UDC 517.947.5.37

MATHEMATICS

V. I. GORBACHUK, M. L. GORBACHUK

EXPANSION IN EIGENFUNCTIONS OF A SECOND-ORDER DIFFERENTIAL EQUATION WITH OPERATOR COEFFICIENTS

(Presented by Academician A. Yu. Ishlinskii, 5 VI 1968)

The results of G. Weyl \((^{1})\) are known, concerning expansion in eigenfunctions of the self-adjoint boundary-value problem

\[ l[y]=y''+p(t)y=\lambda y\quad (0\leq t<\infty),\qquad y'(0)-hy(0)=0, \]

where \(p(t)\) and \(h\) are real. F. S. Rofe-Beketov \((^{2})\) generalized these results to the case where \(p(t)\) and \(h\) are bounded self-adjoint operators in a separable Hilbert space \(H\).

In the present paper we consider the case where \(p(t)\) is an unbounded operator of the form \(p(t)=A-q(t)\), where \(A\) is a self-adjoint operator in \(H\), semibounded below, and \(q(t)\) is a bounded self-adjoint operator. This case includes, for example, eigenvalue problems for certain hyperbolic equations.

1. Let \(H\) be a separable Hilbert space with scalar product \((\cdot,\cdot)\) and norm \(\|\cdot\|\). Denote by \(L_2(H,(0,b))\) \((0<b\leq\infty)\) the set of all vector-functions \(u(t)\) \((0\leq t\leq b)\) with values in \(H\) such that

\[ \int_0^b \|u(t)\|^2\,dt<\infty. \]

As is known, \(L_2(H,(0,b))\) is a complete Hilbert space with scalar product

\[ (u,v)_b=\int_0^b (u(t),v(t))\,dt \qquad (u,v\in L_2(H,(0,b))). \]

Consider the differential equation

\[ l[u]=u''+Au-q(t)u=\lambda u \tag{1} \]

with the boundary condition

\[ u'(0)=0^*, \tag{2} \]

where \(q(t)=q^*(t)\) (* denotes passage to the adjoint operator) is an operator-function, continuous in the uniform operator topology, whose values are bounded operators in \(H\); \(\lambda\) is a complex number; \(A\) is a self-adjoint operator in \(H\), semibounded below. Without loss of generality one may assume that \(A>0\) and the operator \(A^{-1}\) is bounded. We also assume that the functions \(A^{1/2}q(t)A^{-1/2}\) and \(Aq(t)A^{-1}\) are strongly continuous in \(t\).

A vector-function \(u(t)\) is called a strong solution of equation (1) if \(u(t)\), for every \(t\), belongs to \(D(A)\) (\(D(A)\) is the domain of definition of the operator \(A\)), is twice strongly differentiable, and satisfies equation (1).

\[ \text{* The condition (2) is considered for simplicity of exposition. The results are also valid} \]
\[ \text{for a boundary condition of the form }u'(0)=Bu(0),\text{ where }B\text{ is a bounded self-adjoint} \]
\[ \text{operator with the property }BD(A)\subset D(A). \]

On the set \(H_{+}=D(A)\) introduce the scalar product \((f,g)_{+}=(Af,Ag)\). Then \(H_{+}\) is a complete Hilbert space with respect to \((\cdot,\cdot)_{+}\), and it may be regarded as a space with positive norm with respect to \(H_{0}=H\) (see (3)). We denote by \(H_{-}\) the space with negative norm constructed from \(H_{+}\) and \(H_{0}\).

A vector function \(u(t)\) with values in \(H\) will be called a weak solution of equation (1) if \(u(t)\) is twice weakly differentiable in \(H_{-}\) (i.e., the scalar function \((u(t),f)\) is twice differentiable for every \(f\in H_{+}\)) and

\[ \frac{d^{2}}{dt^{2}}(u(t),f)+(u(t),Af)-(u,q(t)f)=\lambda(u(t),f)\qquad (f\in H_{+}). \]

By the method of successive approximations it is not difficult to show that the integral equations

\[ \omega_{1}(t,\lambda)=\cos\sqrt{A-\lambda E}\,t+ \int_{0}^{t}\frac{\sin\sqrt{A-\lambda E}(t-x)}{\sqrt{A-\lambda E}}\, q(x)\omega_{1}(x,\lambda)\,dx, \]

\[ \omega_{2}(t,x,\lambda)= \frac{\sin\sqrt{A-\lambda E}(t-x)}{\sqrt{A-\lambda E}}+ \int_{x}^{t}\frac{\sin\sqrt{A-\lambda E}(t-s)}{\sqrt{A-\lambda E}}\, q(s)\omega_{2}(s,x,\lambda)\,ds \]

