∈Ω} and families of minimal {T_{min,i}}_{i∈Ω} and maximal {T_{max,i}}_{i∈Ω} differential operators. Consider a respective family {T_{i}}_{i∈Ω} of self-adjoint extensions. Further, we introduce a system Hilbert space L^{2} = ⊕_{i∈Ω}L_{i}^{2}, consisting of the vectors f = ⊕_{i∈Ω}f_{i} such that f_{i} ∈ L_{i}^{2} and

∥f∥^{2} = ∑_{i∈Ω} ∥f_{i}∥_{i}^{2} = ∑_{i∈Ω} ∫_{I_{i}} |f_{i}|^{2} dx < ∞ .

In the space L^{2} consider the operator T : D(T) ⊆ L^{2} → L^{2}, defined on the domain

D(T) = {f ⊆ L^{2} : ∑_{i∈Ω} ∥T_{i}f_{i}∥_{i}^{2} < ∞}

by T f = ⊕_{i∈Ω}T_{i}f_{i}.

The operator T is called a differential vector-operator generated by the selfadjoint extensions T_{i}, or a self-adjoint differential vector-operator. If Ω is infinite, the vector-operator T is called infinite. The operators T_{i} are called coordinate operators.

The abstract preliminaries for this work may be found, for instance, in books [14, 15].

Fix i ∈ Ω. For each T_{i} there exists a unique resolution of the identity E_{λ}^{i} and a unitary operator U_{i}, making the isometrically isomorphic mapping of the Hilbert space L_{i}^{2} onto the space L^{2}(M_{i}, μ_{i}), where the operator T_{i} is represented as a multiplication operator. Below, we remind the structure of the mapping U_{i}.

We call ϕ ∈ L_{i}^{2} a cyclic vector if for each z ∈ L_{i}^{2} there exists a Borel function f, such that z = f(T_{i})ϕ. Generally, there is no a cyclic vector in L_{i}^{2} but there is a collection {ϕ^{k}} of them in L_{i}^{2}, such that L_{i}^{2} = ⊕_{k}L_{i}^{2}(ϕ^{k}), where L_{i}^{2}(ϕ^{k}) are T_{i}-invariant subspaces in L_{i}^{2} generated by the cyclic vectors ϕ^{k}. That is L_{i}^{2}(ϕ^{k}) = \overline{\{f(T_{i})ϕ^{k}\}}, for a varying Borel function f, such that ϕ^{k} ∈ D(f(T_{i})).

A vector ϕ ∈ L_{i}^{2} is called maximal relative to the operator T_{i}, if each measure (E^{i}(·)x, x)_{i}, x ∈ L_{i}^{2}, is absolutely continuous with respect to the measure (E^{i}(·)ϕ, ϕ)_{i}.

For each Hilbert space L_{i}^{2}, there exist a unique (up to unitary equivalence) decomposition L_{i}^{2} = ⊕_{k}L_{i}^{2}(φ_{i}^{k}), where φ_{i}^{1} is maximal in L_{i}^{2} relative to T_{i}, and a decreasing set of multiplicity sets e_{k}^{i}, where e_{1}^{i} is the whole line, such that ⊕_{k}L_{i}^{2}(φ_{i}^{k}) is equivalent with ⊕_{k}L^{2}(e_{k}^{i}, μ_{i}), where the measure of the ordered representation is defined as μ_{i}(·) = (E^{i}(·)φ_{i}^{1}, φ_{i}^{1})_{i}. A spectral representation of T_{i} in ⊕_{k}L^{2}(e_{k}^{i}, μ_{i}) is called the ordered representation and it is unique, up to a unitary equivalence. Two operators are called equivalent, if they create the same ordered representation of their spaces.

For i ∈ Ω, we introduce a sliced union of sets M_{i} (see also preliminaries) as a set M, containing all M_{i} on different copies of ∪_{i∈Ω}M_{i}. The sets M_{i} do not intersect in M, but they can superpose, i.e. two sets M_{i} and M_{j} superpose, if their projections in the set ∪_{i∈Ω}M_{i} intersect.

For z_{i} ∈ L_{i}^{2}, i ∈ Ω, define ẑ_{i} = {0, …, 0, z_{i}, 0, …, 0} ∈ L^{2}, where z_{i} is on the i-th place.

For each i ∈ Ω, let δ(T_{i}) denote the subspectrum of the operator T_{i}, i.e. δ(T_{i}) = σ_{pp}(T_{i}) ∪ σ_{cont}^{∗}(T_{i}), where σ_{pp}(T_{i}) is the set of eigenvalues which may be open and σ_{cont}^{∗}(T_{i}) is the continuous spectrum with a removed set of spectral measure zero. σ_{cont}^{∗}(T_{i}) may be also open. Note that \overline{δ(T_{i})} = σ(T_{i}). For instance, the subspectrum of an operator having the complete system of eigenfunctions with eigenvalues being the rational numbers of [0, 1] equals to ℚ ∩ [0, 1]; the subspectrum of an operator having the continuous spectrum [0,1] is assumed to equal to (0,1) without loss of generality. The notion of the subspetrum arises quite naturally. Indeed, let we are given a self-adjoint operator A with a simple spectrum σ(A) = [a, b]. Choosing any point ξ ∈ σ(A) we can obtain σ(A) = [a, ξ) ∪ [ξ, b]. If we are interested in obtaining a formula A_{1} ⊕ A_{2} = A, where σ(A_{1}) = [a, ξ] and σ(A_{2}) = [ξ, b], we have to suppose that ξ ∉ σ_{pp}(A). But if we pass to subspectra, we will not need to care about inessential points appearing as limit points.

