MATHEMATICS

V. N. FUNTAKOV

ON EXPANSION IN EIGENFUNCTIONS OF A NON-SELF-ADJOINT DIFFERENTIAL OPERATOR OF ARBITRARY EVEN ORDER ON THE HALF-AXIS ([0,\infty))

(Presented by Academician A. N. Kolmogorov on 21 I 1960)

The present work is an extension of the results obtained in ((^1)) to the case of operators of arbitrary even order. The work uses methods first applied in ((^1)).

Consider the differential expression:

[
l(y)=y^{(2n)}+p_2(x)y^{(2n-2)}+p_3(x)y^{(2n-3)}+\cdots+p_{2n}(x)y,
\tag{1}
]

where (p_k(x)), (k=2,3,\ldots,2n), are complex-valued functions summable on the interval ([0,\infty)). Denote by (D) the set of all functions (y(x)\in L^2(0,\infty)) such that: 1) the derivatives (y^{(\nu)}(x)), (\nu=1,2,\ldots,2n-1), exist and are absolutely continuous in every finite interval ([0,b]), (b>0); 2) (l(y)\in L^2(0,\infty)).

Denote by (D_\alpha) the set of all functions (y(x)\in D) satisfying the boundary conditions

[
u_\nu(y)=\alpha_{\nu0}y(0)+\alpha_{\nu1}y'(0)+\cdots+\alpha_{\nu,\,2n-1}y^{(2n-1)}(0)=0,
\quad \nu=1,2,\ldots,n,
\tag{2}
]

where (\alpha_{\nu k}) are complex numbers.

Define the operator (L_\alpha) as follows: its domain of definition is (D_\alpha), and for (y\in D_\alpha)

[
L_\alpha y=l(y).
\tag{3}
]

The operator (L_\alpha^*), adjoint to (L_\alpha), is constructed analogously for the differential expression adjoint to (1),

[
l^*(z)=z^{(2n)}+(\bar p_2 z)^{(2n-2)}-(\bar p_3 z)^{(2n-3)}+\cdots+\bar p_{2n}z
\tag{4}
]

and for the boundary conditions adjoint to (2),

[
v_\nu(z)=\beta_{\nu0}z(0)+\beta_{\nu1}z'(0)+\cdots+\beta_{\nu,\,2n-1}z^{(2n-1)}(0)=0,
\quad \nu=1,2,\ldots,n.
\tag{5}
]

Put (\rho^{2n}=-\lambda). Let (\omega_1,\ldots,\omega_{2n}) be the roots of degree (2n) of (-1); divide the complex (\rho)-plane into (2n) equal sectors (S_k), (k=0,1,\ldots,2n-1), defined by the inequality

[
\frac{k\pi}{n}<\arg \rho<\frac{(k+1)\pi}{n}.
]

In each sector (S_k) one can choose such an arrangement of the numbers (\omega_1,\ldots,\omega_{2n}) that for (\rho\in S_k)

[
\operatorname{Re}(\rho\omega_1)\leq \operatorname{Re}(\rho\omega_2)\leq\cdots\leq \operatorname{Re}(\rho\omega_{2n})
]

(see ((^2))). Denote by (T_k) and (T_{k-1}) the boundaries of the sector (S_k). Suppose that the functions (p_k(x)) satisfy the additional condition

[
e^{\varepsilon x}|p_k(x)|\leq c_k;
\tag{6}
]

(c_k) are constants, (\varepsilon_2) is a certain fixed number. It can be shown that the equation (l(y)=\lambda y) has linearly independent solutions (y_k(x,\rho)), (k=1,\ldots,2n), holomorphic with respect to (\rho) for (\rho\in S_k) and satisfying the asymptotic conditions:

as (x\to+\infty),

[
y_k^{(\nu)}(x,\rho)=\rho^\nu e^{\rho\omega_k x}\left[\omega_k^\nu+O(1)\right]
\tag{7}
]

uniformly with respect to (\rho\in S_k);

as (\rho\to\infty),

[
y_k^{(\nu)}(x,\rho)=\rho^\nu e^{\rho\omega_k x}\left[\omega_k^\nu+O\left(\frac1\rho\right)\right]
\tag{8}
]

uniformly with respect to (x\in[0,\infty)).

