UDC 517.43 + 517.94

MATHEMATICS

F. S. ROFE-BEKETOV

SELF-ADJOINT EXTENSIONS OF DIFFERENTIAL OPERATORS IN A SPACE OF VECTOR FUNCTIONS

(Presented by Academician A. A. Dorodnitsyn on 10 VI 1968)

1. In the present note we establish the general form of self-adjoint boundary-value problems on the finite interval \([0,b]\) for differential equations \(l[y]=\lambda y\) of arbitrary order \(m\) with continuous operator coefficients. For a scalar quasi-differential expression of even order with real coefficients, a description of all self-adjoint extensions on the interval \([0,b]\) was given in the work of M. G. Krein \((^1)\) and is presented in the monographs \((^2,^3)\). For the algebraic study of scalar boundary-value problems (Bôcher, etc.) see \((^4)\).

2. The basis of the proposed investigation is the concept, introduced by us, of a Hermitian relation.

Definition. A binary relation \(\theta\) given in some Hilbert space \(H\) is called Hermitian if from \(x\theta x'\), \(y\theta y'\), where \(x,x',y,y'\in H\), it follows that

\[ (x',y)-(x,y')=0, \tag{1} \]

and from the validity of (1) for certain \(x,x'\in H\) with all pairs \(y\theta y'\) it follows that also \(x\theta x'\).

For each pair \(x\theta x'\) construct the vectors \(x^\pm=x'\pm ix\). By the equality \(U_\theta x^+=x^-\) a unitary operator \(U_\theta\) is defined, which we shall call the Cayley transform of the Hermitian relation \(\theta\).

Theorem 1. Whatever the self-adjoint operator \(A\) and the unitary operator \(U\), the relation defined by either of the equations

\[ \cos A\cdot x' - \sin A\cdot x = 0, \tag{2} \]

\[ (U-I)x' + i(U+I)x = 0, \tag{3} \]

is Hermitian. Conversely, every Hermitian relation \(x\theta x'\) is representable in the forms (2) and (3), where the unitary operators \(-e^{2iA}\) and \(U\) determine the relation \(\theta\) uniquely and are its Cayley transform \(U_\theta\).

Corollary. Any Hermitian relation \(x\theta x'\) can be represented in the form \(x'=A_1x+x^\perp\), where \(x\in H_1\), \(H_1\) is some subspace of \(H\), \(x^\perp\in H_1^\perp\), and \(A_1\) is a self-adjoint operator in \(H_1\), possibly unbounded*.

Theorem 2. Let \(B\) and \(C\) be arbitrary bounded operators on all of \(H\). The relation \(\theta\) defined by them,

\[ x\theta x' \leftrightarrow Cx' - Bx = 0 \tag{4} \]

is Hermitian if and only if the operators \(B\pm iC\) are invertible on their ranges and the operator \(U=(B+iC)^{-1}(B-iC)\) is unitary. Under these conditions the relation \(\theta\) can be represented through the operator \(U\) by formula (3).

* Thus, the known general form of self-adjoint boundary conditions for elliptic differential equations \((^5)\) also admits interpretation from the point of view of the concept of Hermitian relations.

Corollary. If \(\dim H<\infty\), relation (4) is Hermitian if and only if \(BC^*=CB^*\) and \(\det(BB^*+CC^*)\ne0\).

We shall say that a relation \(\theta\) is the closure of \(\theta\) if the graph \(\widetilde{\theta}\) in \(H\oplus H\) is the closure of the graph of \(\theta\).

Theorem 3. Let the operators \(B\) and \(C\) in (4) be arbitrary (possibly unbounded, nonclosed, and with domains \(D_B\) and \(D_C\) not dense in \(H\)). In order that the closure \(\widetilde{\theta}\) of the relation \(\theta(4)\) be Hermitian, the following conditions are necessary: 1. The linear span \(D_B\cup D_C\) is dense in \(H\). 2. The operators \(B\pm iC\) are invertible on their ranges. 3. The operator
\[ U_1=(B+iC)^{-1}(B-iC) \]
in \(H_1=D_B\cap D_C\) has a closure \(\overline{U}_1\), which is unitary. 4. \(B\{D_B\cap D_C^\perp\}=\{0\}\), \(C\{D_C\cap D_B^\perp\}=\{0\}\). Conditions 1–4, together with condition 5.
\[ \overline{D_B\cap D_C}=\overline{D_B}\cap\overline{D_C}, \]
become sufficient for the Hermiticity of \(\widetilde{\theta}\), and the Cayley transform of the relation \(\widetilde{\theta}\) is then given by the formula
\[ U_{\widetilde{\theta}}=\overline{U}_1\oplus I_{D_B^\perp}\oplus(-I_{D_C^\perp}). \]