Consider a projecting mapping P : M → ∪_{i∈Ω}M_{i} such that P (δ(T_{i})) = δ(T_{i}). Let Ω = ∪_{k=1}^{K}A_{k}, A_{k} ∩ A_{s} = ∅ for k ≠ s and

A_{k} = {s ∈ Ω : ∀s, l ∈ A_{k}, s ≠ l , P (δ(T_{s})) ∩ P (δ(T_{l})) = B_{sl} ,

where ∥E^{t}(B_{sl})φ_{t}∥_{t}^{2} = 0 for any cyclic φ_{t} ∈ L_{t}^{2}, t = s, l}. From all such divisions of Ω we choose and fix the one, which contains the minimal number of A_{k}. In case when all the coordinate spectra σ(T_{i}) are simple, we define the number Λ = min{K} as the spectral index of the vector-operator T.

The following two lemmas were proved in [9].

Lemma 1.1. The identity resolution {E_{λ}} of the vector-operator T equals to the direct sum of the coordinate identity resolutions {E_{λ}^{i}}, that is {E_{λ}} = ⊕_{i∈Ω}{E_{λ}^{i}}

Lemma 1.2. Let each T_{i} have a cyclic vector a_{i} in L_{i}^{2}. Then the vector-operator T has minimum Λ cyclic vectors {a_{k}}_{k=1}^{Λ}, having the form a_{k} = ∑_{i∈A_{k}} â_{i}.

In the next section we will see what a spectral multiplicity of a vector-operator is. Nevertheless, this notation is intuitively clear. Running ahead, let us present here two examples, which will show the difference between the spectral index and the spectral multiplicity of the vector-operator T. Example 1. We have a three-interval EMZ system {I_{i}, τ_{i}, 1}_{i=1}^{3} (a kinetic energy, a mirror kinetic energy, an impulse):

I_{1} = [0, +∞), τ_{1} = −(\frac{d}{dt})^{2},

D(T_{1}) = {f ∈ D(T_{max,1}) : f(0) + kf′(0) = 0, −∞ < k ⩽ ∞};

I_{2} = [0, +∞), τ_{2} = (\frac{d}{dt})^{2},

D(T_{2}) = {f ∈ D(T_{max,2}) : f(0) + sf′(0) = 0, −∞ < s ⩽ ∞};

I_{3} = [0, 1], τ_{3} = \frac{1}{i} \frac{d}{dt},

D(T_{3}) = {f ∈ D(T_{max,3}) : f(0) = e^{iα}f(1), α ∈ [0, 2π]}.

(a) If k, s ∈ (−∞, 0] ∪ {+∞} then

δ(T_{1}) = (0, +∞), δ(T_{2}) = (−∞, 0), δ(T_{3}) = ⋃_{n=−∞}^{∞}(2πn − α).

For this system we have: {1, 2, 3} = ∪_{k=1}^{2}A_{k} and A_{1} = {1, 2}, A_{2} = {3}. Thus, here the spectral index does not coincide with the spectral multiplicity (which is 1) and equals 2.

(b) The case 0 < k, s < +∞ leads to

δ(T_{1}) = \{−\frac{1}{k^{2}} \} ∪ (0, +∞), δ(T_{2}) = (−∞, 0) ∪ \{\frac{1}{s^{2}} \}, δ(T_{3}) = ⋃_{n=−∞}^{∞}(2πn − α).

If

α ∉ [ ⋃_{n=−∞}^{∞}(2πn + \frac{1}{k^{2}} )] ⋃ [ ⋃_{n=−∞}^{∞}(2πn − \frac{1}{s^{2}} )],

we have A_{1} = {1}, A_{2} = {2}, A_{3} = {3}. That is Λ = 3 but ⊕_{i=1}^{3}T_{i} has a simple spectrum. Example 2. Let us have a vector-operator ⊕_{i=1}^{3}T_{i} with

δ(T_{1}) = ⋃_{n∈ℤ,n⩾0} n, δ(T_{2}) = ⋃_{n∈ℤ,n⩽0} n, δ(T_{3}) = ⋃_{n∈ℤ,n≠0} n.

The spectral index is 3 but spectral multiplicity is 2.

Definition 1.3. A vector-operator T = ⊕_{i∈Ω}T_{i} with the simple coordinate spectra σ(T_{i}) is called distorted if its spectral index does not equal its spectral multiplicity.

Generally it is not possible to build a spectral representation for a distorted vector-operator without applying the division method, but in some cases (Example 1) Theorem 1.2 may lead to the construction of a spectral representation even for some distorted vector-operators. If we want to obtain an ordered spectral representation for any self-adjoint vector-operator, only implementation of the division method can achieve this.