In the domain (\widetilde S_k), defined by the relation

[
\left|\operatorname{Re}(\rho\omega_n)\right|\leq\varepsilon_1,\qquad
0<\varepsilon_1<\varepsilon_2,\qquad
\rho\in S_k \text{ or } S_{k+1},
]

the solution (y_{n+1}) is replaced by (\widehat y_{n+1}), which also satisfies conditions (7), (8).

Denote

[
A(\rho)=
\left|
\begin{array}{cccc}
u_1(y_1)\ldots u_1(y_{n-1}) & u_1(y_n)\
\cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot & \cdot\
\cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot & \cdot\
u_n(y_1)\ldots u_n(y_{n-1}) & u_n(y_n)
\end{array}
\right|,
\qquad
\widetilde A(\rho)=
\left|
\begin{array}{cccc}
u_1(y_1)\ldots u_1(y_{n-1}) & u_1(\widehat y_{n+1})\
\cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot & \cdot\
\cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot & \cdot\
u_n(y_1)\ldots u_n(y_{n-1}) & u_n(\widehat y_{n+1})
\end{array}
\right|.
]

For simplicity we shall assume that

[
A(\rho)\ne0,\qquad \widetilde A(\rho)\ne0
\tag{9}
]

for (\rho\in T_k), and that the eigenvalues of the operator (L_\alpha) are simple.

Theorem 1. The spectrum of the operator (L_\alpha) is continuous for even (n) on the positive semi-axis (for odd (n), respectively, on the negative semi-axis) and is discrete in the entire remaining complex (\lambda)-plane. The eigenvalues of the operator (L_\alpha) form a finite set. For values of (\lambda) not belonging to the spectrum, the resolvent ((L_\alpha-\lambda 1)^{-1}) of the operator (L_\alpha) is a bounded integral operator with kernel (K(x,\xi,\lambda)), satisfying the conditions:

[
\int_0^\infty |K(x,\xi,\lambda)|^2\,d\xi<+\infty,
\qquad
\int_0^\infty |K(x,\xi,\lambda)|^2\,dx<+\infty .
]

Consider the auxiliary boundary-value problem on the interval ([0,b]):

[
l(y)=\lambda y,
]

[
u_\nu(y)=\alpha_{\nu0}y(0)+\alpha_{\nu1}y'(0)+\cdots+\alpha_{\nu,\,2n-1}y^{(2n-1)}(0)=0,
]

[
u_{\mu b}(y)=\gamma_{\mu0}y(b)+\gamma_{\mu1}y'(b)+\cdots+\gamma_{\mu,\,2n-1}y^{(2n-1)}(b)=0,
\tag{10}
]

[
\nu,\mu=1,2,\ldots,n.
]

The boundary-value problem adjoint to (10) is constructed by means of the differential expression (4) and the boundary conditions adjoint to (10):

[
v_\nu(z)=\beta_{\nu0}z(0)+\beta_{\nu1}z'(0)+\cdots+\beta_{\nu,\,2n-1}z^{(2n-1)}(0)=0,
]

[
v_{\mu b}(z)=\delta_{\mu0}z(b)+\delta_{\mu1}z'(b)+\cdots+\delta_{\mu,\,2n-1}z^{(2n-1)}(b)=0,
\tag{11}
]

[
\mu,\nu=1,\ 2,\ldots,n.
]

Denote

[
R_n=
\left|
\begin{array}{ccc}
u_{1b}(y_{n+1}) & u_{1b}(y_{n+2})\ldots & u_{1b}(y_{2n})\
\cdot & \cdot & \cdot\
\cdot & \cdot & \cdot\
u_{nb}(y_{n+1}) & u_{nb}(y_{n+2})\ldots & u_{nb}(y_{2n})
\end{array}
\right|,
\qquad
R_{n+1}=
\left|
\begin{array}{ccc}
u_{1b}(y_n) & u_{1b}(y_{n+2})\ldots & u_{1b}(y_{2n})\
\cdot & \cdot & \cdot\
\cdot & \cdot & \cdot\
u_{nb}(y_n) & u_{nb}(y_{n+2})\ldots & u_{nb}(y_{2n})
\end{array}
\right|.
]

We shall assume that for (\rho\in T_k), (R_n\ne0), (R_{n+1}\ne0). For (\rho\in \bar S_k) define the single-valued holomorphic function (\omega(\rho)) by the relations:

[
\omega(\rho)=\ln\left[\frac{R_n}{R_{n+1}}\frac{A(\rho)}{\bar A(\rho)}\right],
\qquad
\lim_{\rho\to\infty}\omega(\rho)=i\arg\frac{\theta_{-1}}{\theta_1},
\tag{12}
]

