Criteria for the Uniqueness of a Solution to a Differential-Operator Equation with Non-Degenerate Conditions

Abstract

In this article, the symmetric operator L corresponding to the boundary value problem is represented as the difference of two commuting operators A and $B.$ The uniqueness of the solution is guaranteed if the spectra of the operators A and B do not intersect and the domain of the operator B is given by non-degenerate boundary conditions. In contrast to the existing papers, the criterion for the uniqueness of the boundary value problem formulated in this paper is satisfied even when the system of root functions of the operator B does not form a basis in the corresponding space. At the same time, only the closedness of the linear operator A is required.

1. Introduction

In this paper, we consider a differential-operator equation of the form

$$-\frac{{\partial }^{2}u}{\partial t^2}+q\left(t\right)u=Au+f\left(t\right),0<t<T<\infty$$

(1)

with non-degenerate boundary conditions in time

$${\Gamma }_i\left(u\right)=a_{i1}u\left(0\right)+a_{i2}{u}^{\prime }\left(0\right)+a_{i3}{u}^{\prime }\left(T\right)+a_{i4}{u}^{\prime }\left(T\right)=0,i=1,2.$$

(2)

It is also assumed that the operator A does not depend on t and is a closed linear operator on a separable Hilbert space $H.$ In this paper, no other restrictions on the operator A are assumed. Recall that boundary conditions (2) are called non-degenerate if one of the following three requirements is satisfied:

$\left(1\right)\left|\begin{array}{cc}{a}_{14}& a_{12}\\ a_{24}& a_{22}\end{array}\right|\ne 0,$

$\left(2\right)\left|\begin{array}{cc}{a}_{14}& a_{12}\\ a_{24}& a_{22}\end{array}\right|=0,\left|\begin{array}{cc}{a}_{11}& a_{14}\\ a_{21}& a_{24}\end{array}\right|+\left|\begin{array}{cc}{a}_{13}& a_{12}\\ a_{23}& a_{22}\end{array}\right|\ne 0,$

$\left(3\right)\left|\begin{array}{cc}{a}_{14}& a_{12}\\ a_{24}& a_{22}\end{array}\right|=0,\left|\begin{array}{cc}{a}_{11}& a_{14}\\ a_{21}& a_{24}\end{array}\right|+\left|\begin{array}{cc}{a}_{13}& a_{12}\\ a_{23}& a_{22}\end{array}\right|=0,\left|\begin{array}{cc}{a}_{11}& a_{13}\\ a_{21}& a_{23}\end{array}\right|\ne 0.$

Otherwise, the coefficients $a_{ij}$ of the boundary conditions are arbitrary and can be complex numbers. The coefficient $q\left(t\right)$ of the differential expression on the left side of (1) is assumed to be an integrable complex-valued function on $[0,T].$

The main goal of this article is to establish a criterion for the uniqueness of the solution of problems (1) and (2). There are various ways to prove uniqueness. Usually, the maximum principle [1] and its various generalizations like the Hopf [2] and Zaremba–Giraud [3] principles are effective for proving uniqueness. For the given problems (1) and (2), these principles are not applicable. Therefore, we need a different tool instead of the extremum principle.

We first note the work of I.V. Tikhonov [4], which is devoted to the uniqueness theorems in linear nonlocal problems for abstract differential equations. I.V. Tikhonov’s proof is mainly based on the “quotient method” for entire functions of exponential type. In [5], authors studied the uniqueness question of the solution of the heat equation with a nonlocal condition expressed as an integral over time on a fixed interval. They succeeded in giving a complete description of the uniqueness classes in terms of the behavior of solutions as $\left|x\right|\to \infty .$ In this paper, I.V. Tikhonov’s method is adapted for operators whose differential part is a second-order operator with general boundary conditions.

In this paper, the symmetric operator L corresponding to boundary value problem, (1) and (2), is represented as the difference of two commuting operators A and $B.$ The uniqueness of the solution is guaranteed if the spectra of the operators A and B do not intersect and the domain of the operator B is given by non-degenerate boundary conditions. In contrast to the existing papers [6,7,8], the criterion for the uniqueness of the boundary value problem—(1) and (2)—formulated in this paper is satisfied even when the system of root functions of the operator B does not form a basis in the corresponding space. At the same time, only the closedness of the linear operator A is required. For example, in our case, the unbounded operator A may not necessarily be semibounded or have an empty spectrum. Note that in papers [6,7,8], operator A was required to be semibounded, while operator B had to have a system of root functions forming a basis.

The method of proving the uniqueness of the solution of the boundary value problem, (1) and (2), is based on the method of guiding functionals by M.G. Krein [9] with their subsequent estimation when the spectral parameter infinitely increases in the complex region.

