In this chapter, the Sturm-Liouville equation with block-triangular, increasing at infinity operator potential is considered. A fundamental system of solutions is constructed, one of which decreases at infinity, and the second increases. The asymptotic behavior at infinity was found out. The Green’s function and the resolvent for a non-self-adjoint differential operator are constructed. This allows to obtain sufficient conditions under which the spectrum of this non-self-adjoint differential operator is real and discrete. For a non-self-adjoint Sturm-Liouville operator with a triangular matrix potential growing at infinity, an example of operator having spectral singularities is constructed.
- differential operators
- block-triangular operator coefficients
- Green’s function
The question of the generalization of the oscillatory Sturm theorem for scalar equations of higher orders and for equations with matrix coefficients for a long time remained open. Only in recent joint papers by F. Rofe-Beketov and A. Kholkin (see ) a connection was established between spectral and oscillatory properties for self-adjoint operators generated by equations of arbitrary even order with operator coefficients and boundary conditions of general form. Later, a Sturm-type oscillation theorem was proved  for a problem on finite and infinite intervals for a second-order equation with block-triangular matrix coefficients. In the case of non-self-adjoint differential operators, oscillation theorems have not been considered earlier.
Results turning out in self-adjoint and non-self-adjoint cases differentiate substantially. The theory of non-self-adjoint singular differential operators, generated by scalar differential expressions, has been well studied. An overview on the theory of non-self-adjoint singular ordinary differential operators is provided in V.E. Lyantse’s Appendix I to the monograph . In the study of the connection between spectral and oscillation properties of non-self-adjoint differential operators with block-triangular operator coefficients [2, 4] the question arises of the structure of the spectrum of such operators. For scalar non- self-adjoint differential operators these questions were studied in the papers [5, 6, 7, 8]. The theory of singular non-self-adjoint differential operators with matrix and operator coefficients is relatively new. In the context of the inverse scattering problem, for an operator with a triangular matrix potential decreasing at infinity, the first moment of which is bounded, the structure of the spectrum was established in [9, 10]. The theory of equations with block - triangular operator coefficients the first results were published in 2012 in the works of the author [11, 12, 13].
In this works we construct the fundamental system of solutions of differential equation with block-triangular operator potential that increases at infinity, one of that is decreasing at infinity, and the second growing. The asymptotics of the fundamental system of solutions of this equation is established. The Green’s function is constructed for a non-self-adjoint system with a block-triangular potential, the diagonal blocks of which are self-adjoint operators. We obtained a resolvent for a non-self-adjoint differential operator, using which the structure of the operator spectrum is set. Sufficient conditions at which a spectrum of such non-self-adjoint differential operator is real and discrete are obtained. Here the rate of growth elements, not on the main diagonal, is subordinated to the rate of growth of the diagonal elements. In case of infringement of this condition, the operator can have spectral singularities .
2. The fundamental solutions for an non-self-adjoint differential operator with block – triangular operator coefficients.
Let us designate as a finite-dimensional or infinite-dimensional separable Hilbert space with inner product and norm . Denote by Element will be written in the form of , where , , - are identity operators in and accordingly.
We denote by the Hilbert space of vector-valued functions with values in with inner product and the norm .
Now let us consider the equation with block-triangular operator potential in
is a real scalar function such that monotonically, as , and it has monotone absolutely continuous derivative. Also, is a relatively small perturbation, e. g. as or . The diagonal blocks are assumed to be bounded self-adjoint operators in .
In case where
we suppose that coefficients of the Eq. (1) satisfy relations:
In case of , we suppose that the coefficients of the Eq. (1) satisfy the relation
2.1 Construction of the fundamental system of solutions for an operator differential equation with a rapidly increasing at infinity potential
Consider first the case where .
Condition (3) is performed, for example, quickly increasing functions etc.
Rewrite the Eq. (1) in the form
where determined by a formula (cf. with the monograph )
Now let us denote.
It is easy to see that as . These solutions constitute a fundamental system of solutions of the scalar differential equation
in such a way that for all one has.