[
\theta_1=
\left|
\begin{array}{ccc}
\omega_1^{2n-1}\ldots & \omega_{n-1}^{2n-1} & \omega_{n+1}^{2n-1}\
\cdot & \cdot & \cdot\
\cdot & \cdot & \cdot\
\omega_1^n\ldots & \omega_{n-1}^n & \omega_{n+1}^n
\end{array}
\right|
\cdot
\left|
\begin{array}{ccc}
\omega_n^{2n-1} & \omega_{n+2}^{2n-1}\ldots & \omega_{2n}^{2n-1}\
\cdot & \cdot & \cdot\
\cdot & \cdot & \cdot\
\omega_n^n & \omega_{n+2}^n\ldots & \omega_{2n}^n
\end{array}
\right|,
]

[
\theta_{-1}=
\left|
\begin{array}{ccc}
\omega_1^{2n-1}\ldots & \omega_n^{2n-1}\
\cdot & \cdot\
\cdot & \cdot\
\omega_1^n\ldots & \omega_n^n
\end{array}
\right|
\cdot
\left|
\begin{array}{ccc}
\omega_{n+1}^{2n-1}\ldots & \omega_{2n}^{2n-1}\
\cdot & \cdot\
\cdot & \cdot\
\omega_{n+1}^n\ldots & \omega_{2n}^n
\end{array}
\right|.
]

For sufficiently large (b), to each of the eigenvalues (\lambda_1,\lambda_2,\ldots,\lambda_r) of the operator (L_\alpha) there corresponds exactly one eigenvalue (\lambda_1(b),\lambda_2(b),\ldots,\lambda_r(b)) of the boundary-value problem (10) such that (\lambda_k(b)\to\lambda_k) as (b\to\infty). All the remaining eigenvalues of the boundary-value problem (10) satisfy the asymptotic relation

[
\lambda=-\rho_k^{2n},\qquad
\rho_k\omega_n=\frac{k\pi i}{b}+\frac{1}{2b}\omega\left(\frac{k\pi i}{\omega_n b}\right)+\frac{1}{b}o(1)
\tag{13}
]

as (b\to\infty), uniformly with respect to (\rho_k) in the region

[
|\operatorname{Re}(\rho_k\omega_n)|\le \varepsilon_1,\qquad
0\le |\rho_k|\le N.
]

Let (y_k(x)) be an eigenfunction of the operator (L_\alpha) corresponding to the eigenvalue (\lambda_k), (k=1,2,\ldots,r);

[
y_k(x)=-\sum_{i=1}^{n-1}\frac{\Delta_i}{\Delta_0}y_i(x,\rho)+y_n(x,\rho),
\tag{14}
]

where

[
\Delta_i=
\left|
\begin{array}{ccccc}
u_1(y_1)\ldots & u_1(y_{i-1}) & u_1(y_n) & u_1(y_{i+1})\ldots & u_1(y_{n-1})\
\cdot & \cdot & \cdot & \cdot & \cdot\
\cdot & \cdot & \cdot & \cdot & \cdot\
u_{n-1}(y_1)\ldots & u_{n-1}(y_{i-1}) & u_{n-1}(y_n) & u_{n-1}(y_{i+1})\ldots & u_{n-1}(y_{n-1})
\end{array}
\right|,
]

[
\Delta_0=
\left|
\begin{array}{ccc}
u_1(y_1)\ldots & u_1(y_{n-1})\
\cdot & \cdot\
\cdot & \cdot\
u_{n-1}(y_1)\ldots & u_{n-1}(y_{n-1})
\end{array}
\right|,
]

(z_k(x,\rho)), (y_k(x,b)), (z_k(x,b)), (k=1,2,\ldots,r), are the eigenfunctions of the operator (L_\alpha^{*}) and of the boundary-value problems (10) and (11), constructed in an analogous way. Then, as (b\to\infty), the relation

[
\frac{y_k(y,b)\,\bar z_k(\xi,b)}
{\displaystyle\int_0^b y_k(x,b)\bar z_k(\xi,b)\,dx}
=
\frac{y_k(x)\bar z_k(\xi)}
{\displaystyle\int_0^\infty y_k(x)\bar z_k(x)\,dy}
+o(1),
\qquad k=1,2,\ldots,r,
\tag{15}
]

holds.

