MATHEMATICS

V. B. LIDSKII

A THEOREM ON THE SPECTRUM OF A PERTURBED DIFFERENTIAL OPERATOR

(Presented by Academician A. N. Kolmogorov on 27 IX 1956)

We consider the differential operator

[
Lu=-\Delta u+p_1(x)u,
\tag{1}
]

where (\Delta) is the Laplace operator, (x(x_1,x_2,\ldots,x_n)) is a point of the (n)-dimensional real Euclidean space (\mathcal E_n), and (p_1(x)) is a continuous complex function.

It is assumed that the domain of definition of the operator (L) is some linear manifold (D(L)), dense in the Hilbert space of complex functions whose squares are integrable over all of (\mathcal E_n). The Hilbert space is denoted, as usual, by (L_2). It is further assumed that the operator (L) is closed and has a resolvent (R_\lambda) with region of regularity (\sigma). The latter means that for (\lambda \in \sigma) the equation

[
(L-\lambda E)u=f
]

has, for every (f\in L_2), a unique solution belonging to (D(L)), and moreover the linear operator ((L-\lambda E)^{-1}\equiv R_\lambda) is bounded. Along with the operator (1), we consider the perturbed differential operator

[
\widetilde L u=-\Delta u+(p_1(x)+p_2(x))u,
\tag{2}
]

where (p_2(x)) is a continuous function satisfying the condition:

[
\lim_{|x|\to\infty} p_2(x)=0
\qquad
\left(|x|=\sqrt{x_1^2+x_2^2+\cdots+x_n^2}\right).
\tag{3}
]

The perturbed operator (\widetilde L) is defined on the same linear manifold (D(L)) as the unperturbed operator.

The main result of the present work is the following theorem.

Theorem 1. The perturbed operator

[
\widetilde L u=-\Delta u+(p_1(x)+p_2(x))u
]

possesses a resolvent (\widetilde R_\lambda), which is regular everywhere in the region of regularity (\sigma) of the resolvent of the operator (L), with the exception of no more than a countable number of points having no limit points inside (\sigma). These points are poles of (\widetilde R_\lambda).

The theorem just formulated is a generalization of a theorem of I. M. Gel'fand ((^1)), in which the function (p_1(x)) is assumed real, the operator (L) self-adjoint, and the perturbing function (p_2(x)) identically equal to zero outside a finite domain.

For the proof we, as in ((^1)), consider the equation

[
-\Delta u+(p_1(x)+p_2(x))u-\lambda u=f,
\tag{4}
]

where (\lambda\in\sigma,\ f\in L_2), and the solution (u) is sought among the functions of the linear manifold (D(L)). Make the substitution

[
u=R v,
\tag{5}
]

where (v\in L_2). Substituting the expression for (u) from formula (5) into (4), we obtain, with respect to (v), the equation

[
(E+p_2R_\lambda)v=f.
\tag{6}
]

Suppose now that the operator (E+p_2R_\lambda) has a bounded inverse. Then
(v=(E+p_2R_\lambda)^{-1}f) and, by virtue of (5),
(u=R_\lambda(E+p_2R_\lambda)^{-1}f). In this case it is obvious that the solution thus found belongs to (D(L)), is unique, and the operator
(R_\lambda(E+p_2R_\lambda)^{-1}) is bounded.

Thus, in this case

[
\widetilde{R}\lambda=R\lambda(E+p_2R_\lambda)^{-1}.
\tag{7}
]

The theorem will be proved if we show that the operator (p_2R_\lambda) is completely continuous (cf. ((^1))). The point is that, for operators of the form (E+A(\lambda)), where (A(\lambda)) is a completely continuous operator depending analytically on (\lambda), the Fredholm theorems are valid in the same full scope as for operators of the form (E+\lambda P) with completely continuous (P^*). Since the operator (p_2R_\lambda), evidently, is analytic for (\lambda\in\sigma), the validity of the theorem will be established directly from formula (7).

To prove the complete continuity of the operator (p_2R_\lambda), we shall need one proposition, which we state separately.

Lemma. Denote by ((\mathfrak R)) the ball in the (n)-dimensional Euclidean space (\mathcal E_n) with center at the origin and radius (R). Consider on ((\mathfrak R)) the family of functions (M) which coincide on this ball with the functions

[
u=R_\lambda f,
\tag{8}
]

where

[
|f|\leq 1,
\tag{9}
]

and (R_\lambda) is the resolvent of the operator (1).

The family (M) is compact in the sense of convergence in the mean.

Proof. From formula (8) it follows that

[
\int_{\mathcal E_n} |u|^2\,dx=|u|^2\leq |R_\lambda|^2,
\tag{10}
]

whence, a fortiori,

[
\int_{|x|\leq R} |u|^2\,dx\leq |R_\lambda|^2.
\tag{11}
]

We shall now prove, in addition, that

[
\int_{|x|\leq R} |\nabla u|^2\,dx\leq C
\left(|\nabla u|^2=\sum_s \frac{\partial u}{\partial x_s}\frac{\partial \overline{u}}{\partial x_s}\right).
\tag{12}
]

Thereby the compactness of the family (M) will be established (see, for example, ((^2)), p. 191).