3. Let \(\mathcal H(0,b)\) be the Hilbert space of vector-functions with values in the separable Hilbert space \(H\) and with scalar product
\[ \langle x,y\rangle=\int_0^b (x(t),y(t))_H\,dt . \]

Consider in \(\mathcal H(0,b)\) a differential operation \(l[y]\) of order \(m\). For \(m=2n\) put
\[ l[y]=\sum_{k=1}^{n}(-1)^k\{(p_{n-k}y^{(k)})^{(k)} -i[(q_{n-k}y^{(k)})^{(k-1)}+(q_{n-k}y^{(k-1)})^{(k)}]\}+p_ny. \tag{5} \]

Let all operator coefficients be self-adjoint:
\[ p_k(t)=p_k^*(t),\qquad q_k(t)=q_k^*(t) \tag{6} \]
and depend continuously on \(t\), together with their derivatives up to order \(n-k\) inclusive, and suppose that \(p_0^{-1}(t)\) exists and is bounded for \(t\in[0,b]\). Define for the operation (5) the quasiderivatives \(y^{[k]}\) by the formulas*
\[ y^{[j]}=y^{(j)}\quad (j=0,1,\ldots,n-1);\qquad y^{[n]}=p_0y^{(n)}-iq_0y^{(n-1)}, \]
\[ y^{[n+k]}=-\frac{d}{dt}y^{[n+k-1]}+p_ky^{(n-k)} +i\,[q_{k-1}y^{(n-k+1)}-q_ky^{(n-k-1)}] \]
\[ (k=1,\ldots,n;\quad q_n\equiv0;\quad l[y]\equiv y^{[2n]}). \]

Let \(L\) be the operator generated by the expression \(l[y]\) on the set \(D\) of all such \(y(t)\in\mathcal H(0,b)\) with \(m-1\) absolutely continuous (quasi-)derivatives for which \(l[y]\in\mathcal H(0,b)\), and let \(L_0\) be the restriction of \(L\) defined by the conditions
\[ y_0=y_0^{[1]}=\cdots=y_0^{[m-1]}=0,\qquad y_b=y_b^{[1]}=\cdots=y_b^{[m-1]}=0, \tag{7} \]
where \(y_0^{[k]}=y^{[k]}(0)\), \(y_b^{[k]}=y^{[k]}(b)\), \(k=0,1,\ldots\).

Denote
\[ H^m=H\oplus\cdots\oplus H \]
(\(m\) summands), and let \((\cdot,\cdot)_m\) be the scalar product in \(H^m\). To each vector-function \(u(t)\in D\) we associate a pair of vectors \(\hat u,\hat u'\in H^m\) (for \(m=2n\)):
\[ \hat u=\{u_0,u'_0,\ldots,u_0^{(n-1)},u_b,u'_b,\ldots,u_b^{(n-1)}\}, \]
\[ \hat u'=\{u_0^{[2n-1]},u_0^{[2n-2]},\ldots,u_0^{[n]},-u_b^{[2n-1]},-u_b^{[2n-2]},\ldots,-u_b^{[n]}\}. \tag{8} \]

* These requirements are easily weakened by considering \(l\) as a quasidifferential operation.

** For \(q_0=q_1=\cdots=q_{n-1}=0\) our definition coincides with that adopted in \((^{1-3})\).

Then for operation (5) Lagrange’s identity is written in the form

\[ \langle l[x],y\rangle-\langle x,l[y]\rangle=(\hat{x}',\hat{y})_m-(\hat{x},\hat{y}')_m . \tag{9} \]

Lemma 1. The equality \(L_0^*=L\) is valid.

An essential point in the proof of Lemma 1 is

Lemma 2. The manifold of vector solutions \(y=y(t)\) of the homogeneous equation of arbitrary order \(m\):
\[ y^{(m)}+g_1(t)y^{(m-1)}+\cdots+g_m(t)y=0 \]
with arbitrary continuous operator coefficients \(g_k(t)\) forms a subspace in \(\mathcal H(0,b)\) (i.e., a closed one), since the dependence between the solutions \(y(t)\in\mathcal H(0,b)\) and their Cauchy data is mutually continuous.

From (9), Theorem 1, and Lemma 1 there follows

Theorem 4. Between the self-adjoint extensions \(\widetilde L\) of the operator \(L_0\) (5), (7) and the Hermitian relations \(\theta\) in \(H^{2n}\) there exists a one-to-one correspondence, by virtue of which every \(\widetilde L\) is generated by the operation \(l[y]\) (5) and boundary conditions of any of the forms

\[ \cos \hat A\cdot \hat y' - \sin \hat A\cdot \hat y=0, \tag{10} \]

\[ (\hat U-\hat I)\hat y' + i(\hat U+\hat I)\hat y=0, \tag{11} \]

where \(\hat A,\hat U\) are respectively a self-adjoint and a unitary operator in \(H^{2n}\), and \(\hat y,\hat y'\in H^{2n}\) are determined from \(y(t)\in D\) by formula (8). Conversely, any of these boundary conditions determines some self-adjoint extension \(\widetilde L\) of the operator \(L_0\).

