Abstract Generated abstract
This paper studies spectra of self-adjoint extensions of a closed symmetric semibounded operator with infinite deficiency index, under the assumption that the Friedrichs extension has compact inverse and eigenvalues with a prescribed power asymptotic. It proves that for any slower power growth exponent and positive leading constant, there exists a nonnegative self-adjoint extension with discrete spectrum realizing that eigenvalue asymptotic. The general result is applied to the Laplacian in the unit disk, where boundary operators are used to characterize extensions with discrete spectrum and to reduce self-adjointness of boundary-value problems to self-adjointness of associated boundary operators. It also shows that a class of energy-form, or D, extensions has discrete spectrum asymptotic to that of the Dirichlet extension.
Full Text
UDC 517.948
MATHEMATICS
M. M. GEKHTMAN
ON THE SPECTRUM OF SELF-ADJOINT EXTENSIONS OF A SYMMETRIC SEMIBOUNDED OPERATOR
(Presented by Academician I. G. Petrovskii, 23 XII 1968)
1. Consider in a Hilbert space \(\mathscr H\) a closed symmetric operator \(S\), defined on an everywhere dense set \(D(S)\). We shall assume that the operator \(S\) is semibounded from below. Without loss of generality one may suppose that
\[ (Sf,f)\geqslant (f,f),\qquad f\in D(S). \tag{1} \]
From condition (1) it follows that the deficiency indices of the operator \(S\) are equal to each other and coincide with the dimension of the subspace \(U\) of solutions of the equation
\[ S^{*}u=0. \tag{2} \]
In formula (2), \(S^{*}\) is the operator adjoint to the operator \(S\).
In what follows we shall assume that the condition
\[ \dim U=\infty \tag{3} \]
is fulfilled.
As K. O. Friedrichs first showed, there exists a self-adjoint extension \(S_\mu\) of the operator \(S\) whose lower bound is equal to one. Consequently, there exists the inverse operator \(S_\mu^{-1}\), defined on all of \(\mathscr H\). We shall assume that the operator \(S_\mu^{-1}\) is completely continuous, and hence the spectrum of the operator \(S_\mu\) is discrete.
Denote by \(\lambda_n(S_\mu)\) \((n=1,2,\ldots)\) the sequence of eigenvalues of the operator \(S_\mu\), numbered in increasing order. Suppose further that, as \(n\to\infty\),
\[ \lim \frac{\lambda_n(S_\mu)}{n^\alpha}=a,\qquad a>0,\quad \alpha>0. \tag{4} \]
For brevity we shall agree to write condition (4) in the form
\[ \lambda_n(S_\mu)\sim an^\alpha\qquad (n\to\infty). \]
Under the assumptions made, the following is true.
Theorem 1. Whatever the numbers \(0<\beta<\alpha\) and \(b(\beta)>0\) may be, there always exists at least one nonnegative self-adjoint extension \(\widetilde S\) of the operator \(S\) with discrete spectrum, whose eigenvalues \(\lambda_n(\widetilde S)\) satisfy the condition
\[ \lambda_n(\widetilde S)\sim bn^\beta\qquad (n\to\infty). \tag{5} \]
In the proof of the theorem, certain theorems of M. I. Vishik \((^1)\), M. Sh. Birman \((^2)\), and the asymptotic theorem of Fan Zhou \((^3)\) are used in an essential way.
2. Let us indicate an application of the results obtained to the study of the spectrum of self-adjoint extensions of symmetric operators generated by differential expressions.
Let \(D\) be the interior of the unit disk bounded by the circle \(\Gamma\). Denote by \(S_0\) the operator in \(L^2(D)\) generated by the differen-
differential expression \(-(\partial^2/\partial x^2+\partial^2/\partial y^2)\) on the domain of definition \(D(S_0)\), consisting of all functions twice continuously differentiable in \(D+\Gamma\), each of which is equal to zero in some contour strip of the domain \(D\). We shall further denote by \(S\) the closure of the operator \(S_0\) in the metric of \(L^2(D)\), and by \(D(S)\) its domain of definition. It is known that the operator \(S\) is symmetric and semibounded below.
M. Sh. Birman \((^2)\) showed that the domain of definition \(D(\widetilde S)\) of a self-adjoint extension \(\widetilde S\), for which \(\widetilde S^{-1}\) is a completely continuous operator, is described by the relation
\[ D(\widetilde S)=D(S)+(S_\mu^{-1}+B)U. \tag{5'} \]
In this formula \(B\) is a self-adjoint completely continuous operator for which the subspace \(U\) is invariant, and on \(\mathcal H\ominus U\) the operator \(B=0\).
