On the Spectra of General Ordinary Quasi-Differential Operators and Their L2w-Solutions ()
1. Introduction
Akhiezer and Glazman [1] studied that the self-adjoint extension S of the minimal operator
generated by a formally symmetric differential expression
with maximal deficiency indices have resolvents which are Hilbert-Schmidt integral operators and consequently have a wholly discrete spectrum. The relationship between the square-integrable solutions for real values of the spectral parameter and the spectrum of self-adjoint ordinary differential operators of even order with real coefficients and arbitrary deficiency index are studied in [2] [3]. Sobhy E.I. has been extended their results for general ordinary quasi-differential expression
of
th order with complex coefficients [4] [5] [6] [7] [8].
The operators which fulfill the role that the self-adjoint and maximal symmetric operators play in the case of a formally symmetric expression
are those which are regularly solvable with respect to the minimal operators
and
generated by a general quasi-differential expression
and its formal adjoint
respectively, the minimal operators
and
form an adjoint pair of closed, densely-defined operators in the underlying
-space, that is
. Such an operator S satisfies
and for some
,
is a Fredholm operator of zero index, this means that S has the desirable Fredholm property that the equation
has a solution if and only if f is orthogonal to the solution space of
and furthermore the solution space of
and
have the same finite dimension. This notion was originally due to Visik in [5].
Our objective in this paper is an extension of those results in [7] [9] [10] [11] [12] [13] [14] for the general ordinary quasi-differential operators
and to investigate (according to the spectral theory) the location of the point spectra and regularity fields of general ordinary quasi-differential operators in the case of one singular end-point and when all solutions of the equations
and
are in
for some (and hence all
).
We deal throughout this paper with a general quasi-differential expression
of arbitrary order n defined by Shin-Zettl matrices [15] [16] [17] [18], and the minimal operator
generated by
in
, where w is a positive weight function on the underlying interval I. The end-points a and b of I may be regular or singular.
2. Notation and Preliminaries
We begin with a brief survey of adjoint pairs of operators and their associated regularly solvable operators; a full treatment may be found in [2] [11] ( [18], Chapter III) and [19] [20] [21] [22]. The domain and range of a linear operator T acting in a Hilbert space H will be denoted by
and
respectively and
will denote its null space. The nullity of T, written
, is the dimension of
and the deficiency of T, written
, is the co-dimension of
in H; thus if T is densely defined and
is closed, then
. The Fredholm domain of T is (in the notation of [18] ) the open subset
of
consisting of those values of
which are such that
is a Fredholm operator, where I is the identity operator in H. Thus
if and only if
has closed range and finite nullity and deficiency. The index of
is the number
, this being defined for
.
Two closed densely defined operators A and B acting in a Hilbert space H are said to form an adjoint pair if
and, consequently,
; equivalently,
for all
and
, where
denotes the inner-product on H.
Definition 2.1: The field of regularity
of A is the set of all
for which there exists a positive constant
such that:
for all
, (2.1)
or, equivalently, on using the Closed Graph Theorem,
and
are closed.
The joint field of regularity
of A and B is the set of
which are such that
,
and both
and
are finite. An adjoint pair A and B is said to be compatible if
.
Definition 2.2: A closed operator S in H is said to be regularly solvable with respect to the compatible adjoint pair of A and B if
and
, where
.
Definition 2.3: The resolvent set
of a closed operator S in H consists of the complex numbers
for which
exists, is defined on H and is bounded. The complement of
in
is called the spectrum of S and written
. The point spectrum
, continuous spectrum
and residual spectrum
are the following subsets of
(see [5] [7] [12] [14] [18] ).
,
i.e., the set of eigenvalues of S,
;
.
For a closed operator S we have:
(2.2)
An important subset of the spectrum of a closed densely defined operator S in H is the so-called essential spectrum. The various essential spectra of S are defined as in ( [18], Chapter 9) to be the sets:
(2.3)
where
and
have been defined earlier.
Definition 2.4: For two closed densely defined operators A and B acting in H, if
and the resolvent set
of S is nonempty (see [18] ), S is said to be well-posed with respect to A andB.
Note that, if
and
then
and
so that if
and
are finite, then A and B are compatible, in this case S is regularly solvable with respect to A and B. The terminology “Regularly solvable” mentioned by Visik in [4] [5] [6] [7] [8] [19] [20] [22], while the notion of “well posed” was introduced by Zhikhar in [23].