2. On the Spectral Properties of the Sturm–Liouville Operator on a Segment

In this section, we consider the boundary value problem generated on the interval $(0,T)$ by the Sturm–Liouville equation,

$$-w^{″}\left(t\right)+q\left(t\right)w\left(t\right)=\mu w\left(t\right),0<t<T<\infty$$

(3)

and the two following boundary conditions,

$${\Gamma }_i\left(w\right)=a_{i1}w\left(0\right)+a_{i2}{w}^{\prime }\left(0\right)+a_{i3}w\left(T\right)+a_{i4}{w}^{\prime }\left(T\right)=0,i=1,2,$$

(4)

where $q\left(t\right)$ is an integrable complex-valued function and $a_{ik}$ are arbitrary complex numbers. For more details on the generalized Sturm–Liouville-like problems, we refer the reader to [10,11]. Moreover, for practical uses of this kind of problem in fields such as mathematics, quantum physics, etc., mass–heat transfer can be found in [12,13,14,15].

Furthermore, the fundamental system of solutions to Equation (3) determined by the initial data, $c(\mu ,0)=s^{\prime }(\mu ,0)=1,$ $c^{\prime }(\mu ,0)=s(\mu ,0)=0,$ will be denoted by $c(\mu ,t),s(\mu ,t).$

We introduce the characteristic function by the formula

$$\chi \left(\mu \right)=J_{12}+J_{34}+J_{13}s(\mu ,T)+J_{14}{s}^{\prime }(\mu ,T)+J_{32}c(\mu ,T)+J_{42}{c}^{\prime }(\mu ,T),$$

(5)

where $J_{ij}=a_{1i}{a}_{2j}-a_{2i}{a}_{1j}$ is the determinant composed of the i-th and j-th columns of the coefficient matrix of the boundary conditions

$$\left(\begin{array}{cccc}{a}_{11}& a_{12}& a_{13}& a_{14}\\ a_{21}& a_{22}& a_{23}& a_{24}\end{array}\right).$$

Denote by $w_1(t,\mu ),w_2(t,\mu )$ the solutions of the homogeneous Equation (3) with the boundary conditions

$${\Gamma }_1\left(w_1\right)={\Gamma }_2\left(w_2\right)=0,$$

$${\Gamma }_1\left(w_2\right)={\Gamma }_2\left(w_1\right)=\chi \left(\mu \right).$$

The eigenvalue ${\mu }_n$ of the boundary value problem (3)–(4) is called eigenvalue with multiplicity p if ${\mu }_n$ is a root of the function $\chi \left(\mu \right)$ with multiplicity p.

Since

$$\frac{{\partial }^k}{\partial {\mu }^k}{\Gamma }_i\left(w_j\right)={\Gamma }_i\left(\frac{{\partial }^k}{\partial {\mu }^k}{w}_j\right),$$

the functions satisfy

$$w_{i,k}\left(t\right)=\frac{1}{k!}\frac{{\partial }^k}{\partial {\mu }^k}{w}_j(t,\mu )$$

(6)

where $\mu ={\mu }_n$ satisfies both boundary conditions (4), if $0\le k\le p-1.$ The functions $w_{i,0}\left(t\right),\dots ,w_{i,p-1}\left(t\right)(i=1,2)$ form a chain in which the first nonzero function $w_{i,l_i}\left(t\right)$ is its eigenfunctions and the following are its associated functions. Differentiating Equation (3) k times with respect to $\mu ,$ we conclude that the eigenfunction and associated functions of the chain satisfy the equations

$$-w_{i,k}^{″}\left(t\right)+q\left(t\right)w_{i,k}\left(t\right)={\mu }_n{w}_{i,k}\left(t\right)+w_{i,k-1}\left(t\right),0<t<T$$

and boundary conditions (4). To avoid misunderstandings, we emphasize that both chains, $w_{1,0}\left(t\right),\dots ,w_{1,p-1}\left(t\right)$ and $w_{2,0}\left(t\right),\dots ,w_{2,p-1}\left(t\right),$ may consist of the same functions. For us, it is only essential that, in addition to eigenfunctions and associated functions, chains can include only functions that are identically equal to zero.

Denote by B the operator given by the differential expression

$$Bw\left(t\right)=-w^{″}\left(t\right)+q\left(t\right)w\left(t\right)$$

and the domain given by the boundary conditions (4).

Let $\sigma \left(B\right)$ be the spectrum—that is, the set of all eigenvalues ${\mu }_n$ of the boundary value problem (3)–(4), where $p_n$ is their multiplicity. According to the previous function,

$$\frac{1}{k!}\frac{{\partial }^k}{\partial {\mu }^k}{w}_j(t,\mu ){|}_{\mu ={\mu }_n},0\le k\le p_n-1,{\mu }_n\in \sigma \left(B\right),i=1,2$$

are either identically equal to zero or are eigenfunctions or associated functions of this boundary value problem.

