Full Text
F. G. MAKSUDOV
ON THE SPECTRUM OF SINGULAR NON-SELF-ADJOINT DIFFERENTIAL OPERATORS OF ORDER \(2n\)
(Presented by Academician L. S. Pontryagin, 28 VI 1963)
Let us consider the differential expression
\[ l(y) \equiv (-1)^n \bigl(P_0(x)y^{(n)}\bigr)^{(n)} + (-1)^{n-1}\bigl(P_1(x)y^{(n-1)}\bigr)^{(n-1)} + \cdots + P_n(x)y, \qquad 0 \le x < \infty, \tag{1} \]
where \(1/P_0(x),\ P_1(x),\ldots,P_n(x)\) are complex-valued functions locally summable on the interval \(R^+=[0,\infty)\).
Denote by \(D\) the totality of all functions \(y(x)\in L^2(R^+)\) such that:
1) the quasi-derivatives \(y^{[i]}(x)\), \(i=0,1,\ldots,2n-1\), exist and are absolutely continuous on every finite interval \([0,a]\), \(a>0\); 2) \(l(y)\in L^2(R^+)\).
Suppose that boundary conditions are given,
\[ U_i(y) \equiv \sum_{j=0}^{2n-1}\theta_{ij}(y)^{[j]}(0)=0, \qquad i=1,2,\ldots,n, \tag{2} \]
where \(\theta_{ij}\) are complex numbers. Denote by \(D_\theta\) the totality of all functions \(y(x)\in D\) satisfying conditions (2). In the space \(L^2(R^+)\) define the operator \(L_\theta\) as follows: its domain of definition is \(D_\theta\), and for \(y(x)\in D_\theta\), \(L_\theta y=l(y)\).
The non-self-adjoint operator \(L_\theta\) in the case \(n=1,\ P_0(x)\equiv 1\), for complex \(P_n(x)\) and complex \(\theta\), was first considered by M. A. Naimark \((^{1a})\), and since then has been the subject of investigations by many authors. In the present note we set forth some results on the spectrum of the operator \(L_\theta\) under various assumptions concerning \(P_0(x),\ P_1(x),\ldots,P_n(x)\).
\(1^\circ.\) Assume that the functions \(\left(\operatorname{Re}\frac{1}{P_0(x)}\right)'\), \(\operatorname{Im}\frac{1}{P_0(x)}\), \(P_1(x),\ldots,P_n(x)\in L(R^+)\), and let
\[ \lim_{x\to\infty}\operatorname{Re}\frac{1}{P_0(x)}>0. \]
It can be shown that under these conditions the differential equation \(l(y)=\lambda y\) has \(2n\) linearly independent solutions \(y_1(x),y_2(x),\ldots,y_{2n}(x)\) such that, as \(x\to+\infty\),
\[ y_j^{[i]}(x)=c_i(x)\rho_j^i e^{\rho_j\xi(x)}[1+o(1)], \qquad i=0,1,\ldots,2n-1;\ j=1,2,\ldots,2n, \tag{3} \]
where
\[ \xi(x)=\int_0^x \sqrt[2n]{\operatorname{Re}\frac{1}{P_0(t)}}\,dt; \]
\[ c_i(x)= \begin{cases} 1, & \text{for } i\le n-1,\\[4pt] (-1)^{i-n}\left(\operatorname{Re}\dfrac{1}{P_0(x)}\right)^{-1}, & \text{for } i\ge n, \end{cases} \]
and where \(\rho_1,\rho_2,\ldots,\rho_{2n}\) are the distinct roots of degree \(2n\) of \((-1)^n\lambda\).