2. The division method for subspectra (DMS)

Below we present the three theorems (2.1, 2.2 and 2.3) without their complete proofs. Only the structural parts of the proofs essential for the demonstration of the DMS are presented. The complete proof of Theorem 2.1 may be found in [11, 10] and refer to [12] for the proofs of Theorems 2.2 and 2.3.

Theorem 2.1. If θ_{i} and {e_{n}^{i}}_{n=1}^{m_{i}} are measures and multiplicity sets of ordered representations for coordinate operators T_{i}, i ∈ Ω, then there exist processes P r_{1} and P r_{2}, such that the measure

θ = P r_{1}({θ_{i}}_{i∈Ω})

is the measure of an ordered representation and the sets

s_{n} = P r_{2}(\{e_{k}^{i}\}_{i∈Ω; k=\overline{1,m_{i}}})

are the canonical multiplicity sets of the ordered representation of the operator T. Thus, the unitary representation of the space L^{2} on the space ⊕_{n}L^{2}(s_{n}, θ) is the ordered representation and it is unique up to unitary equivalence.

Proof. We divide the proof into units for convenience. Parts (A) and (B) represent the DMS. (A) Let a_{i} be maximal vectors relative to the operators T_{i} in L_{i}^{2}. We want to find a maximal vector relative to the vector-operator T. We know, that the vector ⊕_{i∈Ω}a_{i} does not give a single measure, if a set P (δ(T_{i})) ∩ P (δ(T_{j})) has a non-zero spectral measure for i ≠ j. Consider restrictions T_{i}|_{L_{i}^{2}(a_{i})} = T′_{i}. Since all the operators T′_{i} have single cyclic vectors a_{i}, we can divide Ω into A_{k}, k = \overline{1, Λ} and apply Lemma 1.2 for the operator ⊕_{i∈Ω}T′_{i}. Then we have derive Λ vectors a^{k} = ⊕_{j∈A_{k}}a_{j}, which are maximal in the respective spaces L^{2}(a^{k}) = ⊕_{j∈A_{k}}L_{j}^{2}(a_{j}). (B) Let now 1 < Λ < ∞. Define T^{k} = ⊕_{j∈A_{k}}T′_{j}. For any two operators T^{k} and T^{s}, k ≠ s, let us introduce the sets δ_{k,s} = P (δ(T^{k}))∩P (δ(T^{s})) and δ_{k} = P (δ(T^{k}))\δ_{k,s}. There exist unitary representations

U^{k} : L^{2}(a^{k}) → L^{2}(ℝ, μ_{a^{k}}).

Consider measures μ_{k} and μ_{k,s}, defined as

μ_{k,s}(e) = μ_{a^{k}}(e ∩ δ_{k,s})

and μ_{k}(e) = μ_{a^{k}}(e ∩ δ_{k}), for any measurable set e. For any operator T^{k} (with respect to T^{s}), measures μ_{k} and μ_{k,s} are mutually singular and μ_{k} + μ_{k,s} = μ_{a^{k}}; therefore,

L^{2}(ℝ, μ_{a^{k}}) = L^{2}(ℝ, μ_{k}) ⊕ L^{2}(ℝ, μ_{k,s}).

This means that (according to our designations):

U^{k}^{−1} : L^{2}(ℝ, μ_{a^{k}}) ⟶ L^{2}(a_{k}^{k}) ⊕ L^{2}(a_{k,s}^{k})

and

a^{k} = a_{k}^{k} ⊕ a_{k,s}^{k},  (2.1)

where a_{k}^{k} and a_{k,s}^{k} form the measures μ_{k} and μ_{k,s} respectively. Define also as max{w, ψ} the vector, which is maximal of the two vectors in the brackets (Note that this designation is valid only for vectors, considered on the same set. In order not to complicate the investigation we assume here that any two vectors are comparable in this sense. In order to achieve this, it is enough to decompose each coordinate operator T_{i} into the direct sum T_{i}^{pp} ⊕ T_{i}^{cont}, where the operators have respectively pure point and continuous spectra. Then after redesignation we obtain the equivalent vector-operator to the initial vector-operator ⊕T_{i}).

Consider first two operators T^{1} and T^{2}. It is clear, that the vector

a^{1⊕2} = a_{1}^{1} ⊕ a_{2}^{2} ⊕ max {a_{1,2}^{1}, a_{2,1}^{2}}

is maximal in L^{2}(a^{1}) ⊕ L^{2}(a^{2}). Note that a_{1}^{1} and a_{2}^{2} and they both may equal zero. The maximal vector in L^{2}(a^{1}) ⊕ L^{2}(a^{2}) ⊕ L^{2}(a^{3}) will have the form:

a^{1⊕2⊕3} = a_{1⊕2}^{1⊕2} ⊕ a_{3}^{3} ⊕ max {a_{1⊕2,3}^{1⊕2}, a_{3,1⊕2}^{3}} ,

where a_{1⊕2}^{1⊕2} is the narrowed vector a^{1⊕2}, corresponding to the set which is free from the superposition with δ(T_{3}), as shown in (2.1).