The operator has a dense domain in the space $L_2(0,T).$ Therefore, there is a unique adjoint operator $B^{*}.$ The action of the adjoint operator $B^{*}$ is given by the formula

$$B^{*}\tau \left(t\right)=-{\tau }^{″}\left(t\right)+\overline{q}\left(t\right)\tau \left(t\right),$$

(7)

where $\overline{z}$ means the conjugate of the complex number $z.$

Let the domain of the operator be given by the boundary forms $V_1(·)$ and $V_2(·),$ i.e.,

$$D\left(B^{*}\right)=\left\{\tau \in W_2^2[0,T]:V_1\left(\tau \right)=0,V_2\left(\tau \right)=0\right\}.$$

(8)

Here, the boundary forms of the adjoint problem have the following form:

$$V_i\left(\tau \right)=a_{i1}^{*}\tau \left(0\right)+a_{i2}^{*}{\tau }^{\prime }\left(0\right)+a_{i3}^{*}\tau \left(T\right)+a_{i4}^{*}{\tau }^{\prime }\left(T\right)=0,i=1,2.$$

(9)

Let us introduce a fundamental system of solutions $\{R_1(t,\overline{\mu }),R_2(t,\overline{\mu })\}$ of a homogeneous adjoint equation

$$-R_s^{″}(t,\overline{\mu })+\overline{q}\left(t\right)R_s(t,\overline{\mu })=\overline{\mu }{R}_s(t,\overline{\mu }),0<t<T$$

(10)

satisfying the Cauchy condition at zero

$$R_1(0,\overline{\mu })=1,R_2(0,\overline{\mu })=0,R_1^{\prime }(0,\overline{\mu })=0,R_2^{\prime }(0,\overline{\mu })=1.$$

(11)

Note that all solutions $\{R_i(t,\overline{\mu }),i=1,2\}$ are entire functions of $\overline{\mu }.$ Denote by ${\chi }^{*}\left(\overline{\mu }\right)$ the characteristic determinant given by the formula

$${\chi }^{*}\left(\overline{\mu }\right)=det\left(V_{\nu }\left(R_j\right)\right).$$

The zeros, taking into account their multiplicities of the characteristic determinant ${\chi }^{*}\left(\overline{\mu }\right)$, represent the eigenvalues of the adjoint operator $B^{*}.$

We also introduce ${\tau }_i(t,\overline{\mu })$ for $i=1,2$ solutions of the homogeneous adjoint Equation (10) with heterogeneous conditions

$$V_j\left({\tau }_s\right)={\delta }_{j,s}·{\chi }^{*}\left(\overline{\mu }\right),j=1,2,$$

(12)

where ${\delta }_{j,s}$ is the Kronecker symbol.

Let ${\mu }_0$ be the zero of the characteristic determinant $\chi \left(\mu \right)$, where its multiplicity equals $m_0.$ Then, for any $s=1,2$ in the ordered row

$$\left[{\tau }_s(t,\overline{{\mu }_0}),\frac{1}{1!}\frac{\partial }{\partial \overline{\mu }}{\tau }_s(t,\overline{{\mu }_0}),\dots ,\frac{1}{(m_o-1)!}\frac{{\partial }^{m_0-1}}{\partial {\overline{\mu }}^{m_0-1}}{\tau }_s(t,\overline{{\mu }_0})\right]$$

(13)

the first non-zero function represents the eigenfunction of the operator $B^{*},$ and the subsequent members of the row give the chain of associated functions generated by it.

In what follows, the eigenvalues of the operator $B^{*}$ will be denoted by ${\overline{\mu }}_{\nu },\nu \ge 1$, and the corresponding eigenvalues and associated functions by ${\tau }_{\nu }\left(t\right),\nu \ge 1.$

In [16], the following assertion was proved.

Theorem 1. 

Let the domain of the operator B be given by non-degenerate boundary conditions. Then, the domain of definition of the adjoint operator $B^{*}$ is also given by non-degenerate boundary conditions.

We also need the following assertion from [16].

Theorem 2. 

Let operator B be generated by non-degenerate boundary conditions. Then, the system of eigenfunctions and associated functions of operator B is a complete system in the space $L_2(0,T).$

Applying Theorems 1 and 2 to the adjoint operator $B^{*},$ we can formulate the following assertion.

Theorem 3. 

Let one of the requirements 1–3 be satisfied. Then, the system of eigenfunctions and associated functions of operator $B^{*}$ is complete in space $L_2(0,T).$

For further purposes, it is convenient for us to reformulate Lemmas 1 and 2, as well as Corollaries 1 and 2, from the monograph [16] in the following form.

Lemma 1. 