We divide the complex \(s\)-plane into \(2n\) equal sectors \(S_k\), defined by the inequality
\[ \frac{k\pi}{n}<\arg s<\frac{k+1}{n}\pi . \]
Let \(\omega_1,\omega_2,\ldots,\omega_{2n}\) be all the distinct roots of the \(2n\)-th degree of unity. In each sector \(S_k\) one can choose an arrangement of the numbers \(\omega_1,\omega_2,\ldots,\omega_{2n}\) such that, for \(s\in S_k\),
\[ \operatorname{Re}s\omega_1<\operatorname{Re}s\omega_2<\cdots<\operatorname{Re}s\omega_{2n}. \]
It follows from formulas (3) that, for \(s\in S_k\), among the functions \(y_1(x),y_2(x),\ldots,\) \(\ldots,y_{2n}(x)\) exactly \(n\) functions belong to \(L^2(R^+)\), while the remaining \(n\) functions, and no nonzero linear combination of them, will belong to \(L^2(R^+)\). Let \(y_1(x),y_2(x),\ldots,y_n(x)\) be those solutions of the equation \(l(y)=\lambda y\) which belong to \(L^2(R^+)\). It is clear that the eigenfunctions of the operator \(L_\theta\) can only be linear combinations of these functions:
\(y(x)=c_1y_1(x)+c_2y_2(x)+\cdots+c_ny_n(x)\). Since \(y(x)\in D_\theta\), the corresponding eigenvalues of the operator \(L_\theta\) are determined from the equation
\[ A(s)= \begin{vmatrix} U_1(y_1) & U_1(y_2) & \cdots & U_1(y_n)\\ U_2(y_1) & U_2(y_2) & \cdots & U_2(y_n)\\ \cdots & \cdots & \cdots & \cdots\\ U_n(y_1) & U_n(y_2) & \cdots & U_n(y_n) \end{vmatrix} =0. \]
Let the complex number \(\lambda=(-1)^n s^{2n}\), \(s\in S_k\), not be an eigenvalue of the operator \(L_\theta\); taking into account that
\(y_1(x),y_2(x),\ldots,\ldots,y_n(x)\in L^2(R^+)\), while \(y_{n+1}(x),y_{n+2}(x),\ldots,y_{2n}(x)\notin L^2(R^+)\), and using the boundary conditions (2), we compute the resolvent of the operator \(L_\theta\):
\[ R_\lambda f=\int_0^\infty K(x,t,\lambda)f(t)\,dt, \]
where \(f(x)\in L^2(R^+)\),
\[ K(x,t,\lambda)= \begin{cases} \displaystyle \sum_{k=1}^{n} y_k(t,s)\,[h_k(x,s)-v_k(x,s)], & \text{for } x<t,\\[1.2em] \displaystyle \sum_{k=1}^{n} y_k(x,s)\,[h_k(t,s)-v_k(t,s)], & \text{for } x>t; \end{cases} \]
moreover \(h_k(x,s)\in L^2(R^+)\), \(k=1,2,\ldots,n\), and the functions \(v_k(x,s)\) have the form
\[ v_k(x,s)=a_k e^{-s\omega_k\xi(x)}[1+o(1)],\qquad k=1,2,\ldots,n, \]
where \(a_k\) are certain constants.
Theorem 1. Suppose the functions \(\left(\operatorname{Re}\frac{1}{P_0(x)}\right)'\), \(\operatorname{Im}\frac{1}{P_0(x)}\), \(P_1(x),\ldots,P_n(x)\) belong to \(L(R^+)\), and suppose
\[ \lim_{x\to\infty}\operatorname{Re}\frac{1}{P_0(x)}>0. \]
Then the spectrum of the operator \(L_\theta\) is continuous on the positive half-axis and discrete in the entire remaining complex \(\lambda\)-plane; moreover, the limit points of the nonreal spectrum and, for \(n>1\), the eigenvalues of the operator \(L_\theta\) may lie on the positive half-axis. For the remaining values of \(\lambda\) not belonging to the spectrum, the resolvent \(R_\lambda\) of the operator \(L_\theta\) is an integral operator with kernel \(K(x,t,\lambda)\) satisfying the conditions
\[ \int_0^\infty |K(x,t,\lambda)|^2\,dx<\infty,\qquad \int_0^\infty |K(x,t,\lambda)|^2\,dt<\infty. \tag{4} \]
Suppose \(\operatorname{Re}\frac{1}{P_0(x)}\) is not differentiable, but there exists a number \(a_0>0\) such that
\[ \operatorname{Re}\frac{1}{P_0(x)}-a_0\in L(R^+), \]
and suppose
\[ \operatorname{Im}\frac{1}{P_0(x)},\ P_1(x),\ldots,P_n(x)\in L(R^+). \]
then the equation \(l(y)=\lambda y\) has \(2n\) linearly independent solutions such that, as \(x\to+\infty\),
\[ y_j^{[i]}(x)=c_i(s\omega_j)^i e^{\rho\omega_j(x)}[1+o(1)],\qquad \rho=s\sqrt[2n]{\frac{1}{a_0}}, \]
\[ i=0,1,\ldots,2n-1;\qquad j=1,2,\ldots,2n; \]
where \(c_i\) are constants. In this case the assertion of Theorem 1 also remains valid.