3. Quasi-Differential Expressions
The general quasi-differential expressions are defined in terms of a Shin-Zettl matrix Q on an interval I. The set
of Shin-Zettl matrices on I consists of
-matrices
whose entries are complex-valued functions on I which satisfy the following conditions:
(3.1)
For
, the quasi-derivatives associated with Q are defined by:
(3.2)
where the prime
denotes differentiation.
The general quasi-differential expression
associated with Q is given by:
(3.3)
this being defined on the set:
where
denotes the set of functions which are absolutely continuous on every compact subinterval of
.
The formal adjoint
of
is defined by the matrix
given by:
, for all
(3.4)
where
, the quasi-derivatives associated with the matrix
in
,
, for each r and s; (3.5)
are therefore:
(3.6)
Note that:
and so
. We refer to [11] [13] [16] - [22] [24] for a full account of the above and subsequent results on quasi-differential expressions.
For
,
and
, we have Green,s formula,
(3.7)
where,
(3.8)
see [4] [10] - [18] [24]. Let the interval
have end-points
, and let
be a non-negative weight function with
and
(for almost all
). Then
denotes the Hilbert function space of equivalence classes of Lebesgue measurable functions\ such that
; the inner-product is defined by:
The equation:
(3.9)
is said to be regular at the left end-point
, if for all
,
,
,
.
Otherwise (3.9) is said to be singular at a. If (3.9) is regular at both end-points, then it is said to be regular; in this case we have:
,
,
.
We shall be concerned with the case when a is a regular end-point of (3.9), the end-point b being allowed to be either regular or singular. Note that, in view of (3.5), an end-point of I is regular for (3.9), if and only if it is regular for the equation:
(3.10)
Note that: At a regular end-point a, say,
,
is defined for all
. Set:
(3.11)
The subspaces
and
of
are domains of the so-called maximal operators
and
respectively, defined by:
,
and
,
.
For the regular problem the minimal operators
and
, are the restrictions of
and
to the subspaces:
(3.12)
respectively. The subspaces
and
are dense in
and
and
are closed operators (see ( [6], Section 3) and [18] [19] [20] [21] [22] ).
In the singular problem we first introduce the operators
and
;
being the restriction of
to the subspace:
(3.13)
and with
defined similarly. These operators are densely-defined and closable in
; and we define the minimal operators
and
to be their respective closures (see [5] [11] ( [13], Section 5) [18] [22] ) and We denote the domains of
and
by
and
respectively. It can be shown that:
(3.14)
because we are assuming that a is a regular end-point. Moreover, in both regular and singular problems, we have:
(3.15)
see ( [13], Section 5) in the case when
and compare with treatment in ( [18], Section III.10.3) and [20] in general case.
For the case of one singular end-point, we consider our interval to be
and denote by
and
the minimal and maximal operators. We see from (3.15) that
and hence
and
form an adjoint pair of closed densely defined operators in
.
Lemma 3.1: For
,
is constant and:
.
In the problem with one singular end-point,
for all
.
In the regular problem,
for all
.
Proof: The proof is similar to that in [3] [4] [15] [16] [17] [18] [21] [22], and therefore omitted.
For
, we define r, s and m as follows:
,
(3.16)
and:
. (3.17)
Also,
. (3.18)
For
the operators which are regularly solvable with respect to
and
are characterized by the following theorem which proved for a general quasi-differential operator in [7] [11] [12] ( [18], Theorem 10.15) [23] [24].
Theorem 3.2: For
. Let r, s and m be defined by (3.16) and (3.17), and let
,
be arbitrary functions satisfying:
1)
are linearly independent modulo
and
are linearly independent modulo
2)
,
.
Then the set:
(3.19)
is the domain of an operator S which is regularly solvable with respect to
and
and the set:
(3.20)
is the domain of the operator
; moreover
.
Conversely, if S is regularly solvable with respect to
and
and
, then with r, s and m defined by (3.16) and (3.17) there exist functions
which satisfy 1) and 2) and are such that (3.19) and (3.20) are the domains of S and
respectively.
S is self-adjoint if, and only if,
,
and
; S is J-self-adjoint if
(J is a complex conjugate),
and
.
Proof: The proof is entirely similar to that of [7] [11] [13] [18] [23] [24] and therefore omitted.