For all functions $f\left(t\right)\in L_1(0,T)$, the equalities are true

$$\underset{\left|\rho \right|\to \infty }{lim}{e}^{-\left|Im\rho T\right|}{\int }_0^{T}f\left(t\right)cos\rho tdt=\underset{\left|\rho \right|\to \infty }{lim}{e}^{-\left|Im\rho T\right|}{\int }_0^{T}f\left(t\right)sin\rho tdt=0.$$

(14)

Denote by $R_{1T}(t,\lambda ),R_{2T}(t,\lambda ),(\lambda =\overline{\mu })$ the solution of Equation (11) with initial data

$$\begin{array}{c}{R}_{1T}(T,\lambda )=R_{2T}^{\prime }(T,\lambda )=1,\\ R_{1T}^{\prime }(T,\lambda )=R_{2T}(T,\lambda )=0.\end{array}$$

Corollary 1. 

For all functions $f\left(x\right)\in L_1(0,T)$, the equalities are true

$$\underset{\left|\rho \right|\to \infty }{lim}{e}^{-\left|Im\rho T\right|}{\int }_0^{T}f\left(t\right)R_1(t,{\rho }^2)dt=\underset{\left|\rho \right|\to \infty }{lim}{e}^{-\left|Im\rho T\right|}{\int }_0^{T}f\left(t\right)R_{1T}(t,{\rho }^2)dt=0.$$

$$\underset{\left|\rho \right|\to \infty }{lim}{e}^{-\left|Im\rho T\right|}{\int }_0^{T}f\left(t\right)\rho R_2(t,{\rho }^2)dt=\underset{\left|\rho \right|\to \infty }{lim}{e}^{-\left|Im\rho T\right|}{\int }_0^{T}f\left(t\right)\rho R_{2T}(t,{\rho }^2)dt=0.$$

(15)

Corollary 2. 

For all functions $f\left(t\right)\in L_1(0,T)$, the equalities are true

$$\underset{\left|\rho \right|\to \infty }{lim}{e}^{-\left|Im\rho T\right|}{\int }_0^{T}f\left(t\right){\tau }_i(t,{\rho }^2)dt=0,i=1,2.$$

(16)

Lemma 2. 

If the boundary conditions in the boundary value problem (3)–(4) are non-degenerate, then there exists a constant $C>0$ and a sequence of infinitely expanding contours $K_n$ on the ρ–planes on which the inequalities hold

$$|{\chi }^{*}\left({\overline{\rho }}^2\right){|>|\rho |}^{-1}C{e}^{|Im\overline{\rho }T|},\overline{\rho }\in K_n.$$

(17)

3. Main Result and Its Proof

In this section, a criterion for the uniqueness of the solution of the boundary value problem in (1) and (2) is formulated and proved. In accordance with the notation of Section 1, the boundary value problem in (1) and (2) can be written in operator form

$$Bu\left(t\right)=Au\left(t\right)+f\left(t\right),t\in (0,T).\left(18\right)$$

(18)

Here, operator B acts on the variable t and its spectral properties are given in Section 1. The operator A is a closed linear operator in a separable Hilbert space H and does not depend on $t.$

Theorem 4. 

Let the matrix of boundary coefficients

$$J=\left(\begin{array}{cccc}{a}_{11}& a_{12}& a_{13}& a_{14}\\ a_{21}& a_{22}& a_{23}& a_{24}\end{array}\right)$$

be rank 2, which means that at least one of the numbers $J_{42},J_{14}+J_{32}$, and $J_{13}$ is different from zero, where $J_{kj}=a_{1k}{a}_{2j}-a_{2k}{a}_{1j}$, and we assume that operator A is a closed linear operator in a separable Hilbert space H and does not depend on $t.$ Then, the homogeneous operator equation

$$Bu=Au$$

has the trivial solution $u\in D\left(B\right)\cap D\left(A\right)$ if and only if

$$\sigma \left(B\right)\cap \sigma \left(A\right)=⌀,$$

where $\sigma \left(B\right)$ and $\sigma \left(A\right)$ are the spectra of operators B and A, respectively.

Proof. 

Proof of necessity. Let ${\mu }_n$ be some eigenvalue of operator B (with own function $w_n\left(t\right)$) and also an eigenvalue of operator A—that is, there is an eigenvalue ${\lambda }_s$ of operator $A,$ which is the same as the eigenvalue ${\mu }_n$ of operator $B.$ Suppose that the eigenvalue ${\lambda }_s$ of operator A corresponds to the eigenvalue $v_s$. Then, the function $u\left(t\right)=w_n\left(t\right)·v_s$ will be a non-trivial solution of the homogeneous Equation (17). The necessity of Theorem 3 is proved. □

Proof. 