\(2^\circ\). Let \(\operatorname{Re} P_n(x)\to+\infty\) as \(x\to+\infty\), and suppose that the following conditions are satisfied:
a) for sufficiently large \(x_0\), the functions \((\operatorname{Re}P_n(x))'\) and \((\operatorname{Re}P_n(x))''\) do not change sign on the interval \([x_0,\infty)\);
b) as \(x\to\infty\),
\[ (\operatorname{Re}P_n(x))'=o\left(|\operatorname{Re}P_n(x)|^\alpha\right), \qquad 0<\alpha<1+\frac{1}{2n}; \]
c) the functions
\[ \left(\operatorname{Re}\frac{1}{P_0(x)}\right)^{-1} \cdot \left(\operatorname{Re}\frac{1}{P_0(x)}\right)', \qquad \left(\operatorname{Re}\frac{1}{P_0(x)}\right)' \operatorname{Im}\frac{1}{P_0(x)} |\operatorname{Re}P_n|^{\frac{1}{2n}}, \]
\[ P_1(x)|\operatorname{Re}P_n(x)|^{-\frac{1}{2n}}, \ldots, P_{n-1}(x)|\operatorname{Re}P_n(x)|^{\frac{-2n+3}{2}}, \operatorname{Im}P_n(x)|\operatorname{Re}P_n(x)|^{\frac{-2n+1}{2}} \]
are summable on the interval \(R^+\).
Then the equation \(l(y)=\lambda y\) has \(2n\) linearly independent solutions \(y_1(x),y_2(x),\ldots,y_{2n}(x)\) such that, as \(x\to+\infty\),
\[ y_j^{[i]}(x)=c_i(x)\omega_j^i e^{\omega_j\xi(x)}[1+o(1)], \tag{5} \]
where
\[ c_i(x)= \begin{cases} \left(\operatorname{Re}\dfrac{1}{P_0(x)}\right)^{1/2} \rho^{\frac{-2n-1}{2}+i}, & \text{for } i\le n-1,\\[1.2ex] (-1)^{i-n}\left(\operatorname{Re}\dfrac{1}{P_0(x)}\right)^{-1/2} \rho^{\frac{-2n-1}{2}+i}, & \text{for } i\ge n; \end{cases} \]
\[ \xi(x)=\int_{x_0}^{x}\rho(t)\,dt = \int_{x_0}^{x} \sqrt[2n]{(-1)^n(\lambda-\operatorname{Re}P_n(t))\operatorname{Re}\frac{1}{P_0(t)}}\,dt. \]
Here \(\arg\rho(x)\) is determined by continuity from a single-valued choice of its value at a fixed value of \(x\).
Theorem 2. Let \(\operatorname{Re}P_n(x)\to\infty\) as \(x\to+\infty\), and let conditions a), b), c) be satisfied. Then the spectrum of the operator \(L_\theta\) is discrete and has no finite limit points. For all other values of \(\lambda\) not belonging to the spectrum, the resolvent \(R_\lambda\) is an integral operator with a kernel satisfying conditions (4).