(\(E\) is the identity operator in \(H\)) have solutions in the class of strongly continuous operator functions. For fixed \(t,x\in[0,b]\), the solutions \(\omega_{1}(t,\lambda)\) and \(\omega_{2}(t,x,\lambda)\) are entire functions of \(\lambda\).

If \(f\in D(A)\), \(g\in D(A^{1/2})\), then \(\omega_{1}(t,\lambda)f\) and \(\omega_{2}(t,0,\lambda)g\) are strong solutions of equation (1), satisfying the initial data

\[ \omega_{1}(0,\lambda)f=f,\qquad \omega_{1}'(0,\lambda)f=0;\qquad \omega_{2}(0,0,\lambda)g=0,\qquad \omega_{2}'(0,0,\lambda)g=g. \]

If \(f,g\in H\), then \(\omega_{1}(t,\lambda)f\) and \(\omega_{2}(t,0,\lambda)g\) are weak solutions of this equation.

Theorem 1. The Cauchy problem \(u(0)=f,\ u'(0)=g\) \((f,g\in H)\) for equation (1) has a unique weak solution. This solution has the form

\[ u(t,\lambda)=\omega_{1}(t,\lambda)f+\omega_{2}(t,0,\lambda)g. \]

1. Denote by \(D'\) the set of all functions \(u(t)\in L_{2}(H,(0,b))\) that are twice strongly differentiable in \(H\), for each fixed \(t\in[0,b]\) belong to \(D(A)\), satisfy condition (2), and are such that \(l[u]\in L_{2}(H,(0,b))\). On \(D'\) define the operator \(L'\): \(L'u=l[u]\). Denote also by \(D_{0}'\) the set of functions \(u(t)\in D'\) finite in a neighborhood of the point \(b\), and by \(L_{0}'\) the restriction of \(L'\) to \(D_{0}'\). The operator \(L_{0}'\) is Hermitian and \(L'\subset L_{0}'{}^{*}\). Let \(L_{0}\) be the closure of \(L_{0}'\).

Theorem 2. \(D(L_{0}^{*})\) consists of functions \(u(t)\in L_{2}(H,(0,b))\) having two weak derivatives in \(H_{-}\) and such that \(u'(0)=0\). If \(L_{0}^{*}u=u^{*}\), then

\[ u(t)=\omega_{1}(t,0)f+\int_{0}^{t}\omega_{2}(t,x,0)u^{*}(x)\,dx \qquad (f=u(0)\in H). \tag{3} \]

In the case \(b<\infty\), the operator \(L_{0}^{*}\) coincides with the closure of the operator \(L'\) in \(L_{2}(H,(0,b))\), and \(D(L_{0})\) consists of functions \(u(t)\in L_{2}(H,(0,b))\), twice weakly differentiable in \(H_{-}\), and such that \(u'(0)=u(b)=u'(b)=0\).

1. Denote by \(\mathfrak N_{\lambda}\) the defect subspace of the operator \(L_{0}\). The subspace \(\mathfrak N_{\lambda}\) consists of vectors of the form \(\omega_{1}(t,\overline{\lambda})f\), where \(f\in H\) is such that

\[ \int_{0}^{b}\|\omega_{1}(t,\overline{\lambda})f\|^{2}\,dt<\infty. \]

In what follows we shall assume that \(b<\infty\). In this case \(L_{0}\) has infinite defect numbers.

Put

\[ I_{\lambda}=I_{\lambda,b}=\int_{0}^{b}\omega_{1}^{*}(t,\lambda)\omega_{1}(t,\lambda)\,dt. \]

Lemma 1. The operator \(I_{\lambda}\) is invertible for every complex \(\lambda\).