4. The Square Integrable Solutions
The following Lemma is very important for the proof of the main results in this section.
Lemma 4.1 (cf. [8]: Gronwall’s inequality). Let
and
be two continuous and non-negative functions on the interval
,
be a constant. The classical Gronwall’s inequality states that, if:
Then:
(4.1)
Theorem 4.2: Suppose
, and suppose that the conditions (3.1) are satisfied. Then, given any complex numbers
,
and
, there exists a unique solutions of
(
) which satisfies:
Proof: The proof is similar to that in ( [21], part II, Theorem 16.2.2) and therefore omitted.
Theorem 4.3: (cf. [18] [21] ). Let
be a regular quasi-differential expression of order n on the interval
. For
, the equation
has a solution
satisfying:
If and only if f is orthogonal in
to solution space of
, i.e.,
Corollary 4.4 (cf. [12] ), As a result from Theorem 4.2, we have that:
Let
be the solutions of the homogeneous equation:
(4.2)
satisfying:
for all
for fixed
,
. Then
is continuous in
for
,
, and for fixed t it is entire in
. Let
denote the solutions of the adjoint homogeneous equation:
(4.3)
satisfying:
for all
.
Suppose
, by [21], a solution of the product equation:
(4.4)
satisfying
is giving by:
where
stands for the complex conjugate of
and for each
,
is constant which is independent of
(but does depend in general on
).
The next lemma is a form of the variation of parameters formula for a general quasi-differential equation is giving by the following Lemma.
Lemma 4.5: Suppose
locally integrable function and
is the solution of the Equation (4.4) satisfying:
for
,
is giving by:
(4.5)
for some constants
, where
and
are solutions of the Equations (4.2) and (4.3) respectively,
is a constant which is independent of t.
Proof: The proof is similar to that in [8] - [16] and [17] - [22].
Lemma 4.5: Contain the following lemma as a special case.
Lemma 4.6: Suppose
locally integrable function and
be the solution of the Equation (4.4) satisfying:
for
,
.
Then:
(4.6)
for
. We refer to [13] [22] for more details.
Lemma 4.7: Suppose that for some
all solutions of the equations:
(4.7)
are in
. Then all solutions of the Equations in (4.7) are in
for every complex number
.
Proof: The proof is similar to that in [16] - [22].
Lemma 4.8: Suppose that for some complex number
all solutions of the Equations in (4.7) are in
. Suppose
. Then all solutions of the Equation (4.4) are in
for all
.
Proof: The proof is similar to that in [8] - [16] [22].
Remark: Lemma 4.8 also holds if the function f is bounded on
.
Lemma 4.9: Suppose that for some
all solutions of the Equations in (4.7) are in
. Then all solutions of the Equations (4.2) and (4.3) are in
for every complex number
.
Proof: The proof is similar to that in [16] - [22].
Lemma 4.10: If all solutions of the equation
are bounded on
and
for some
,
. Then all solutions of the equation
are also bounded on
for every complex number
.
Lemma 4.11: Suppose that for some complex number
all solutions of the Equations in (4.7) are in
. Suppose
, then all solutions of the Equation (4.4) are in
for all
.
Proof: Let
,
be two sets of linearly independent solutions of the Equations (4.7). Then for any solutions
of the equation
(
) which may be written as follows:
and it follows from (4.5) that:
(4.8)
for some constants
. Hence:
(4.9)
Since
and
for some
, then
, for some
and
.
Setting:
(4.10)
then:
(4.11)
On application of the Cauchy-Schwartz inequality to the integral in (4.11), we get:
(4.12)
From the inequality
it follows that:
(4.13)
By hypothesis there exist positive constant
and
such that:
and
;
. (4.14)
Hence:
(4.15)
Integrating the inequality in (4.15) between a and t, we obtain:
(4.16)
where:
(4.17)
Now, on using Gronwall’s inequality (Lemma 4.1), it follows that:
(4.18)
Since,
for some
and for
, then
.
Remark: Lemma 4.11 also holds if the function
is bounded on
.
Lemma 4.12: Let
. Suppose for some
that:
1) All solutions of
are in
.
2)
are bounded on
.
Then
for any solution
of the equation
for all
.
Proof: The proof is similar to that in ( [20], Theorem 4.2).
Lemma 4.13: Let
. Suppose for some
that:
1) All solutions of
are in
.
2)
are bounded on
for some
.
Then
for any solution
of the equation
for all
.