uniformly with respect to (x,\ 0\leq x\leq c,\ c>0). Denote by (y(x,\rho_k)), (z(x,\rho_k)), (k=1,2,\ldots), the eigenfunctions of problems (10), (11) corresponding to the eigenvalues (13). Then, as (b\to\infty),

[
\frac{1}{b}\int_0^b y(x,\rho_k)\,\overline{z}(x,\rho_k)\,dx
=
-\left[
\frac{A(\rho_k)}{\widetilde A(\rho_k)}
+
\frac{B(\rho_k)}{\widetilde B(\rho_k)}
\right]+o(1),
\tag{16}
]

where (B(\rho)), (\widetilde B(\rho)) are functions constructed analogously to the functions (A(\rho)), (\widetilde A(\rho)) for (L_\alpha^*).

Let (G(x,\xi,\lambda)) be the resolvent kernel of the boundary-value problem (10). If (\lambda) does not belong to the spectrum of the operator (L_\alpha), then, as (b\to\infty),

[
G(x,\xi,\lambda)=K(x,\xi,\lambda)+o(1)
\tag{17}
]

uniformly with respect to (x,\xi) in every finite square (0\leq x,\xi\leq c,\ c>0).

Denote

[
\widetilde y(x,\rho)
=
-\sum_{i=1}^{n-1}\frac{\Delta_i}{\Delta_0}y_i(x,\rho)
+y_n(x,\rho)
+\frac{A(\rho)}{\widetilde A(\rho)}
\left[
\sum_{i=1}^{n-1}\frac{\Delta_i'}{\Delta_0}y_i(x,\rho)
-\widehat y_{n+1}(x,\rho)
\right],
\tag{18}
]

where

[
\Delta_i'
=
\left|
\begin{array}{cccccc}
u_1(y_1)&\cdots&u_1'(y_{i-1})&u_1(\widehat y_{n+1})&u_1(y_{i+1})&\cdots u_1(y_{n-1})\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\
u_{n-1}(y_1)&\cdots&u_{n-1}(y_{i-1})&u_{n-1}(\widehat y_{n+1})&u_{n-1}(y_{i+1})&\cdots u_{n-1}(y_{n-1})
\end{array}
\right|,
]

and let (\widetilde z(x,\rho)) denote the function constructed analogously for the operator (L_\alpha^*). Then the following holds.

Theorem 2. Suppose that conditions (6) and (9) are satisfied, and let (K(x,\xi,\lambda)) be the resolvent kernel of the operator (L_\alpha). For any point (\lambda) not belonging to the spectrum of the operator (L_x):

[
K(x,\xi,\lambda)
=
\sum_{k=1}^{r}
\frac{y_k(x)\overline{z_k}(\xi)}
{(\lambda-\lambda_k)\displaystyle\int_0^\infty y_k(x)\overline{z_k}(x)\,dx}
+
\frac{\omega_n}{\pi i}
\int_{\Gamma_k}
\frac{\widetilde y(x,\rho)\widetilde z(\xi,\rho)\,d\rho}
{(\rho^{2n}+\lambda)
\left[
\frac{A(\rho)}{\widetilde A(\rho)}
+
\frac{B(\rho)}{\widetilde B(\rho)}
\right]},
\tag{19}
]

where the integral converges absolutely and uniformly with respect to (x,\xi) in the region (0\leq x,\xi<\infty).

Denote by (\mathfrak M_\alpha) the totality of all functions (g(x)) satisfying the conditions: 1) (g(x)), (l(g)) are summable on the interval ([0,\infty)); 2) (g^{(\nu)}(x)), (\nu=1,\ldots,2n-1), exist and are absolutely continuous on each finite interval ([0,a]); 3) (u_\nu(g)=0,\ \nu=1,\ldots,n). Then, with the aid of formula (19) and by repeating the arguments given in (1), it is easy to obtain the expansion of the function (g(x)) in the eigenfunctions of the operator (L_\alpha) and an analogue of Parseval’s equality.

In conclusion, the author expresses his deep gratitude to M. A. Naimark for valuable advice and comments.

Moscow Institute of Physics and Technology

Received
28 XII 1959

CITED LITERATURE

1. M. A. Naimark, Trudy Moskovskogo matematicheskogo obshchestva, 3, 181 (1954).
2. M. A. Naimark, Linear Differential Operators, 1954.