Continuing this process, we obtain a maximal vector in the main space L^{2}:

a^{1⊕···⊕Λ} = a_{1⊕···⊕Λ−1}^{1⊕···⊕Λ−1} ⊕ a_{Λ}^{Λ} ⊕ max {a_{1⊕···⊕Λ−1,Λ}^{1⊕···⊕Λ−1}, a_{Λ,1⊕···⊕Λ−1}^{Λ}} .  (2.2)

Let Λ = ∞. We obtain a^{1⊕···⊕Λ} as a vector which satisfies the following equality:

∥[⊕_{i∈Ω}E^{i}(·)] a^{1⊕···⊕Λ}∥^{2} = lim_{L→∞} ∥[⊕_{j=1}^{L}E^{j}(·)] a^{1⊕···⊕L}∥^{2} ,  (2.3)

since the limit on the right side exists. (C) The next step is to build the measure of the ordered representation for the vector-operator. From Lemma 1.1 and the reasonings above, it follows that such a measure will be

θ(·) = ([⊕_{i∈Ω}E^{i}(·)] a^{1⊕···⊕Λ}, a^{1⊕···⊕Λ}) .

(D) The canonical multiplicity sets s_{n} of the vector-operator have the form:

s_{n} = [ ⋃_{i} P (e_{n}^{i})] ⋃ [ ⋃ ; ∑ m_{i}⩾n ⋂ P (e_{m_{i}}^{i}\e_{m_{i}+1}^{i}) ].  (2.4)
□

Example 3. Let us have a vector-operator T = ⊕_{i=1}^{5}T_{i}, generated by five coordinate operators with simple continuous spectra:

δ(T_{1}) = (0, 2), δ(T_{2}) = (1, 3), δ(T_{3}) = (2, 4), δ(T_{4}) = (3, 6), δ(T_{5}) = (0, 4).

Divide Ω = {1, 2, 3, 4, 5} into A_{k}:

A_{1} = {1, 3}, A_{2} = {2, 4}, A_{3} = {5}.

The spectral index Λ is 3 and we pass to the three operators

T^{1} = T_{1} ⊕ T_{3}, T^{2} = T_{2} ⊕ T_{4}, T^{3} = T_{5},

with maximal elements respectively:

a^{1} = a_{1} ⊕ a_{3}, a^{2} = a_{2} ⊕ a_{4}, a^{3} = a_{5}.

Consider first two sub-vector-operators T^{1} and T^{2}. Find elements a_{1}^{1}, a_{1,2}^{1}, a_{2}^{2}, a_{2,1}^{2}. Since the spectra are continuous, we may assume that

max{a_{1,2}^{1}, a_{2,1}^{2}} = a_{1,2}^{1}.

We derive a maximal vector a^{1⊕2} in the space ⊕_{i=1}^{4}L_{i}^{2}:

a^{1⊕2} = a_{1}^{1} ⊕ a_{1,2}^{1} ⊕ a_{2}^{2} = a^{1} ⊕ a_{2}^{2}.

Apply the DMS to the vectors a^{1⊕2} and a^{3}. We obtain the vecotrs

a_{1⊕2}^{1⊕2}, a_{1⊕2,3}^{1⊕2}, a_{3,1⊕2}^{3} = a^{3}, a_{3}^{3} = 0.

Eventually, we find the maximal element in ⊕_{i=1}^{5}L_{i}^{2} = ⊕_{k=1}^{3}L^{2}(a^{k}):

a^{1⊕2⊕3} = a_{1⊕2}^{1⊕2} ⊕ max{a_{1⊕2,3}^{1⊕2}, a^{3}} ⊕ 0 = a_{1⊕2}^{1⊕2} ⊕ a_{1⊕2,3}^{1⊕2} ⊕ 0 = a^{1⊕2} ⊕ 0 = a^{1} ⊕ a_{2}^{2} ⊕ 0.

It is easy to see that the multiplicity sets for the initial vector-operator are: s_{1} = ℝ, s_{2} = (0, 4), s_{3} = (1, 4).

Let us return to Examples 1 and 2. For the distorted vector-operator T_{1}⊕T_{2}⊕T_{3} from Example 1, a spectral measure will be constructed on the maximal vector a^{1⊕2⊕3}. The multiplicity sets s_{i}, i ⩾ 2 have measures zero. For the vector-operator from Example 2 two spectral measures are constructed on a^{1⊕2⊕3} and

min{a_{1,2}^{1}, a_{2,1}^{2}} ⊕ min{a_{2,3}^{2}, a_{3,2}^{3}} ⊕ min{a_{3,1}^{3}, a_{1,3}^{1}},

where the sense of the minimums is clear. The multiplicity set s_{2} will be

[P (δ(T_{1})) ∩ P (δ(T_{2}))] ∪ [P (δ(T_{2})) ∩ P (δ(T_{3}))] ∪ [P (δ(T_{3})) ∩ P (δ(T_{1}))].

Now the term ’distorted vector-operator’ is clearly explained by the form of the cyclic vectors for such an operator.