Also, there exists increasing at infinity operator solutionsatisfying the conditions
Proof a. Eq. (7) equivalently to integral equation
where . Thus
and since with one has , we deduce that
Now consider another block-triangular operator solution that increases at infinity diagonal blocks which are defined by.
are the diagonal blocks of operator solution as in Section a). In view (16) and the definition of the functions can be proved that
Since and are the operator solutions of Eq. (1) that increase at infinity,
The solution given by is subject to first from condition (13). Use (12) to differentiate (27), then find the asymptotes of as similarly to (21) to obtain the second part of formula (13). Theorem is proved.
In this section, the fundamental system of solution is constructed for an operator differential equation with a rapidly increasing at infinity potential.
2.2 Asymptotic of the fundamental system solutions of equation with block-triangular potential
where determined by a formula
There solutions constitute a fundamental system of solutions of the scalar differential equation , in such a way that for all one has .
We are about to establish the asymptotics1 of as :
After expanding here the integral, we obtain the exponential as follows
In case , i.e. , this expression after integration acquires the form:
The asymptotics of as is as follows:
In particular, for , has the following asymptotics at infinity:
In case we set , with being the integral part of to obtain the following asymptotics for at infinity:
In particular, with one has
A similar procedure allows to establish the asymptotics of as If , i.e. , then
With , this becomes
In case , we set to get the asymptotics.
In case , one has
Proof is similar to Theorem 2.1. Moreover, note that
As , one has , and that is why
The remaining statements of Theorem 2.1 are proved similarly.
Corollary 2.1 If, i.e., then, under condition (6), the solutionsandhave common (known) asymptotics, as in the qualityandyou can take the following functions.
If, i.e. the coefficient, and the condition (6) holds, then.
Remark 2.1 It is known that scalar equation
for has the solution , where is the Chebyshev – Hermitre polynomial, that at has next asymptotics . Hence the solution of the Eq. (49) at will have the following asymptotics at infinity: .
In the case of in (2), the Eq. (1) is splitting into infinity system scalar equations of the form (49). The operator solution will be diagonal in this case. Denote by the diagonal elements of the operator . Then, by Corollary 2.1, the solution will have the following asymptotics at infinity: . In particular, for , this yields the solution proportional to .
In this section, the asymptotics of the fundamental system of solutions for the Sturm-Liouville equation with block-triangular operator potential, increasing at infinity is established. One of the solutions is found decreasing at infinity, the other one increasing.
3. Green’s function for an operator differential equation with block – triangular coefficients
Let us suppose that at the given boundary conditions
where - the block-triangular operator of the same structure as the coefficients of the differential equation, - bounded self-adjoint operators in , which satisfy the conditions
with the boundary conditions
Let denote the minimal differential operator generated by differential expression (1) and the boundary condition (50), and let , denote the minimal differential operator on generated by differential expression and the boundary conditions (53). Taking into account the conditions on coefficients, as well as sufficient smallness of perturbations and conditions (51), we conclude that, for every symmetric operator , there is a case of limit point at infinity. Hence their self-adjoint extensions are the closures of operators respectively. The operators are semi-bounded below, and their spectra are discrete.
Let denote the operator extensions , by requiring that be the domain of operator .
The following theorem is proved in .
Theorem 3.1 Suppose that, for Eq. (1)conditions (3)-(5) are satisfied foror condition (6) for. Then the discrete spectrum of the operatoris real and coincides with the union of spectra of the self-adjoint operators, i.e.,.
Comment 3.1 Note that this theorem contains a statement of the discrete spectrum of the non-self-adjoint operator only and no allegations of its continuous and residual spectrum.
Along with the Eq. (1) we consider the equation
(is adjoint to the operator ). If the space is finite-dimensional, then the Eq. (54) can be rewritten as
where and the equation is called the left.
For operator -functions let
Because the operator function satisfies equation
the operator function is a solution to the left of the equation
and satisfies the initial conditions .