Let \(\widetilde L\) be a self-adjoint extension in \(L_2(H,(0,b))\) of the operator \(L_0\). Then \(D(\widetilde L)\) consists of elements of the form

\[ y=u+\omega_1(\cdot,\widetilde\lambda_0)I_{\lambda_0}^{-1/2}UI_{\lambda_0}^{1/2}f-\omega_1(\cdot,\lambda_0)f, \tag{4} \]

where \(u\in D(L_0)\); \(f\in H\); \(U\) is a unitary operator in \(H\); \(\lambda_0\) is a fixed nonreal number \((\operatorname{Im}\lambda_0>0)\). Conversely, if \(U\) is a unitary operator in \(H\), then formula (4) defines the domain of a certain self-adjoint extension \(\widetilde L\) of the operator \(L_0\) in \(L_2(H,(0,b))\).

Theorem 3. A vector-function \(y(t)\in D'\cap D(\widetilde L)\) if and only if it satisfies the boundary condition

\[ [\omega_1^*(b,\lambda_0)-I_{\lambda_0}^{1/2}UI_{\bar\lambda_0}^{-1/2}\omega_1^*(b,\bar\lambda_0)]y'(b)- \]

\[ -[{\omega_1'}^{*}(b,\lambda_0)-I_{\lambda_0}^{1/2}UI_{\bar\lambda_0}^{-1/2}{\omega_1'}^{*}(b,\bar\lambda_0)]y(b)=0. \tag{5} \]

Between the set of all self-adjoint extensions of \(L_0\) in \(L_2(H,(0,b))\) and the set of all unitary operators \(U\) in \(H\) there exists a one-to-one correspondence.

1. We shall say that an operator-function \(M(z)\) in \(H\) belongs to the class \(R\) if it is analytic in the upper half-plane and \(\operatorname{Im}M(z)=[M(z)-M^*(z)]/2i\ge0\) for \(\operatorname{Im}z>0\). An operator-function \(M(z)\) of class \(R\) is represented uniquely in the form

\[ M(z)=P+Qz+\int_{-\infty}^{\infty}\frac{1+tz}{t-z}\,d\sigma(t), \tag{6} \]

where \(P\) is a bounded self-adjoint operator and \(Q\) is a bounded positive operator; \(\sigma(t)\) \((\sigma(0)=0,\ \sigma(t-0)=\sigma(t)\) in the strong sense) is a monotonically increasing operator function such that \(\sigma(+\infty)\) is a bounded operator.

Theorem 4. There exists an operator-function \(M(z)\in R\) such that

\[ \chi(b,\lambda)=\omega_2(b,0,\lambda)-\omega_1(b,\lambda)M(\lambda)=0. \]

Denote

\[ G(t,s,\lambda)= \begin{cases} \omega_1(t,\lambda)\chi^*(s,\bar\lambda), & \text{for } 0\le t\le s\le b,\\ \chi(t,\lambda)\omega_1^*(s,\bar\lambda), & \text{for } 0\le s\le t\le b, \end{cases} \]

where \(\chi(t,\lambda)=\omega_2(t,0,\lambda)-\omega_1(t,\lambda)M(\lambda)\). The function \(G(t,s,\lambda)\) has the following properties:

1) for fixed \(s\) and \(t\), \(G(t,s,\lambda)\) is an analytic function of \(\lambda\) \((\operatorname{Im}\lambda\ne0)\);

2) for fixed \(\lambda\), \(G(t,s,\lambda)\) \((0\le t,s\le b)\) is a strongly continuous function jointly in the variables \(t\) and \(s\);

3) for fixed \(s\) \((0\le s\le b)\), the vector-function \(G(t,s,\lambda)f,\ f\in H\), has a weak derivative with respect to \(t\) in \(H\) in each of the intervals \([0,s)\) and \((s,b]\), and, at \(t=s\),

\[ \frac{\partial}{\partial t}G(s+0,s,\lambda)f-\frac{\partial}{\partial t}G(s-0,s,\lambda)f=f; \]

4) in each of the intervals \([0,s)\) and \((s,b]\), \(G(t,s,\lambda)f,\ f\in H\), as a function of \(t\), is a weak solution of equation (1), satisfying the conditions \(u'(0)=0,\ u(b)=0\).