We shall assume henceforth that the operator \(B\) maps smooth harmonic functions into smooth ones. Under this assumption it follows from formula (5′) that \(\widetilde S\) is the closure of its restriction to sufficiently smooth functions belonging to \(D(\widetilde S)\). In what follows, without stating this separately, when speaking of the domain of definition \(D(\widetilde S)\) we shall mean precisely the smooth functions belonging to \(D(\widetilde S)\). The subspace \(U\) coincides with the totality of all harmonic functions \(u(x,y)\in L^2(D)\). Consequently, condition (3) is fulfilled. In addition, the operator \(S_\mu^{-1}\) is completely continuous. Thus all the hypotheses of Theorem 1 are satisfied. Following M. I. Vishik \((^1)\), we define the boundary operators \(\gamma_1\) and \(\gamma_2\) as follows:
\[ \gamma_1 f=f|_\Gamma;\qquad \gamma_2 f=\left.\frac{\partial f}{\partial n}\right|_\Gamma-\left.\frac{\partial \bar f}{\partial n}\right|_\Gamma;\qquad f\in D(\widetilde S). \tag{6} \]
In formulas (6) the function \(\bar f\) is defined as the solution of the Dirichlet problem
\[ \Delta \bar f=0;\qquad \bar f|_\Gamma=\gamma_1 f. \]
Define in \(L^2(\Gamma)_{2\pi}\) the operator \(A\) by the formula
\[ Af=\int_0^{2\pi} G(s,\eta)f(\eta)\,d\eta. \tag{7} \]
In formula (7) the function \(G(s,\eta)\) is defined by the relation
\[ G(s,\eta)=\iint_D K(\bar x,s)K(\bar x,\eta)\,dx\,dy,\qquad \bar x=(x,y); \]
\(K(\bar x,s)\) is the Poisson kernel for the disk. It is easy to verify that the operator \(A\) so defined is self-adjoint, positive, and completely continuous. It is not difficult to show that there exists an operator
\[ C=(\gamma_2 S_\mu^{-1}\gamma_1^{-1})^{-1}, \tag{8} \]
and, moreover, if we denote \(f_1(s)=\gamma_1 f\), \(f_2(s)=\left.\dfrac{\partial f}{\partial n}\right|_\Gamma\), then
\[ C\gamma_2 f=H_1 f_1+H_2 f_2. \tag{9} \]
The operators \(H_1\) and \(H_2\) in the case of the disk can be computed explicitly. We note that \(H_1=A^{-1}H_0\), \(H_2=A^{-1}\).
Theorem 2. Let \(T\) be an arbitrary self-adjoint nonnegative completely continuous operator in \(L^2(\Gamma)\). In order that the spectrum of the self-adjoint extension \(\widetilde S\) of the operator \(S\) be discrete, it is sufficient that \(f(\bar x)\in D(\widetilde S)\) satisfy the relation
\[ (A^{1/2}-TA^{1/2}H_1)f_1=TA^{1/2}H_2 f_2. \tag{10} \]
If \(\lambda_n(T) \sim 1/bn^\beta\), then
\[ \lambda_n(\widetilde S) \sim bn^\beta \qquad (0<\beta<\alpha). \tag{11} \]
- Suppose now it is known that some extension \(\widetilde S\) of the operator \(S\) is determined by boundary operators \(L\) and \(M\) in \(L^2(\Gamma)\), i.e., by the relation
\[ L_1 f_1 = M f_2. \tag{12} \]
The question is whether \(\widetilde S\) will be a self-adjoint extension of \(S\), and whether the spectrum of \(\widetilde S\) is discrete.
Theorem 3. If the operator
\[ T=A^{1/2}(MH_0+L)^{-1}MA^{1/2} \tag{13} \]
is self-adjoint and completely continuous in \(L^2(\Gamma)\), then the extension \(\widetilde S\) of the operator \(S\), determined by the boundary conditions (12), will be self-adjoint in \(L^2(D)\), and the spectrum of this extension is discrete.
Thus, the question of the self-adjointness of the boundary-value problem (12) is reduced to the question of the self-adjointness in \(L^2(\Gamma)\) of the operator \((MH_0+L)^{-1}M\). It is not difficult to indicate sufficient conditions for the self-adjointness of this operator.
- We shall call \(D\)-extensions of the operator \(S\) those self-adjoint extensions \(\widetilde S\) for which
\[ (\widetilde S f,f)=\iint_D \left(\left|\frac{\partial f}{\partial x}\right|^2+\left|\frac{\partial f}{\partial y}\right|^2\right)\,dx\,dy. \tag{14} \]
Note that the self-adjoint extension \(A_0\) of the operator \(S\), characterized by the boundary condition \(f|_\Gamma=0\), is a \(D\)-extension.
Theorem 4. The spectrum of any \(D\)-extension is discrete and
\[ \lambda_n(D) \sim \lambda_n(A_0) \qquad (n\to\infty). \]
- An obvious generalization of the results obtained is possible for arbitrary domains \(D\) and elliptic differential operators.
In conclusion, the author considers it his pleasant duty to express gratitude to A. G. Kostyuchenko for posing the problem and to R. S. Ismagilov for valuable advice and discussion of the results obtained.
Received
13 XII 1968
CITED LITERATURE
- M. I. Vishik, Tr. Mosk. matem. obshch., 1, 187 (1952).
- M. Sh. Birman, Matem. sborn., 38, No. 4, 431 (1956).
- I. Ts. Gokhberg, M. G. Krein, Introduction to the Theory of Linear Non-Self-Adjoint Operators in Hilbert Space, “Nauka,” 1965.