Let I = ⋁_{i∈Ω} I_{i} denote the sliced union of intervals I_{i}. Similarly, I^{k} = ⋁_{j∈A_{k}} I_{j}. If x_{i} are variables on I_{i}, then ∨x_{i} will designate a variable either on I or I^{k} depending on the context. This notation shows, that a vector-function

z = {z_{1}(x_{1}), …, z_{n}(x_{n}), … }

on I or I^{k} may be written as z(∨x_{i}). In particular, we may also write z(∨x_{i}) instead of z = ⊕_{i∈Ω}z_{i}.

Let us introduce the space ⊕_{i∈Ω}L^{∞}(I_{i}^{n}). Here, z(∨x_{i}) ∈ ⊕_{i∈Ω}L^{∞}(I_{i}^{n}) means that

sup_{i∈Ω} {ess sup_{x_{i}∈I_{i}^{n}} |z_{i}(x_{i})|} < ∞,

where for each i, families {I_{i}^{n}}_{n=1}^{∞} represent compact subintervals of I_{i}, such that ∪_{n=1}^{∞}I_{i}^{n} = I_{i}. In [4, Lemma 2.1], it was shown that ⊕_{i∈Ω}L^{∞}(I_{i}^{n}) = (⊕_{i∈Ω}L^{1}(I_{i}^{n}))^{∗}, where the space of Lebesgue-integrable vector-functions ⊕_{i∈Ω}L^{1}(I_{i}^{n}) is defined analogously to L^{2}.

We also need to introduce a symbolic integral ∫_{⋁ J_{i}} f(∨x_{i}) d(∨x_{i}) defined by:

∫_{⋁ J_{i}} f(∨x_{i}) d(∨x_{i}) = ⊕_{i} ∫_{J_{i}} f_{i}(x_{i}) dx_{i},

where f(∨x_{i}) is understood to be measurable relative to d(∨x_{i}), if f_{i}(x_{i}) are measurable relative to Lebesgue measures dx_{i}.

Theorem 2.2. Let T be a self-adjoint vector-operator, generated by an EMZ system {I_{i}, τ_{i}}_{i∈Ω}. Let U be an ordered representation of the space L^{2} = ⊕_{i∈Ω}L^{2}(I_{i}) relative to T with the measure θ and the multiplicity sets s_{k}, k = \overline{1, m}. Then there exist kernels Θ_{k}(∨x_{i}, λ), measurable relative to d(∨x_{i})×θ, such that Θ_{k}(∨x_{i}, λ) = 0 for λ ∈ ℝ \ s_{k} and (⊕_{i∈Ω}τ_{i} − λ)Θ_{k}(∨x_{i}, λ) = 0 for each fixed λ. Moreover for any bounded Borel set Δ,

∫_{Δ} |Θ_{k}(∨x_{i}, λ)|^{2} dθ(λ) ∈ ⊕_{i∈Ω}L^{∞}(I_{i}^{n}) ∀n ∈ ℕ.  (2.5)
(Uw)^{k}(λ) = lim_{n→∞} ∫_{I^{n}} w(∨x_{i}) \overline{Θ_{k}(∨x_{i}, λ)} d(∨x_{i}), w ∈ L^{2},  (2.6)

where the limit exists in L^{2}(s_{k}, θ). The kernels {Θ_{k}(∨x_{i}, λ)}_{k=1}^{n}, n ⩽ m, are linearly independent as vector-functions of the first variable almost everywhere relative to the measure θ on s_{n}.

Proof. We again need the DMS to prove this theorem. Fix i. If θ_{i} and {e_{p}^{i}}_{p=1}^{m_{i}} are respectively the measure and the multiplicity sets of an ordered representation for T_{i}, then there exists the decomposition L_{i}^{2} = ⊕_{p=1}^{m_{i}}L^{2}(e_{p}^{i}, θ_{i}), which implies T_{i} = ⊕_{p=1}^{m_{i}}T_{i}^{p} and L^{2}(e_{p}^{i}, θ_{i}) are T_{i}^{p}-invariant. For vector-operator (⊕_{i∈Ω}⊕_{p=1}^{m_{i}}T_{i}^{p}) → redesignate → ⊕_{s}T_{s}, s = {i, p} ∈ Ω_{1}, we may write Ω_{1} = ∪_{k=1}^{Λ}A_{k}.

Let us separate the proof into units for convenience. (A) For each T_{j}, j ∈ A_{k} and k = \overline{1, Λ}, there exists a single cyclic vector a_{j} ∈ L_{j}^{2} and [15, XII.3, Lemma 9 and XIII.5, Theorem 1(I)] a function W_{j}(x_{j}, λ) defined on I_{j} × e_{j} (note, that for a fixed i ∈ Ω, I_{j} = I_{i} for all p = \overline{1, m_{i}}) and measurable relative to dx_{j} × μ_{a_{j}}, such that W_{j}(x_{j}, λ) = 0, λ ∈ ℝ \ e_{j} and for any bounded Δ ⊂ e_{j}: ∫_{Δ} |W_{j}(x_{j}, λ)|^{2} dμ_{a_{j}}(λ) ∈ L^{∞}(I_{j}^{n}), n ∈ ℕ. Also

(E^{j}(Δ)F_{j}(T_{j})a_{j}) (x_{j}) = ∫_{Δ} W_{j}(x_{j}, λ)F_{j}(λ) dμ_{a_{j}}(λ),  (2.7)

for any F_{j} ∈ L^{2}(e_{j}, μ_{a_{j}}). On I^{k} = ⋁_{j∈A_{k}} I_{j}, we construct the vector-function

W^{k}(∨x_{j}, λ) = {W_{1}(x_{1}, λ), …, W_{n}(x_{n}, λ), … },

which is obviously measurable relative to d(∨x_{j})×∑ μ_{a_{j}}. Separate arguments show that this vector-function is a correctly constructed generalized eigenfunction and thus satisfies the statement of the theorem within each A_{k}.