Operator solutions of Eq. (54) decreasing and increasing at infinity will be denoted by , , and the corresponding solutions of the Eq. (59) denote by and . For the system operator solutions of the Eqs. (1) and (59), respectively, will take the form of Wronskian and do not depend on .
Let us designate
In the following theorem it is proved that the operator function possesses all the classical properties of the Green’s function.
Theorem 3.2 The operator functionis the Green’s function of the differential operator, i.e.:
The functionis continuous for all values;
For any fixed, the functionhas a continuous derivative with respect toon each of the intervalsand, and atit has the jumpE61
For a fixed, the functionof the variableis an operator solution of Eq. (1) on each of the intervals, , and it satisfies the boundary condition (50) , and at a fixedfunctionof the variableis an operator solution of the Eq. (59) on each of the intervals, and it satisfies the boundary condition
Proof The function is continuous with respect to at each of the intervals and . Similarly to the variable . To prove the continuity of the function for all , it is sufficient that the identity shown as
is satisfied for all This identity shown as
which is equivalent to
This follows from the fact that .
To make sure that the jump in the first derivative at is equal to , i.e., that the equality (61) holds, it is sufficient to prove the identity
Now we consider the function
which is an analogue of the Cauchy function. This function is the solution of Eq. (1) of the variable and it is the solution of Eq. (59) of the variable . By (62), we have . But in this case , and, therefore, , i.e.,
It shows that we obtain (61).
Similarly, operator solution of Eq. (59) can be represented in the form
Similarly we get . Thus,
By Theorem 2.1, on the asymptotic behavior of functions and at infinity, we have
This completes the proof of the formula (61), and with it the theorem 3.1.
Corollary. By the definition (60) , functionis meromorphic of the parameterwith the poles coincide with the eigenvalues of the operator.
We constructed Green’s function for the non-self-adjoint differential operator.
4. Resolvent for an non-self-adjoint operator differential equation with block – triangular coefficients
We consider the operator defined in by the relation
Theorem 4.1 The operatoris the resolvent of the operator.
Proof One can directly verify that, for any function , the vector-function is a solution of the equation whenever . We will prove that .
Similarly, the operator solution of Eq. (59) can be represented in the following form
Let us show that each of these vector-functions belongs to . Since the operator solution decays fairly quickly as , then . It follows that
By virtue of the asymptotic formulas for the operator solutions and we obtain that
Let us rewrite this relation in the following form
By using the definition of the functions and (see (9)) and by applying the Cauchy-Bunyakovskii inequality we obtain
Since , we get , and then the latter estimate for can be rewritten as follows
Replacing variables , we get
Since the inequality holds for all and with sufficiently large , we have
Hence it follows that , therefore In case, where and and where , the proof is similar.
Thus, for any function .
Since the resolvent is a meromorphic function of , the poles of which coincide with the eigenvalues of the operator , the statement of Theorem 3.1 can be refined.
Theorem 4.2 If the conditions (3)-(5)) whereor condition (6) whereare satisfied for the Eq. (1), then the spectrum of the operatoris real, discrete and coincides with the union of spectra of self-adjoint operators, i.e..
In this section, a resolvent for a non-self-adjoint differential operator with a block-triangular operator potential, increasing at infinity, is constructed. Sufficient conditions under which the spectrum is real and discrete are obtained.
5. Spectral singularities of differential operator with triangular matrix coefficients
Example 5.1 Consider the equation:
with the boundary condition
with the boundary conditions.
As above, denote by the differential operator generated by the differential expression (93) and the boundary condition (94), and by denote the minimal symmetric operators on , generated by the differential expressions and the boundary conditions (97). Their self-adjoint extensions are the closures of the operators , respectively. The operators are semi-bounded; let us denote their spectra by .
Denote by the extension of the operator generated by the requirement on the functions from the domain of the operator to belong to , and by its spectrum.
Denote by the matrix solution of the Eq. (93), satisfying the initial conditions .