Proof of sufficiency. Let none of $\{{\mu }_n,n\ge 1\}$ eigenvalues of operator B be an eigenvalue of operator $A.$ In other words, if ${\lambda }_s$ is an arbitrary eigenvalue of operator $A,$ then $\chi \left({\lambda }_s\right)\ne 0.$

Let us show that the solution $u\left(t\right)$ of the homogeneous operator Equation (18) is identically equal to zero in the space $L_2((0,T);H).$

To do this, we introduce functions with values in the Hilbert space H for $j=1,2$

$$F_j\left(\overline{\mu }\right)={\int }_0^T\overline{{\tau }_j(t,\overline{\mu })}u\left(t\right)dt.$$

(19)

Type $F_1\left(\overline{\mu }\right)$ and $F_1\left(\overline{\mu }\right)$ functions, described for the first time by M.G. Krein in [9], are called guiding functionals.

According to the Lagrange formula [17], function $A{F}_j\left(\overline{\mu }\right)$ for $j=1,2$ can be rewritten as follows:

$$A{F}_j\left(\overline{\mu }\right)={\int }_0^T\overline{{\tau }_j(t,\overline{\mu })}·Au\left(t\right)dt={\int }_0^T\overline{{\tau }_j(t,\overline{\mu })}·Bu\left(t\right)dt=$$

$${\int }_0^{T}u\left(t\right)\left(\overline{-{\tau }_j^{\prime \prime }(t,\overline{\mu })+\overline{q\left(t\right)}{\tau }_j(t,\overline{\mu })}\right)dt+{\Gamma }_1\left(u\right)\overline{V_4\left({\tau }_j\right)}+{\Gamma }_2\left(u\right)\overline{V_3\left({\tau }_j\right)}+$$

$${\Gamma }_3\left(u\right)\overline{V_2\left({\tau }_j\right)}+{\Gamma }_4\left(u\right)\overline{V_1\left({\tau }_j\right)},$$

(20)

where ${\Gamma }_3(·){\Gamma }_4(·)$ are linear forms [18], and ${\Gamma }_1(·){\Gamma }_2(·)$ are complementary linear forms, up to a Dirichlet system of order 4. In [18], it is stated that the system of linear forms $\{V_1(·),V_2(·),V_3(·),V_4(·)\}$ determined by $\{{\Gamma }_1(·){\Gamma }_2(·),{\Gamma }_3(·){\Gamma }_4(·)\}$ is unique and forms a Dirichlet system of order 4.

Since ${\Gamma }_1\left(u\right)={\Gamma }_2\left(u\right)=0,$$V_1\left({\tau }_2\right)=V_2\left({\tau }_1\right)=0,V_1\left({\tau }_1\right)=V_2\left({\tau }_2\right)={\chi }^{*}\left(\overline{\mu }\right),$ then relation (20) takes the form

$$A{F}_j\left(\overline{\mu }\right)=\mu F_j\left(\overline{\mu }\right)+{\Gamma }_{5-j}\left(u\right)·\overline{{\chi }^{*}\left(\overline{\mu }\right)},j=1,2.$$

(21)

If ${\mu }_0$ is an arbitrary zero of multiplicity $m_0$ of the characteristic function $\chi \left(\mu \right),$ then the last relation (21) implies the equalities

$$A{F}_j\left(\overline{{\mu }_0}\right)={\mu }_0{F}_j\left(\overline{{\mu }_0}\right),$$

$$A\frac{d{F}_j\left(\overline{{\mu }_0}\right)}{d\overline{\mu }}={\mu }_0\frac{d{F}_j\left(\overline{{\mu }_0}\right)}{d\overline{\mu }}+F_j\left(\overline{{\mu }_0}\right),$$

(22)

$$\dots$$

$$A\frac{d^{m_0-1}}{d{\overline{\mu }}^{m_0-1}}{F}_j\left(\overline{{\mu }_0}\right)={\mu }_0\frac{d^{m_0-1}{F}_j\left(\overline{{\mu }_0}\right)}{d{\overline{\mu }}^{m_0-1}}+\frac{d^{m_0-2}{F}_j\left(\overline{{\mu }_0}\right)}{d{\overline{\mu }}^{m_0-2}}.$$

Since ${\mu }_0\overline{\in }\sigma \left(A\right),$ relation (22) implies the equalities

$$\frac{d^s{F}_j\left(\overline{{\mu }_0}\right)}{d{\overline{\mu }}^s}\equiv 0\mathrm{at}s=0,1,\dots ,m_0-1.$$

(23)

Then, for $j=1,2$, the relations ${\displaystyle \frac{F_j\left(\overline{\mu }\right)}{{\chi }^{*}\left(\overline{\mu }\right)}}$ are entire functions of $\overline{\mu },$ since at the point $\overline{\mu }={\overline{\mu }}_0$, relations ${\displaystyle \frac{F_j\left(\overline{\mu }\right)}{{\chi }^{*}\left(\overline{\mu }\right)}}$ have a removable singularity.