Note that since for all p = \overline{1, m_{i}} there exists the equality (τ_{i} − λ)W_{i}^{p} = 0 (see [15, XIII.5, Theorem 1]), it is obvious that (⊕_{j∈A_{k}}τ_{j} − λ)W^{k} = 0, where τ_{j} = τ_{i} for a fixed i and all p = \overline{1, m_{i}}. If P (δ(T_{i})) ∩ P (δ(T_{j})) has zero spectral measures for all i, j ∈ Ω, then A_{k} : Ω_{1} = ∪_{k=1}^{Λ_{1}}A_{k} may be constructed such that A_{k} contains of indices \{i, k\}, i ∈ Ω, k = \overline{1, max_{i}\{m_{i}\}}. (B) Consider the set of indices Ω_{2} = {j ∈ Ω_{1} : j = {i, 1}, i ∈ Ω}. Construct A_{k} : Ω_{2} = ∪_{k=1}^{Λ_{2}}A_{k}. Apply the reasonings used in (A), considering everywhere Ω_{2} instead of Ω_{1}. Hence, for each A_{k} and we find a vector-function W_{1}^{k}(∨x_{j}, λ) which is the solution of the equation (⊕_{j∈A_{k}}τ_{j} −λ)y = 0. Consider W_{1}^{k} and W_{1}^{s} for s ≠ k. For a^{k} there exists the decomposition a^{k} = a_{k}^{k} ⊕ a_{k,s}^{k} (see the proof of Theorem 2.1). This fact induces the decomposition for W_{1}^{k}: W_{1}^{k} = W_{1,k}^{k} ⊕ W_{1,k,s}^{k}. It is clear that being the restrictions of W_{1}^{k}, the vector-functions W_{1,k}^{k} and W_{1,k,s}^{k} are also the solutions of the equation (⊕_{j∈A_{k}}τ_{j} − λ)y = 0. They, along with a_{k}^{k} and a_{k,s}^{k} define unitary transformations U_{k}^{k} and U_{k,s}^{k}, such that: U_{k}^{k} : L^{2}(a_{k}^{k}) → L^{2}(ℝ, μ_{k}) and U_{k,s}^{k} : L^{2}(a_{k,s}^{k}) → L^{2}(ℝ, μ_{k,s}) (see the proof of Theorem 2.1). This implies, that the decomposition W^{k} = W_{1,k}^{k} ⊕ W_{1,k,s}^{k} is correct.

Define as max{W_{1,k,s}^{k}, W_{1,s,k}^{s}} the vector-function, which corresponds to the vector max{a_{k,s}^{k}, a_{s,k}^{s}}, respectively min{W_{1,k,s}^{k}, W_{1,s,k}^{s}} as the vector-function which corresponds to that a_{k,s}^{k} or a_{s,k}^{s}, which is not maximal of the two. (C) Without loss of generality, suppose that k = 1 and s = 2. From the reasonings presented in Part (A) of this proof, it follows that

Θ_{1}^{1⊕2} = W_{1,1}^{1} ⊕ W_{1,2}^{2} ⊕ max {W_{1,1,2}^{1}, W_{1,2,1}^{2}}

is correctly constructed vector-function satisfying the statement of the theorem for the case T = [⊕_{j∈A_{1}}T_{j}] ⊕ [⊕_{q∈A_{2}}T_{q}]. Apply the above described process to Θ_{1}^{1⊕2} and W_{1}^{3} to obtain the correctly constructed vector-function:

Θ_{1}^{1⊕2⊕3} = Θ_{1,1⊕2}^{1⊕2} ⊕ W_{1,3}^{3} ⊕ max {Θ_{1,1⊕2,3}^{1⊕2}, W_{1,3,1⊕2}^{3}} .

Continuing this process, we finally obtain:

Θ_{1}(∨x_{i}, λ) = Θ_{1}^{1⊕···⊕Λ_{2}}

= Θ_{1,1⊕···⊕Λ_{2}−1}^{1⊕···⊕Λ_{2}−1} ⊕ W_{1,Λ_{2}}^{Λ_{2}} ⊕ max {Θ_{1,1⊕···⊕Λ_{2}−1,Λ_{2}}^{1⊕···⊕Λ_{2}−1}, W_{1,Λ_{2},1⊕···⊕Λ_{2}−1}^{Λ_{2}}},

where in the case of Λ_{2} = ∞, Θ_{1}^{1⊕···⊕Λ_{2}} is the function which satisfies (analogously to (2.3)):

