Abstract
Full Text
UDC 517.941.92
MATHEMATICS
R. A. SHIRIKYAN
ON THE SPECTRUM OF NON-SELF-ADJOINT DIFFERENTIAL OPERATORS
(Presented by Academician I. M. Vinogradov on April 2, 1968)
- In the papers \((^1,^2)\), the eigenvalues of the operator \(l_h\) (in \(L_2=L_2(0,\infty)\)) were studied:
\[ ly=-y''+q(x)y,\qquad y'(0)-hy(0)=0, \]
where \(h\) is a complex number, \(q(x)\) a complex-valued function. The spectral analysis of the operator \(L_h\) was carried out by M. A. Naimark \((^1)\). In particular, he proved that if \(q(x)\) satisfies the condition
\[ \sup_x |q(x)|\exp(\varepsilon x)<\infty,\qquad \varepsilon>0, \tag{1} \]
then the operator \(l_h\) has a finite number of eigenvalues. B. S. Pavlov \((^2)\) obtained a more precise result, namely: condition (1) can be replaced by the condition
\[ \sup_x |q(x)|\exp(\varepsilon \sqrt{x})<\infty,\qquad \varepsilon>0. \tag{2} \]
In the present note, analogous questions are considered for the equation
\[ -y''=k^2 C(x)y-kQ(x)y,\qquad x\ge 0, \tag{3} \]
where \(C(x)\) and \(Q(x)\) are square matrices of order \(n\), and the eigenvalues of the matrix \(C(x)\) are positive and distinct.
It is proved in the paper that if
\[ \|C'(x)\|\le C_1\exp(-\varepsilon_1\sqrt{x}),\qquad \|Q(x)\|\le C_2\exp(-\varepsilon_2\sqrt{x}), \tag{4} \]
where \(C_1, C_2, \varepsilon_1, \varepsilon_2>0\), then the number of eigenvalues of the operator \(L\) is finite (the operator \(L\) is defined below).
- We formulate the basic conditions on equation (3). The matrices \(C(x)\) and \(Q(x)\) satisfy the following conditions:
\(1^\circ.\ \lim\limits_{x\to\infty} C(x)=C(+\infty)\) exists, is finite and nondegenerate.
\(2^\circ.\ \|C'(x)\|^2+\|C''(x)\|\in L_1=L_1(0,\infty).\)
\(3^\circ.\ \lim\limits_{x\to\infty}\|C'(x)\|=0,\qquad \|C'(x)\|\in L_1.\)
\(4^\circ.\ \|Q(x)\|\in L_1.\)
Lemma. If conditions \(1^\circ\)—\(4^\circ\) are fulfilled, then equation (3) has \(2n\) linearly independent solutions \(y_1,y_2,\ldots,y_{2n}\). These solutions, for any fixed \(x\ge x_0\), are regular for \(\operatorname{Im} k>0\) and continuous for \(\operatorname{Im} k\ge 0\). The solutions \(y_1,y_2,\ldots,y_n\in L_2\), while the solutions \(y_{n+1},\ldots,y_{2n}\), and no nontrivial combination of them, belong to \(L_2\).
Introduce the operator
\[ l=-d^2/dx^2-k^2C(x)+kQ(x). \]
Let \(D\) be the collection of all vector-functions \(y\in L_2\), all components of which are absolutely continuous on every finite interval \([0,a]\), \(a>0\), and such that \(ly\in L_2\). Denote by \(D_L\) the collection of all vector-functions \(y\in D\) such that
\[ Ay(0,k)+By'(0,k)=0, \tag{5} \]
where \(A\) and \(B\) are constant square matrices of order \(n\), for which one of the following conditions is satisfied:
1) if \(B=0\), then \(\det A\ne 0\).
2) if \(B\ne 0\), then also \(\det B\ne 0\).
By \(L\) we denote the operator in \(L_2\) with domain \(D_L\) and \(Ly=ly\) for \(y\in D_L\).
- Introduce the function
\[ D(k)=\det\bigl[(AY(x)+BY_x'(x))|_{x=0}\bigr], \tag{6} \]
where \(Y(x)=(y_1,y_2,\ldots,y_n)\).
Following B. S. Pavlov \((^2)\), we shall call the point \(k=k_0\) a singular point of the operator \(L\) if \(D(k_0)=0\). The multiplicity of the singular point \(k=k_0\) will be called the multiplicity of the root \(D(k_0)=0\). The set of all singular points of the operator \(L\) will be denoted by \(E\), the set of all eigenvalues by \(E_0\), the set of all singular points lying on the real axis \((-\infty,\infty)\) by \(E_1\), the set of singular points of infinite multiplicity by \(E_2\), and the set of all accumulation points of eigenvalues by \(E_3\).
Theorem 1. Suppose that conditions \(1^0\)—\(4^0\) are satisfied. Then:
1) The set of eigenvalues, counted with multiplicities, satisfies the condition
\[ \sum |\operatorname{Im} k_0|<\infty . \]
2) \(E_3\subset E_1\).
3) The set \(E_1\) is bounded, closed, has measure zero, and satisfies the condition
\[ \sum |l_k|\ln |l_k|>-\infty, \]
where \(|l_k|\) is the length of the interval of adjacency \(l_k\) to the set \(E_1\), and the summation extends over all bounded intervals of adjacency.
Theorem 2. Suppose that condition \(1^0\)—\(4^0\) is satisfied. In addition, let
\[ \|C'(x)\|\le C_1\exp(-\varepsilon_1x^\alpha),\qquad \|Q(x)\|\le C_2\exp(-\varepsilon_2x^\alpha), \]
where \(C_1,C_2,\varepsilon_1,\varepsilon_2>0,\ 0<\alpha<1/2\).
Then:
1) \(E_1\subset E_2\).
2) The set \(E_2\) is bounded, closed, has measure zero, and satisfies the condition
\[ \sum |l_k|^{(1-2\alpha)/(1-\alpha)}<\infty, \]
where \(|l_k|\) is the length of the interval of adjacency \(l_k\) to the set \(E_2\), and the summation extends over all bounded intervals of adjacency.
Theorem 3. If all the conditions of Theorem 2 are satisfied and \(\alpha=1/2\), then the number of eigenvalues of the operator \(L\) is finite.
The methods of proof are analogous to the methods used in papers \((^2,^3)\).
I express my deep gratitude to M. V. Fedoryuk for posing the problem and for constant attention to the work.
Moscow Institute of Physics and Technology
Received
19 III 1968
REFERENCES
\(^1\) M. A. Naimark, Trudy Moskov. matem. obshch., 3, 181 (1954).
\(^2\) B. S. Pavlov, Problems of Mathematical Physics, issue 1, Leningrad, 1966.
\(^3\) M. V. Fedoryuk, DAN, 169, No. 2, 288 (1966).