Example 1. Separated self-adjoint boundary conditions for \(t=0\) are always reduced to the form*

\[ \cos \hat A_0\cdot \hat y_0' - \sin \hat A_0\cdot \hat y_0=0, \tag{12} \]

where \(\hat A_0\) is an arbitrary self-adjoint operator in \(H^n\),
\[ \hat y_0=\{y_0,y_0,\ldots,y_0^{(n-1)}\},\qquad \hat y_0'=\{y_0^{[2n-1]},y_0^{[2n-2]},\ldots,y_0^{[n]}\}. \]

Example 2. Generalized periodic conditions:
\[ y_b^{[2n-k]}=U_k y_0^{[2n-k]},\qquad y_b^{(k-1)}=U_k y_0^{(k-1)},\quad k=1,\ldots,n; \]
\(U_k\) are unitary operators in \(H\).

4. Let now \(l\) be an operation of odd order \(m=2n+1\)

\[ l[y]=\sum_{k=0}^{n}(-1)^k\left\{\,i\left[(q_{n-k}y^{(k)})^{(k+1)}+(q_{n-k}y^{(k+1)})^{(k)}\right]+(p_{n-k}y^{(k)})^{(k)}\right\}, \tag{13} \]

where \(q_0^{-1}(t)\) exists and is continuous for \(t\in[0,b]\), and (6) is satisfied. For operation (13) we define quasiderivatives

\[ y^{[j]}=y^{(j)}\quad (j=0,1,\ldots,n-1),\qquad y^{[n]}=-iq_0y^{(n)}, \]

\[ y^{[n+k+1]}=-\frac{d}{dt}y^{[n+k]}+p_k y^{(n-k)} +i\,[q_k y^{(n-k+1)}-q_{k+1}y^{(n-k-1)}] \]

\[ (k=0,1,\ldots,n;\ q_{n+1}\equiv0;\ l[y]\equiv y^{[2n+1]}). \]

Let \(H_t^\pm\) be invariant subspaces of the operator \(q_0(t)\) such that \(q_0(t)>0\) on \(H_t^+\) and \(q_0(t)<0\) on \(H_t^-\), and let \(P_t^\pm\) be the orthoprojectors onto \(H_t^\pm\), respectively. Put
\[ q_\pm(t)=\pm q_0(t)P_t^\pm,\qquad Q_t^\pm=q_+^{1/2}(t)\pm q_-^{1/2}(t). \]
From the properties of \(q_0(t)\) it follows that \(\dim H_b^\pm=\dim H_0^\pm\). Let \(U_q\) be an arbitrary but fixed unitary operator in \(H\) carrying \(H_b^\pm\) into \(H_0^\pm\). To each \(v(t)\in D\) (see item 3 for \(m=2n+1\)) we associate a pair \(\hat v,\hat v'\in H^{2n+1}\):

\[ \hat v=\{Q_0^+v_0^{(n)}+U_qQ_b^+v_b^{(n)},\ v_0,\ v_0',\ldots,v_0^{(n-1)},\ v_b,\ v_b',\ldots,v_b^{(n-1)}\}, \tag{14} \]

\[ \text{* For a scalar real equation of even order, conditions of the form (12) are given as self-adjoint in }{}^{(6)}. \text{ For Schrödinger’s operator potential equation } (*)\ -y''+q(t)y=\lambda y,\text{ condition (12) is used in }{}^{(7)}. \text{ The general form of self-adjoint separated boundary conditions for equation }(*)\text{ was obtained by another method and in another form in }{}^{(8)}. \]

\[ \hat v'=\{iU_qQ_bv_b^{(n)}-iQ_0^-v_0^{(n)},\ v_0^{[2n]},\ v_0^{[2n-1]},\ldots,\ v_0^{[n+1]},\ -v_b^{[2n]},\ -v_b^{[2n-1]},\ldots, \ldots,\ -v_b^{[n+1]}\}. \tag{14} \]

Lemma 3. For the operator \(l\) (13), Lagrange’s identity has the form (9), where \(\hat x,\hat y,\hat x',\hat y'\) are defined by formulas (14).

Theorem 5. For the operator \(L_0\) (13), (7) of odd order, Theorem 4 is valid with the replacement in its formulation of \(l[y]\) (5) by \(l[y]\) (13), \(H^{2n}\) by \(H^{2n+1}\), and formulas (8) by (14).