∥[⊕_{i∈Ω}E^{i}(Δ)] ∫_{Δ} Θ_{1}^{1⊕···⊕Λ_{2}}dθ(λ)∥^{2} = lim_{L→∞} ∥[⊕_{j=1}^{L}E^{j}(Δ)] ∫_{Δ} Θ_{1}^{1⊕···⊕L} dθ_{L}(λ)∥^{2},

(2.8)

for any bounded Borel set Δ, where

θ_{L}(·) = ([⊕_{j=1}^{L}E^{j}(·)] a^{1⊕···⊕L}, a^{1⊕···⊕L})

is the measure of the ordered representation of the space ⊕_{j=1}^{L}L_{j}^{2}. The limit on the right side exists and in fact it appears that

∫_{Δ} Θ_{1}^{1⊕···⊕L} dθ_{L}(λ) → ∫_{Δ} Θ_{1}^{1⊕···⊕Λ_{2}}dθ(λ),

as L → ∞. (D) Define Ω_{3} = {j ∈ Ω_{1} : j = {i, 2}, i ∈ Ω}. Construct A_{k} : Ω_{3} = ∪_{k=1}^{Λ_{3}}A_{k}. Apply processes (B) and (C) of this proof, substituting everywhere Ω_{3} instead of Ω_{2}. We obtain a vector-function Θ_{2}^{1⊕···⊕Λ_{3}}, which is defined on the set ∪_{i}P (e_{2}^{i}). But, as we know (see (2.4)), the set s_{2} also includes the sets where there are non-empty superpositions of δ(T_{i}). Therefore, designating

Θ_{2}^{1} = Θ_{2}^{1⊕···⊕Λ_{3}}, Θ_{2}^{2} = min{W_{1,1,2}^{1}, W_{1,2,1}^{2}}, …,
Θ_{2}^{Λ_{2}+1} = min {Θ_{1,1⊕···⊕Λ_{2}−1,Λ_{2}}^{1⊕···⊕Λ_{2}−1}, W_{1,Λ_{2},1⊕···⊕Λ_{2}−1}^{Λ_{2}}},

we may again use the process (C) to build the vector-function Θ_{2}(∨x_{i}, λ) defined on s_{2} and Θ_{2}(∨x_{i}, λ) = 0 for λ ∈ ℝ \ s_{2}. Using processes (B), (C), (D) and formula (2.4), we finally obtain Θ_{m}(∨x_{i}, λ). (E) The linear independence is proved by separate arguments. □

Theorem 2.3 (Eigenfunction expansions). For any w ∈ L^{2}, there exists a decomposition

w = ∑_{k=1}^{m} lim_{n→∞} ∫_{−n}^{+n}(Uw)^{k}(λ)Θ_{k}(∨x_{i}, λ) dθ(λ),

Since the kernels from Theorem 2.2 are only measurable relative to λ, the following theorem is important:

Theorem 2.4. Each kernel Θ_{k}(∨x_{i}, λ), k = \overline{1, m}, may be decomposed as

Θ_{k}(∨x_{i}, λ) = ∑_{s=1}^{M_{k}} γ_{sk}(λ)σ_{sk}(∨x_{i}, λ),  (2.9)

where the M_{k} are finite for each k and σ_{sk}(∨x_{i}, λ) depend analytically on λ.

Proof. We separate the proof in parts which will correspond to the analogous parts of the proof of Theorem 2.2. (A*) Each kernel W_{j}(x_{j}, λ) from the part (A) of the proof of Theorem 2.2 may be decomposed:

W_{j}(·, λ) = ∑_{s=1}^{n_{j}} α_{js}(λ)σ_{js}(·, λ),

where α_{js} are supposed to equal zero on ℝ \ e_{j}, see [15, p. 1351]. Supplementing the defining systems with zeros where necessary, we obtain:

W^{k}(∨x_{j}, λ) = ⊕_{j∈A_{k}}W_{j}(x_{j}, λ)

= ⊕_{j∈A_{k}} ∑_{s=1}^{n_{j}} α_{js}(λ)σ_{js}(x_{j}, λ)

= ∑_{q=1}^{N_{k}} ⊕_{j∈A_{k}}α_{jq}(λ)σ_{jq}(x_{j}, λ)

= ∑_{q=1}^{N_{k}} α_{q}^{k}(λ)σ_{q}^{k}(∨x_{j}, λ),

where

N_{k} = max_{j∈A_{k}} n_{j}, α_{q}^{k}(λ) = ∑_{j∈A_{k}} α_{jq}(λ), σ_{q}^{k}(∨x_{j}, λ) = ⊕_{j∈A_{k}}σ_{jq}(x_{j}, λ).

Since e_{j} and e_{k} do not intersect almost everywhere for j, k ∈ Ω_{2}, j ≠ k, the series ∑_{j∈A_{k}} α_{jq}(λ) converges almost everywhere on ∪_{j∈Ω_{2}}P (e_{j}).