Now, we proceed to the second step of the proof. Since H is a separable Hilbert space, there is a counting system of elements $v_1,v_2,\dots ,$ whose linear span is dense in $H.$

Function $F_j\left(\overline{\mu }\right)$ multiplies scalar by element $v_k$, and we denote them by

$$G_j^k\left(\overline{\mu }\right)\equiv {\langle F_j\left(\overline{\mu }\right),v_k\rangle }_H,j=1,2,k=1,2,\dots ,$$

(24)

where ${\langle ·,·\rangle }_H$ is a dot product in Hilbert space $H.$

The multiplicity of zeros of the functional $G_j^k\left(\mu \right)$ is not less than multiplicities of zeros of functions $F_j\left(\mu \right).$ Therefore, the relationship

$$Q_j^k\left(\overline{\mu }\right)\equiv \frac{G_j^k\left(\overline{\mu }\right)}{{\chi }^{*}\left(\overline{\mu }\right)}$$

(25)

defines entire functions from $\overline{\mu }.$

Further analysis of entire functions $Q_j^k\left(\overline{\mu }\right)$ is based on the technique of estimating the order of growth and the type of entire functions. Note that the entire function $Q_j^k\left(\overline{\mu }\right)$ does not depend on the choice of the fundamental system of solutions of the homogeneous Equation (11).

According to Lemma 1, there exists a constant M and a sequence of infinitely expanding closed contours ${\Omega }_n$ such that $\rho \in {\Omega }_n$,

$$|Q_j^k\left({\overline{\rho }}^2\right)|\le M|\rho \left|\right|G_j^k\left({\overline{\rho }}^2\right)|e^{-\left|Im\rho \right|T}$$

for all admissible index values k and j.

The last estimate and Corollary 2 imply the limit equalities

$$\underset{n\to \infty }{lim}\underset{\rho \in {\Omega }_n}{max}\left|\frac{1}{\rho }{Q}_j^k\left({\overline{\rho }}^2\right)\right|=0,j=1,2,\forall \ge 1.$$

It follows that the entire functions $Q_j^k\left({\overline{\rho }}^2\right)$ at $\rho \to \infty$ grow slower than the first degree $\left|\rho \right|$. Then, by the Liouville theorem, we obtain that

$$Q_j^k\left({\overline{\rho }}^2\right)\equiv f_j^k,j=1,2,\forall k\ge 1,$$

where $f_j^k$ are some constants.

For this reason,

$$G_j^k\left({\overline{\mu }}^2\right)=f_j^k·{\chi }^{*}\left(\overline{\mu }\right),\forall \mu \in \mathbb{C}.$$

(26)

This is because

$${\tau }_1(t,\overline{\mu })=\left|\begin{array}{cc}{R}_1(t,\overline{\mu })& R_2(t,\overline{\mu })\\ V_2\left(R_1\right)& V_2\left(R_2\right)\end{array}\right|,{\tau }_2(t,\overline{\mu })=\left|\begin{array}{cc}{V}_1\left(R_1\right)& V_1\left(R_2\right)\\ R_1(t,\overline{\mu })& R_2(t,\overline{\mu })\end{array}\right|,$$

then relation (26) takes the form

$$\left\{\begin{array}{c}{V}_2\left(R_2\right){\alpha }_1^k\left(\overline{\mu }\right)-V_2\left(R_1\right){\alpha }_2^k\left(\overline{\mu }\right)=f_1^k{\chi }^{*}\left(\overline{\mu }\right),\\ -V_1\left(R_2\right){\alpha }_1^k\left(\overline{\mu }\right)-V_1\left(R_1\right){\alpha }_2^k\left(\overline{\mu }\right)=f_2^k{\chi }^{*}\left(\overline{\mu }\right),\end{array}\right.$$

(27)

where

$${\alpha }_j^k\left(\overline{\mu }\right)={\langle {\int }_0^T\overline{R_j(t,\overline{\mu })}u\left(t\right)dt,v_k\rangle }_H,j=1,2.$$

Consequently, from system (27), we obtain the equality

$$\{V_1\left(R_1\right)V_2\left(R_2\right)-V_1\left(R_2\right)V_2\left(R_1\right)\}{\alpha }_2^k\left(\overline{\mu }\right)={\chi }^{*}\left(\overline{\mu }\right)\{f_1^k{V}_1\left(R_2\right)+f_2^k{V}_2\left(R_2\right)\}.$$

This is because

$${\chi }^{*}\left(\overline{\mu }\right)=V_1\left(R_1\right)V_2\left(R_2\right)-V_1\left(R_2\right)V_2\left(R_1\right),$$

and

$${\alpha }_2^k\left(\overline{\mu }\right)=f_1^k{V}_1\left(R_2\right)+f_2^k{V}_2\left(R_2\right).$$