If some , and - is the corresponding eigenfunction of the operator , then the vector function is the eigenfunction of the operator , corresponding to the eigenvalue , i.e. . Moreover, is the eigenvalue of the operator if and only if the solution of the equation
satisfying the initial conditions , belongs to . Let be the solutions of the Eq. (95), satisfying the initial conditions , and let - be the Cauchy function of the Eq. (95). Then the solution is given by
Choose the coefficient , where (for instance, ), and show that the integral diverges and, consequently, . Indeed, since the solution has finitely many zeros, we conclude that, for any ,
and the Cauchy function decays no faster than . Hence, if , we have
There arises the question on the nature of such values .
Consider the equation with a triangular matrix potential:
where are scalar functions, are real functions and monotonically as .
Let the boundary condition is given at :
where is a triangular matrix, .
Consider the separated system
with the boundary conditions
Let be the differential operator generated by the differential expression (103) and the boundary condition (104), and let be minimal symmetric operators on generated by the differential expressions and the boundary conditions (106), (108) respectively. Denote by the self-adjoint extensions of the operators respectively. The operators are semi-bounded; let us denote their spectra by and respectively. Denote by the extension of the operator and by its spectrum.
Let be the solutions of the Eq. (104) with the boundary conditions The general solution of the Eq. (104) has the form up to a constant. Choose an such that the condition holds true. This equality is valid for (the solution has finitely many zeros for a fixed , hence whenever is sufficiently large). Put . Since for the operator there is the case of a limit point, then, as is known, has a unique limit as , and the solution of the Eq. (104) satisfies . Similarly we obtain that the solution of the Eq. (105) satisfies .
Denote by the matrix solution of the Eq. (103) satisfying the initial conditions . We have ; as .
The solution is given by , where is the Cauchy function of the Eq. (104).
Further, we have as . Put .
Together with the Eq. (102), we consider the left equation.
The matrix solutions of the Eq. (108) will be denoted by and .
The function is the Green function of the operator generated by the problem (102), (103), , which spectrum coincides with the union of spectra of the operators generated by the problems (104), (106), and (105), (107), respectively. Eigenvalues of the operators and tend to ones of the operators and respectively as , , and
Poles of the Green function of the operator coincide with the zero set of the determinant , where.
Since the matrices are triangle, we have /, where . On the other hand, zeros of the function are eigenvalues of the self-adjoint operator . Hence the poles of the Green function of the operator are situated on the real axis, and their set coincides with the union of spectra of the operators and .
Consider the operator defined on by.
One can directly verify that the operator is the resolvent of the operator .
Let be an arbitrary vector function square integrable on . Choose a sequence of finite continuous vector functions converging in mean square to . Substituting for in (114) and letting first and then , we obtain the following formula for the resolvent of the operator : , where the Green function of the operator is defined by the formula (112).
Theorem 5.1 The operatoris the resolvent of the operator. The resolvent’s poles coincide with the union of the spectra of the self-adjoint operatorsand.
Remark 5.2 As in Example 5.1, ifand, thenis the pole of the resolventof the operatorbut it is not the eigenvalue of this operator, i.e.,is the point of the spectral singularity of the operator.
Theorem 5.1 implies that, if the rate of the coefficient’s growth of the Eq. (102) is subordinated to one of and , then the operator has no spectral singularities, and its spectrum is real and coincides with the union of the spectra of the operators and .
For a non-self-adjoint Sturm-Liouville operator with a triangular matrix potential growing at infinity, an example of operator having spectral singularities is constructed. A special role of these points was found first by M.A. Naimark in . The notion “spectral singularity” was introduced later due to J. Schwartz  (see also Supplement I in the monograph ).
We consider the Sturm-Liouville equation with block-triangular, increasing at infinity operator potential. For him, built a fundamental system of solutions, one of which is decreasing at infinity, and the second is growing. The asymptotics of these solutions at infinity is defined. For non-self-adjoint operator generated by such differential expression obtained the Green’s function. A resolvent of such an operator is constructed. Sufficient conditions at which a spectrum of such non-self-adjoint differential operator is real and discrete are obtained.
- For α=1 and α=12, i.e., for vx=x2 and vx=x, the asymptotics of the functions γ0xλ and γ∞xλ is known.