(B*) and (C*) Now pass to the part (B). There we obtained the decompositions W_{1}^{k} = W_{1,k}^{k} ⊕ W_{1,k,s}^{k} and W_{1}^{s} = W_{1,s}^{s} ⊕ W_{1,s,k}^{s}. Let us totally order the set {T^{j}}_{j=1}^{Λ_{2}} saying that T^{k} ⪯ T^{s} if max{W_{1,k,s}^{k}, W_{1,s,k}^{s}} = W_{1,k,s}^{k}. At that, T^{k} ≃ T^{s} if and only if T^{k} ⪯ T^{s} and T^{s} ⪯ T^{k}. According to this, we build ⊕_{j=1}^{Λ_{2}}T^{j}, where T^{j} ⪯ T^{j+1}, j = \overline{1, Λ_{2} − 1} if Λ_{2} ⩾ 2. The obtained vector-operator is obviously equivalent to the initial vector-operator (comprising unordered operators). Note that

W_{1}^{k}(∨x_{i}, λ) = ∑_{q=1}^{N_{k}} α_{1q}^{k}(λ)σ_{1q}^{k}(∨x_{j}, λ)

and analogously

W_{1}^{s}(∨x_{i}, λ) = ∑_{p=1}^{N_{s}} α_{1p}^{s}(λ)σ_{1p}^{s}(∨x_{j}, λ).

All the above leads to the following equality:

Θ_{1}^{1⊕2} = W_{1,1}^{1} ⊕ W_{1,2}^{2} ⊕ max {W_{1,1,2}^{1}, W_{1,2,1}^{2}} = W_{1}^{1} ⊕ W_{1,2}^{2}

= ( ∑_{q=1}^{N_{1}} α_{1q}^{1}(λ)σ_{1q}^{1}(∨x_{j}, λ)) ⊕ ( ∑_{p=1}^{N_{2}} α_{1p}^{2}(λ)χ_{δ_{2}}(λ)σ_{1p}^{2}(∨x_{j}, λ))

= ∑_{s=1}^{N^{1⊕2}} α_{1s}^{1⊕2}(λ)σ_{1s}^{1⊕2}(∨x_{j}, λ) ,

where N^{1⊕2} = max{N_{1}, N_{2}}; α_{1s}^{1⊕2}(λ) = α_{1s}^{1}(λ) + α_{1s}^{2}(λ)χ_{δ_{2}}(λ),

σ_{1s}^{1⊕2}(∨x_{j}, λ) = σ_{1s}^{1}(∨x_{j}, λ) ⊕ (σ_{1s}^{2}(∨x_{j}, λ)χ_{δ_{2}}(λ)),

s = \overline{1, N^{1⊕2}}.

Continuing this process until the finite Λ_{2}, we obtain:

Θ_{1}(∨x_{i}, λ) = Θ_{1}^{1⊕···⊕Λ_{2}} = ∑_{s=1}^{N^{1⊕···⊕Λ_{2}}} α_{1s}^{1⊕···⊕Λ_{2}}(λ)σ_{1s}^{1⊕···⊕Λ_{2}}(∨x_{j}, λ),  (2.10)

where N^{1⊕···⊕Λ_{2}} = max\{N_{1}, N_{2}, … N_{Λ_{2}}\} and for s = \overline{1, N^{1⊕···⊕Λ_{2}}}:

\begin{matrix}α_{1s}^{1⊕···⊕Λ_{2}}(λ) = α_{1s}^{1}(λ) + ∑_{i=2}^{Λ_{2}} α_{1s}^{i}(λ)χ_{δ_{i}}(λ); \\ σ_{1s}^{1⊕···⊕Λ_{2}}(∨x_{j}, λ) = σ_{1s}^{1}(∨x_{j}, λ) ⊕ (⊕_{i=2}^{Λ_{2}}σ_{1s}^{i}(∨x_{j}, λ)χ_{δ_{i}}(λ)) . \end{matrix}  (2.11)

In the case of infinite Λ_{2}, N^{1⊕···⊕Λ_{2}} is clearly finite. The series in the right side of (2.11) pointwise converges, since it consists of items defined on non-intersecting sets. σ_{1s}^{1⊕···⊕Λ_{2}}(∨x_{j}, λ) is defined by induction. (D*) Borrowing the designations from (D) and using processes described in (A*) and (C*), we shall come to the decomposition of Θ_{2}^{1⊕···⊕Λ_{3}}:

Θ_{2}^{1⊕···⊕Λ_{3}} = ∑_{s=1}^{N^{1⊕···⊕Λ_{3}}} α_{2s}^{1⊕···⊕Λ_{3}}(λ)σ_{2s}^{1⊕···⊕Λ_{3}}(∨x_{j}, λ).

To obtain Θ_{2}(∨x_{i}, λ), as in (D), we repeat part (C*) for

Θ_{2}^{1} = Θ_{2}^{1⊕···⊕Λ_{3}}, Θ_{2}^{2} = W_{1,2,1}^{2}, …, Θ_{2}^{Λ_{2}+1} = W_{1,Λ_{2},1⊕···⊕Λ_{2}−1}^{Λ_{2}}.

Finally, the same way we obtain decompositions for all Θ_{k}(∨x_{i}, λ), k = \overline{1, m}, which will have the form (2.9). □

Acknowledgments. The author is grateful to Professor R. R. Ashurov for his attention to this work

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Maksim S. Sokolov ICTP Affiliated Center, Mechanics and Mathematics Department, National University of Uzbekistan, Tashkent 700095, Uzbekistan

E-mail address: sokolovmaksim@hotbox.ru