Let us recall definition (9) of the boundary forms $V_1$ and $V_2$ of the adjoint problem. Then, we have the relation

$${\alpha }_2^k\left(\overline{\mu }\right)=(f_1^k{a}_{12}^{*}+f_2^k{a}_{22}^{*})+(f_1^k{a}_{13}^{*}+f_2^k{a}_{23}^{*})R_2(T,\overline{\mu })+$$

$$(f_1^k{a}_{14}^{*}+f_2^k{a}_{24}^{*})R_2^{\prime }(T,\overline{\mu }).$$

(28)

Consider relation (28) for $\mu \to \pm \infty$. Recall [16] the asymptotic formulas

$$R_2(T,\overline{\mu })=\frac{sin\sqrt{\overline{\mu }}T}{\sqrt{\overline{\mu }}}+{\int }_0^T{K}_1\left(\xi \right)\frac{sin\sqrt{\overline{\mu }}\xi }{\sqrt{\overline{\mu }}}d\xi ,$$

$$R_2^{\prime }(T,\overline{\mu })=cos\sqrt{\overline{\mu }}T+\frac{1}{2}{\int }_0^T\overline{q\left(\xi \right)}d\xi ·\frac{sin\sqrt{\overline{\mu }}T}{\sqrt{\overline{\mu }}}+{\int }_0^T{K}_2\left(\xi \right)\frac{sin\sqrt{\overline{\mu }}\xi }{\sqrt{\overline{\mu }}}d\xi ,$$

(29)

where $K_1(·)$ and $K_1(·)$ are some integrated functions.

Using Corollary 1 and Formula (29), relation (28) can be represented in the following form:

$$\frac{\delta \left(\sqrt{\overline{\mu }}\right)}{\sqrt{\overline{\mu }}}\equiv (f_1^k{a}_{12}^{*}+f_2^k{a}_{22}^{*})+(f_1^k{a}_{13}^{*}+f_2^k{a}_{23}^{*})\left(\frac{sin\sqrt{\overline{\mu }}T}{\sqrt{\overline{\mu }}}+\frac{{\epsilon }_1\left(\sqrt{\overline{\mu }}T\right)}{\sqrt{\overline{\mu }}}\right)+$$

$$(f_1^k{a}_{14}^{*}+f_2^k{a}_{24}^{*})\left(\cos \sqrt{\overline{\mu }}T+{\epsilon }_2(\sqrt{\overline{\mu }})\right),$$

(30)

where functions $\delta ,{\epsilon }_1$, and ${\epsilon }_2$ tend to zero at $\mu \to \infty$.

This is possible if and only if the numbers

$$(f_1^k{a}_{12}^{*}+f_2^k{a}_{22}^{*}),(f_1^k{a}_{13}^{*}+f_2^k{a}_{23}^{*}),(f_1^k{a}_{14}^{*}+f_2^k{a}_{24}^{*}).$$

are identically equal to zero.

Consequently, from (28), it follows that ${\alpha }_2^k\left(\overline{\mu }\right)\equiv 0.$

From the last equality, it follows that

$${\langle {\int }_0^T\overline{R_2(t,\overline{\mu })}u\left(t\right)dt,v_k\rangle }_H=0,\forall k\ge 1.$$

Since the linear span of the system $\{v_k,\forall k\ge 1\}$ is dense in H, we obtain the relation

$${\int }_0^T\overline{R_2(t,\overline{\mu })}u\left(t\right)dt=0,\forall \mu \in \mathbb{C}.$$

The required equality follows from the last relation

$$u\left(t\right)=0,\forall t\in (0,T).$$

In order to verify this, it is necessary to repeat the arguments from Ref. [16] (page 42). Thus, Theorem 3 is completely proved.

Let us give the following examples, as applied to Theorem 3, for some operators A in Equation (1). □

Example 1. 

Let $\Omega \subseteq {\mathbb{R}}^N$ be some bounded area with a smooth boundary $\partial \Omega$. In work [19], operator A is defined $A(x,D)={\displaystyle \sum _{\left|\alpha \right|\le 2l}}{a}_{\alpha }\left(x\right)D^{\alpha }$, which is an arbitrary formally self-adjoint elliptic differential operator of order $2l$ with sufficiently smooth coefficients $a_{\alpha }\left(x\right),$ where $\alpha =({\alpha }_1,\dots ,{\alpha }_N)$ is multi-index and $D=(D_1,\dots ,D_N),$$D_j=\frac{\partial }{\partial x_j}.$

The domain of operator A is given by the following boundary conditions in x:

$$B_{j}u(x,t)=\sum _{\left|\alpha \right|⩽m_j}{b}_{\alpha ,j}\left(x\right)D^{\alpha }u(x,t)=0,0\le m_j\le 2l-1,j=1,2,\dots ,l,x\in \partial \Omega .$$

(31)

where coefficients $b_{\alpha ,j}\left(x\right)$ are sufficiently smooth given functions. The following conclusion follows from Theorem 3.