Proof: The proof is the same up to (4.11). By using Lemma 4.3, (3.11) becomes:
(4.19)
Applying the Cauchy Schwartz inequality to the integral in (4.19), we get:
(4.20)
From the inequality
it follows that:
(4.21)
Since
for some
and
are bounded on
for some
by hypothesis, then there exist a positive constants
and
such that:
and
. (4.22)
Hence,
(4.23)
By integrating the inequality in (4.23) between a and t, and by using Lemma 4.1 (Gronwall’s inequality), we have the result.
5. The Spectra of Differential Operators
In this subsection we deal with the various components of the spectra of quasi-differential operators
and
.
We see from (3.15) and Theorem 4.2 that
and hence
and
form an adjoint pair of closed, closed-densely operators in
.
We shall now investigate in the case of one singular end-point that the resolvent of all well-posed extensions of the minimal operator
and we show that in the maximal case, i.e., when:
for all
these resolvent are integral operators, in fact they are Hilbert-Schmidt integral operators by considering that the function
be in
, i.e., is quadratically integrable over the interval
.
Theorem 5.1: Suppose for an operator
with one singular end-point that,
for all
,
and let S be an arbitrary closed operator which is a well-posed extension of the minimal operator
and
, then the resolvents
and
of S and
respectively are Hilbert-Schmidt integral operators whose kernels
and
are continuous functions on
and satisfy:
and
. (5.1)
where:
and
,
for
, and for all
.
Remark: An example of a closed operator which is a well-posed with respect to a compatible adjoint pair is given by the Visik extension ( [7], Theorem 1) (see ( [18], Theorem III.3.3) and [21] ). Note that if S is well-posed, then
and
are compatible adjoint pair and S is regularly solvable with respect to
and
.
Proof: Let
for all
, then we choose a fundamental system of solutions
,
of the equations,
(5.2)
so that
,
belong to
i.e., they are quadratically integrable in the interval
. Let
be the resolvent of any well-posed extension of the minimal operator
. For
we put
then
and consequently has a solution
in the form,
(5.3)
for some constants
(see Lemma 4.5). Since
and
for some
, then
,
for some
and hence the integral in the right-hand of (5.3) will be finite.
To determine the constants
, let
be a basis for
, then because
, we have from Theorem 3.2 that,
(5.4)
and hence from (5.3), (5.4) and on using Lemma 4.7, we have:
(5.5)
By substituting these expressions into the conditions (5.4), we get:
This implies that the system:
(5.6)
in the variable
. The determinant of this system does not vanish (see [9] and [12] ). If we solve the system (5.6) we obtain:
(5.7)
where
is a solution of the system:
(5.8)
Since, the determinant of the above system (5.8) does not vanish, and the functions
are continuous in the interval
, then the functions
are also continuous in the interval. By substituting in formula (5.3) for the expressions
we get,
(5.9)
Now, we put:
(5.10)
Formula (5.9) then takes the form:
for all
, (5.11)
i.e.,
is an integral operator with the kernel
operating on the functions
. Similarly, the solutions
of the equation
has the form:
(5.12)
where
and
are solutions of the Equations in (5.2). The argument as before leads to,
for
, (5.13)
i.e.,
is an integral operator with the kernel
operating on the functions
, where:
(5.14)
and
is a solution of the system:
(5.15)
From definitions of
and
, it follows that:
(5.16)
for any continuous functions
and by construction (see (5.10) and (5.14)),
and
are continuous functions on
and (5.16) gives us:
for all
. (5.17)
Since
for
and for fixed s,
is a linear combination of
while, for fixed t,
is a linear combination of
Then we have:
and (5.17) implies that,
Now, it is clear from (5.8) that the functions
belong to
since
is a linear combination of the functions
which lie in
and hence
belong to
. Similarly
belong to
. By the upper half of the formula (5.10) and (5.14), we have:
,
for the inner integral exists and is a linear combination of the products
,
and these products are integrable because each of the factors belongs to
. Then by (5.17), and by the upper half of (5.14),
Hence, we also have:
,
and the theorem is completely proved for any well-posed extension.