\(3^\circ\). Suppose now that \(\operatorname{Re}P_n(x)\to-\infty\) as \(x\to+\infty\). From the asymptotic formulas (5) it follows that the number of linearly independent solutions of the equation \(l(y)=\lambda y\) belonging to \(L^2(R^+)\) will be \(n+1\) or \(n\), depending on whether the integral
\[ \int_{0}^{\infty}|\operatorname{Re}P_n(t)|^{-1+\frac{1}{2n}}\,dt \tag{6} \]
converges or diverges.
Theorem 3. Let \(\operatorname{Re}P_n(x)\to-\infty\) as \(x\to+\infty\), and let the integral (6) diverge; moreover, let conditions a), b), and c) be satisfied. Then the spectrum of the operator \(L_\theta\) fills the entire real axis, while in the remaining part of the complex \(\lambda\)-plane the spectrum can only be discrete. For all other values of \(\lambda\) not belonging to the spectrum, the resolvent \(R_\lambda\) is an integral operator with a kernel satisfying conditions (4).
In the case when the integral (6) converges, instead of the operator \(L_\theta\) another operator \(\hat L\) is considered, defined as follows: denote by \(D^*\) the totality of functions \(z(x)\in L^2(R^+)\), analogous to \(D\), but constructed for the adjoint differential expression
\[ l^*(y)=(-1)^n(\overline{P_0}(x)z^{(n)})^{(n)} +(-1)^{n-1}(\overline{P_1}(x)z^{(n-1)})^{(n-1)} +\cdots+\overline{P_n}(x)z . \]
Let \(y(x)\in D,\ z(x)\in D^*\); then from Lagrange’s formula
\[ \int_\alpha^\beta [\,l(y)\overline z-y\overline{l^*(z)}\,]\,dt=[y,z]_\alpha^\beta, \]
where
\[ [y,z]=\sum_{k=1}^{2n-1} y^{[2n-k]}\overline{z^{[k-1]}} -y^{[k-1]}\overline{z^{[2n-k]}}, \]
it follows that \([y,z]_0^\infty\) exists. Choose functions \(z_1(x),z_2(x),\ldots,z_n(x)\) from \(D^*\) such that the determinant
\[ \Delta(\lambda)= \left| \begin{array}{cccc} [y_1,z_1]_0^\infty & [y_2,z_2]_0^\infty & \cdots & [y_1,z_{n+1}]_0^\infty\\ [y_2,z_1]_0^\infty & [y_2,z_2]_0^\infty & \cdots & [y_2,z_{n+1}]_0^\infty\\ [y_{n+1},z_1]_0^\infty & [y_{n+1},z_2]_0^\infty & \cdots & [y_{n+1},z_{n+1}]_0^\infty \end{array} \right| \]
does not vanish identically. Denote by \(\hat D\) the set of all functions \(y(x)\in D\) satisfying the conditions \([y,z_1]_0^\infty=0,\ [y,z_2]_0^\infty=0,\ldots,[y,z_{n+1}]_0^\infty=0\), and by \(\hat L\) the operator with domain \(\hat D\) such that \(\hat Ly=l(y)\) for \(y\in\hat D\).
Computing the resolvent \(\hat R_\lambda\) of the operator \(L_\theta\), we arrive at the following result.
Theorem 4. Let \(\operatorname{Re} P_n(x)\to-\infty\) as \(x\to+\infty\), let the integral (6) converge, and let conditions a), b), and c) be fulfilled. Then the spectrum of the operator \(\hat L\) is discrete and has no finite limit points. For values of \(\lambda\) not belonging to the spectrum, the resolvent \(\hat R_\lambda\) of the operator \(\hat L\) is an integral operator with a Hilbert–Schmidt kernel.
In conclusion, the author expresses gratitude to Prof. M. A. Naimark for his attention to the present work.
Institute of Mathematics and Mechanics
Academy of Sciences of the Azerbaijan SSR
Received
16 V 1963
CITED LITERATURE
- M. A. Naimark, a) DAN, 85, No. 1 (1952); b) Trudy Moskovsk. matem. obshch., 3, 18 (1954); c) Linear Differential Operators, Moscow, 1954.