Let \(L_1\) be the closure in \(L_2(H,(0,b))\) of the operator \(L_1'\), defined on the set of functions \(y(t)\in D'\) satisfying the conditions \(y'(0)=0,\ y(b)=0\), by means of the differential expression \(l[y]\). \(L_1\) is a self-adjoint extension of the operator \(L_0\), and its resolvent, by virtue of properties 1)—4), admits the integral representation

\[ R_\lambda y=(L_1-\lambda E)^{-1}y=\int_0^b G(\cdot,s,\lambda)y(s)\,ds. \tag{7} \]

1. Introduce the \(\omega_1\)-transform of a function \(y(t)\in L_2(H,(0,b))\):

\[ \widetilde y(\lambda)=\int_0^b \omega_1^*(t,\bar\lambda)y(t)\,dt. \]

An operator function \(\rho(\lambda)\) \((\rho(\lambda-0)=\rho(\lambda),\ \rho(0)=0)\) will be called a distribution function if \(\rho(\Delta)=\rho(\lambda+\Delta)-\rho(\lambda)\) is a positive bounded operator in \(H\).

Theorem 5. There exists an operator distribution function \(\rho(\lambda)\) \((-\infty<\lambda<\infty)\) such that, for any \(y(t),z(t)\in L_2(H,(0,b))\),

\[ \int_0^b (y(t),z(t))\,dt=\int_{-\infty}^{\infty}(d\rho(\lambda)\,\widetilde y(\lambda),\widetilde z(\lambda)) \tag{8} \]

\[ \bigl(d\rho(\lambda)=(1+\lambda^2)d\sigma(\lambda),\ \text{where } \sigma(\lambda)\text{ is the function from the representation }(6),\text{ uniquely determined by }M(z)\bigr). \]

An operator function \(\rho(\lambda)\) for which (8) holds will be called the spectral function of problem (1)—(2) on the interval \([0,b]\).

1. Consider the case when \(b=\infty\). If \(\beta_1<\beta_2<\infty\), then \(0<I_{\lambda,\beta_1}<I_{\lambda,\beta_2}\); on the basis of (3), \(I_{\lambda,\beta_1}^{-1}\ge I_{\lambda,\beta_2}^{-1}\), i.e. \(I_{\lambda,\beta}^{-1}\) \((0\le\beta<\infty)\) is a monotonically decreasing operator function in \(H\). Therefore, as \(\beta\to\infty\), it converges to a positive bounded operator \(\Gamma_\lambda\). Similarly to how this is done in \(^4\), one can prove the following theorems.

Theorem 6. For each \(\lambda\) \((\operatorname{Im}\lambda\ne0)\) there exists an operator solution \(x(t,\lambda)\) of equation (1) such that, for all \(f\in H\),

\[ \int_0^\infty \|x(t,\lambda)f\|^2dt<\infty. \]

Theorem 7. The dimension of the orthogonal complement to the null manifold of the operator \(\Gamma_\lambda\) \((\operatorname{Im}\lambda\ne0)\) coincides with the dimension of the defect subspace \(\mathfrak N_\lambda\) of the operator \(L_0\), and hence is the same for all \(\lambda\) from a connected component of the regularity field of the operator \(L_0\).

Using Theorem 5 for arbitrary \(\beta<\infty\), by the method of stretching the interval (see \(^5\)) we arrive at the assertion.

Theorem 8. There exists a spectral function \(\rho(\lambda)\) of problem (1)—(2) on the half-axis \((0,\infty)\), i.e. one such that, for any \(y,z\in L_2(H,(0,b))\),

\[ \int_0^\infty (y(t),z(t))\,dt = \int_{-\infty}^{\infty}(d\rho(\lambda)\,\widetilde y(\lambda),\widetilde z(\lambda)) \left( \int_{-\infty}^{\infty}\frac{(d\rho(\lambda)f,f)}{1+\lambda^2}<\infty \right). \]

In conclusion the authors express their gratitude to Yu. M. Berezanskii for his attention to this work.

Institute of Mathematics
Academy of Sciences of the Ukrainian SSR

Received
30 V 1968

REFERENCES

1. H. Weyl, Math. Ann., 68, 222 (1910).
2. F. S. Rofe-Beketov, Matem. sborn., 51 (93), No. 3, 293 (1960).
3. Yu. M. Berezanskii, Expansion in Eigenfunctions of Self-Adjoint Operators, Kiev, 1965.
4. M. L. Gorbachuk, Ukr. matem. zhurn., 18, No. 2, 3 (1966).
5. B. M. Levitan, Expansion in Eigenfunctions of Second-Order Differential Equations, Moscow—Leningrad, 1950.