Conclusion 1. 

Let operator B satisfy the requirements of Theorem 3. Then, the homogeneous operator equation

$$-\frac{{\partial }^{2}u}{\partial t^2}+q\left(t\right)u=Au,x\in \Omega ,t\in (0,T)$$

(32)

with initial-boundary conditions (2) and (31) has a trivial solution $u\in D\left(B\right)\cap D\left(A\right)$ if and only if

$$\sigma \left(B\right)\cap \sigma \left(A\right)=⌀,$$

where $\sigma \left(B\right)$ and $\sigma \left(A\right)$ are spectra of operators B and A, respectively.

This conclusion strengthens the main result of [19], since Conclusion 1 is valid for operator B with non-degenerate boundary conditions. At the same time, in [19], the operator B required that the boundary conditions be strongly regular in the sense of Birkhoff [17]. The class of non-degenerate boundary conditions is wider than the class of strongly Birkhoff-regular boundary conditions.

Example 2. 

Operator A is generated by the standard wave equation $Av(·)=v_{xx}(·)-v_{yy}(·)$ in the two-dimensional region Ω bounded by the segment $OB:0\le x\le 1$ with axis $y=0$ and characteristics $OC:x+y=0,$ $BC:x-y=1.$

The domain of operator A is given by the following boundary conditions with a shift along $(x,y)$

$$u(\theta ,0;t)=0,0\le \theta \le 1,$$

$$u\left(\frac{\theta }{2},-\frac{\theta }{2};t\right)=au\left(\frac{\theta +1}{2},\frac{\theta -1}{2};t\right),0\le \theta \le \frac{1}{2},0<t<T.$$

(33)

The following conclusion follows from Theorem 3.

Conclusion 2. 

Let the conditions of Theorem 3 be satisfied for operator B. Then, the following homogeneous operator equation

$$-\frac{{\partial }^{2}u}{\partial t^2}+q\left(t\right)u=u_{xx}(x,y;t)-u_{yy}(x,y;t),x\in \Omega ,t\in (0,T)$$

(34)

with initial-boundary conditions (2) and (33) has a trivial solution $u\in D\left(B\right)\cap D\left(A\right)$ if and only if the spectra of operators B and A do not intersect.

This conclusion strengthens the main result of the work [20].

Example 3. 

Operator A is generated by the Tricomi equation. Let $\Omega \in {\mathbb{R}}^2$ be a finite domain bounded for $y>0$ by the Lyapunov curve $\sigma ,$ ending in a neighborhood of points $O(0,0)$ and $B(1,0)$, which are small arcs of the “normal curve” ${\sigma }_0$; at $y<0$, we have the characteristic $OC:x-\frac{2}{3}{(-y)}^{3/2}=0$; $BC:x+\frac{2}{3}{(-y)}^{3/2}=1$ equations

$$Av(·)=y{v}_{xx}(·)+v_{yy}(·).$$

The boundary conditions for the Tricomi operator are given by the Dirichlet condition on the elliptic part and the fractional derivative traces of the solution along the characteristics:

$$u(x,y;t){|}_{{\sigma }_0}=0,{\sigma }_0:{\left(x-\frac{1}{2}\right)}^2+\frac{4}{9}{y}^3=\frac{1}{4},$$

(35)

$$x^{5/6}{D}_{0+}^{1/6}\left(u\left({\chi }_0\left(x\right)\right)x^{-2/3}\right)+{(1-x)}^{5/6}{D}_{1-}^{1/6}\left(u\left({\chi }_1\left(x\right)\right){(1-x)}^{-2/3}\right)=0,$$

(36)

where

$$u\left({\chi }_0\left(x\right)\right)=u\left(x,-{\left[\frac{3x}{2}\right]}^{2/3}\right),0\le x\le \frac{1}{2},$$

$$u\left({\chi }_1\left(x\right)\right)=u\left(x,-{\left[\frac{3(1-x)}{2}\right]}^{2/3}\right),\frac{1}{2}\le x\le 1.$$

Application of Theorem 3 leads to the following conclusion.

Conclusion 3. 

Let the conditions of Theorem 3 be satisfied for operator B. Then, the following homogeneous operator equation

$$-\frac{{\partial }^{2}u}{\partial t^2}+q\left(t\right)u=y{u}_{xx}(x,y;t)+u_{yy}(x,y;t),x\in \Omega ,t\in (0,T)$$

(37)

with initial-boundary conditions (2), (34), and (35) has a trivial solution $u\in D\left(B\right)\cap D\left(A\right)$ if and only if the spectra of operators B and A do not intersect.

This conclusion strengthens the main result of the work [21].