Remark: It follows immediately from Theorem 5.1 that, if for an operator
with one singular end-point that
for all
and S is well-posed with respect to
and
with
then
is a Hilbert-Schmidt integral operator. Thus it is a completely continuous operator, and consequently its spectrum is discrete and consists of isolated eigenvalues having finite algebraic (so geometric) multiplicity with zero as the only possible point of accumulation. Hence, the spectra of all well-posed operators S are discrete, i.e.,
, for
. (5.18)
We refer to [3] [5] [7] [14] ( [18], Theorem IX.3.1) [21] for more details.
An example of a closed operator which is a well-posed with respect to a compatible adjoint pair is given by the Visik extension ( [7], Theorem 1) (see ( [18], Theorem III.3.3) [21] ). Note that if S is well-posed, then
and
are compatible adjoint pair and S is regularly solvable with respect to
and
.
Lemma 5.2: The point spectra
and
of the operators
and
are empty.
Proof: Let
. Then there exists a nonzero element
, such that:
In particular, this gives:
From Theorem 4.2, it follows that
and hence
. Similarly
.
Theorem 5.3: 1)
,
2)
,
3)
.
Proof: 1) Since
is a proper closed subspace of
, then the resolvent set
is empty.
2) Since
is closed, then the continuous spectrum of
is empty set, i.e.,
.
3) From 1) and 2) and Lemma 5.2, it follows that
.
Corollary 5.4: 1)
,
2)
and
.
Proof: From Theorem 4.2 and since
, it follows that
is closed for every
, see ( [3], Theorem 1.3.7). Also, we have:
and:
.
1) Since
is closed and
, then
and this yields that:
2) Since
for every
, then we have
. It also follows that
and hence
.
Lemma 5.5: (cf. ( [18], Lemma IX.9.1)). If
, with
then for any
, the operator
has closed range, zero nullity and deficiency
. Hence,
(5.19)
Proof: The proof is similar to that in ( [7], Lemma 4.9) and [18].
Corollary 5.6: Let
with:
(5.20)
Then,
, for
. (5.21)
of all regularly solvable extensions S with respect to the compatible adjoint pair
and
.
Proof: Since:
for all
.
Then we have from ( [18], Theorem III.3.5) that,
Thus S is an
-dimensional extension of
and so by [5] [7] and ( [18], Corollary IX.4.2).
(5.22)
From Lemma 5.2 and Lemma 5.5, we get,
(5.23)
Hence, by (5.22) and (5.23) we have that,
Remark: If S is well-posed (say the Visik’s extension, see [5] [6] ) we get from (5.19) and (5.22) that:
On applying (5.22) again to any regularly solvable operator S under consideration, hence (5.21).
Corollary 5.7: Let
with:
(5.24)
Then,
, for
. (5.25)
for all regularly solvable operators S with respect to the compatible adjoint pair
and
.
Proof: Since:
for all
.
Then we have from ( [18], Theorem III.3.5) that,
Thus S is an
-dimensional extension of
and so by [5] [7] [18],
(5.26)
From Lemma 5.2 and Lemma 5.5, we get,
(5.27)
Hence, by (5.26) and (5.27) we have that,
Remark: If S is well-posed (say the Visik extension, (see [5] [7] )), we get from (5.21) and (5.26) that:
On applying (5.26) again to any regularly solvable extensions S under consideration, hence (5.25).
Corollary 5.8: If for some
, there are n linearly independent solutions of the equations:
(5.28)
in
, and hence,
and
,
,
where
is the joint essential spectra of
defined as the joint field of regularity
.
Proof: Since all solutions of the equations in (5.28) are in
for some
then,
, for some
.
From Lemma 4.12, we have that
has no eigenvalues and so
exists and its domain
is a closed subspace of
. Hence, since
is a closed operator, then
is bounded and hence
. Similarly
. Therefore
and hence,
, for all
.
From Corollary 5.7, we have for any regularly solvable extension S of
that
,
and by (5.22), we get
,
. Similarly
,
. Hence,
Remark: If there are n linearly independent solutions of the Equations (5.28) in
for some
then the complex plane can be divided into two disjoint sets:
We refer to [3] [4] [5] [7] [11] [12] [13] [14] [18] [19] [20] [21] [22] for more details.
Conclusion. We have investigated (according to the spectral theory) the location of the point spectra and regularity fields of general ordinary quasi-differential operators in the case of one singular end-point and when all solutions of the equations
and
are in the space
for some (and hence all
).
Acknowledgements
I am grateful to the Public Authority of Applied Education and Training (PAAET) in Kuwait for supporting scientific research and encouragement to the researchers.