Theorem 6. For the existence of separated self-adjoint boundary conditions for the operation \(l[y]\) (13) of order \(2n+1\) in the cases \(\dim H<\infty\) or \(\dim H=\infty\), but \(n=0\), it is necessary and sufficient that \(\dim H_0^+=\dim H_0^-\le\infty\). In this case all such boundary conditions are representable (for \(t=0\)) in the form (12), where \(\hat A_0\) is a self-adjoint operator in \(H^{n+}=H^n\oplus H_0^+\),

\[ \hat y_0=\{(q_+^{1/2}(0)+V_0q_-^{1/2}(0))y_0^{(n)},\ y_0,\ y_0',\ldots,\ y_0^{(n-1)}\}, \]

\[ \hat y_0'=\{i(V_0q_-^{1/2}(0)-q_+^{1/2}(0))y_0^{(n)},\ y_0^{[2n]},\ y_0^{[2n-1]},\ldots,\ y_0^{[n+1]}\}, \tag{15} \]

\(V_0\) is an arbitrarily fixed isometric operator mapping \(H_0^-\) onto \(H_0^+\). If, however, \(\dim H=\infty\) and \(n\ge1\), then separated self-adjoint conditions always exist and have the same form (12), but now one must, generally speaking, take

\[ \hat y_0=\{y_0^+ + V_1y_0^-,\ y_0,\ y_0',\ldots,\ y_0^{(n-2)}\}\in H^{n+}, \]

\[ \hat y_0'=\{iV_1y_0^- - iy_0^+,\ y_0^{[2n]},\ y_0^{[2n-1]},\ldots,\ y_0^{[n+2]}\}\in H^{n+}, \tag{16} \]

where

\[ y_0^+=\{q_+^{1/2}(0)y_0^{(n)},\ \tfrac12(y_0^{[n+1]}-iy_0^{(n-1)})\}\in H_0^+\oplus H, \]

\[ y_0^-=\{q_-^{1/2}(0)y_0^{(n)},\ \tfrac12(y_0^{[n+1]}+iy_0^{(n-1)})\}\in H_0^-\oplus H, \]

\(V_1\) is an arbitrarily fixed isometric operator mapping \(H_0^-\oplus H\) onto \(H_0^+\oplus H\). If, in particular, \(\dim H_0^+=\dim H_0^-\), then here too one may use formulas (15) instead of (16).

Corollary. For odd \(\dim H<\infty\), for the operation \(l[y]\) (13) of odd order there do not exist separated self-adjoint boundary conditions.

Example 3. For the operation \(i(P-P^\perp)\dfrac d{dt}y\), where \(P\) and \(P^\perp\) are orthoprojectors onto \(H_1\subseteq H\) and onto \(H_1^\perp\), separated self-adjoint boundary conditions exist if \(\dim H_1=\dim H_1^\perp\), and in this case reduce to the form \(Py(0)=V_1P^\perp y(0)\), \(Py(b)=V_2P^\perp y(b)\), where \(V_1,V_2\) are arbitrary isometric operators mapping \(H_1^\perp\) onto \(H_1\).

Example 4. Generalized periodic conditions for the operation \(l[y]\) (13) of order \(2n+1\):

\[ y_b^{[2n-k]}=U_ky_0^{[2n-k]},\qquad y_b^{(k)}=U_ky_0^{(k)},\qquad q_\pm^{1/2}(b)y_b^{(n)} \]

\[ =V^\pm q_\pm^{1/2}(0)y_0^{(n)},\qquad k=0,1,\ldots,n-1, \]

\(U_k\) are unitary operators in \(H\), and \(V^\pm\) isometrically map \(H_0^\pm\) onto \(H_b^\pm\), respectively.

Physico-Technical Institute of Low Temperatures
Academy of Sciences of the Ukrainian SSR

Received
6 VI 1968

CITED LITERATURE

1. M. G. Krein, Mat. sbornik, 21, 3, 365 (1947).
2. N. I. Akhiezer, I. M. Glazman, Theory of Linear Operators, Moscow, 1966.
3. M. A. Naimark, Linear Differential Operators, Moscow, 1954.
4. E. A. Coddington, N. Levinson, Theory of Ordinary Differential Equations, IL, 1958.
5. M. I. Vishik, Tr. Mosk. matem. obshch., 1, 187 (1952).
6. M. G. Krein, DAN, 74, No. 1, 9 (1950).
7. F. S. Rofe-Beketov, DAN, 156, No. 5, 1029 (1964).
8. M. L. Gorbachuk, Ukr. matem. zhurn., 18, No. 2, 3 (1966).