To prove inequality (12), multiply the relation
(-\Delta u+(p_1(x)-\lambda)u=f), which is equivalent to (8), by the function
(\overline{u}(1-r^2/4R^2)^2) and integrate both sides of the equality over the ball (|x|\leq 2R). The integral

[
-\int_{|x|\leq 2R} \Delta u\,\overline{u}(1-r^2/4R^2)^2\,dx
]

(*) In paper ((^1)) it is indicated that this remark belongs to M. V. Keldysh.

integrate by parts, putting in the formula

[
\Delta g\cdot h=\operatorname{div}(\nabla g\cdot h)-\nabla g\nabla h,
]

[
g=u,\qquad h=\bar u\left(1-r^2/4R^2\right)^2 .
]

In this case the surface integral vanishes, and we arrive at the equality

[
\int_{|x|\leqslant 2R}\nabla u\nabla\bar u\left(1-\frac{r^2}{4R^2}\right)^2\,dx
-\int_{|x|\leqslant 2R}\nabla u\nabla r\,\bar u\left(1-\frac{r^2}{4R^2}\right)\frac{r}{R^2}\,dx+
]

[
+\int_{|x|\leqslant 2R}(p-\lambda)|u|^2\left(1-\frac{r^2}{4R^2}\right)^2\,dx
=\int_{|x|\leqslant 2R} f\bar u\left(1-\frac{r^2}{4R^2}\right)^2\,dx,
\tag{13}
]

from which we obtain a uniform estimate for the first integral on the left. For this purpose denote the real part of the second of the integrals on the left by (I) and estimate it. In view of the fact that

[
\nabla u\nabla r=\frac{\partial u}{\partial r},\qquad
\frac{\partial u}{\partial r}\bar u+\frac{\partial\bar u}{\partial r}u
=\frac{\partial}{\partial r}|u|^2,
]

we obtain

[
I=\operatorname{Re}\left{-\int_{|x|\leqslant 2R}\nabla u\nabla r\,\bar u\left(1-\frac{r^2}{4R^2}\right)\frac{r}{R^2}\,dx\right}
=-\int_{|x|\leqslant 2R}\frac{r}{R^2}\left(1-\frac{r^2}{4R^2}\right)\frac{\partial}{\partial r}|u|^2\,dx .
]

In the last integral we pass to spherical coordinates and integrate with respect to (r) from (0) to (2R) by parts:

[
I=-\int_{|x|=1}\left(\int_0^{2R}\frac{r^n}{R^2}\left(1-\frac{r^2}{4R^2}\right)\frac{\partial}{\partial r}|u|^2\,dr\right)d\omega=
]

[
=-\int_{|x|=1}\left(\int_0^{2R}\frac{|u|^2}{R^2}
\left[n-(n+2)\frac{r^2}{4R^2}\right]r^{n-1}\,dr\right)d\omega .
]

Here it is essential that the second term in the integration by parts is equal to zero.

Thus,

[
I=-\frac1{R^2}\int_{|x|\leqslant 2R}|u|^2
\left(n-(n+2)\frac{r^2}{4R^2}\right)\,dx,
]

and, by virtue of inequality (10), it follows from this that

[
|I|\leqslant C_1 .
\tag{14}
]

We also note that, in view of the boundedness of the function (p(x)) for (|x|\leqslant 2R) and relations (9) and (10), the inequalities

[
\left|\int_{|x|\leqslant 2R}(p-\lambda)\left(1-\frac{r^2}{4R^2}\right)^2|u|^2\,dx\right|\leqslant C_2,\qquad
\left|\int_{|x|\leqslant 2R}f\bar u\left(1-\frac{r^2}{4R^2}\right)^2\,dx\right|\leqslant C_3.
\tag{15}
]

hold. Separating now the real parts in equality (13), we obtain, by virtue of (14) and (15),

[
\int_{|x|\leqslant 2R}\nabla u\nabla\bar u\left(1-\frac{r^2}{4R^2}\right)^2\,dx\leqslant C_4 .
\tag{16}
]

Let us note, however, that

[
\int_{|x|\leqslant R}\nabla u\nabla\bar u\,dx
\leqslant \frac{16}{9}
\int_{|x|\leqslant 2R}\nabla u\nabla\bar u\left(1-\frac{r^2}{4R^2}\right)^2\,dx,
\tag{17}
]

for on the ball (|x|\leq R) the obvious inequality (1-r^2/4R^2\geq 3/4) holds. Inequalities (17) and (16) prove the validity of formula (12). Thus the lemma is proved.

Using the lemma just proved, it is already not difficult to show that the operator (p_2 R_\lambda) is completely continuous. Indeed, let us show that the family of functions

[
p_2 R_\lambda f,
\tag{18}
]

where (|f|\leq 1), is compact in (L_2). Put (R_\lambda f=u). Let (\varepsilon>0) be given. Choose (R) so large that for (|x|\geq R) one has (|p_2(x)|\leq \sqrt{\varepsilon/2}\,|R_\lambda|^{-1}). By virtue of formula (10) we then obtain

[
\int_{|x|\geq R} |p_2|^2 |u|^2\,dx \leq \frac{\varepsilon}{2}.
\tag{19}
]

Since on the ball (|x|\leq R) the family of functions (u=R_\lambda f) is compact, and the function (p_2(x)) is bounded, the family of functions (p_2 R_\lambda f) on the ball under consideration is also compact. Choose for this family a finite (\varepsilon/2)-net on the ball and extend each function of the net by zero to all of (\mathcal E^n). In view of estimate (19), in this way we obtain a finite (\varepsilon)-net for the family (p_2 R_\lambda f). Since (\varepsilon) is arbitrary, this family is compact in (L_2). This proves the complete continuity of the operator (p_2 R_\lambda), and with it the theorem as well.

Let us note a corollary following from the theorem proved.

The spectra of the operators (L) and (\widetilde L) differ from one another by no more than a countable set of points belonging to the discrete part of the spectrum.

Indeed, if (\lambda_0) is not a point of the spectrum of the operator (L), then it either does not belong to the spectrum of the perturbed operator, or is a point of the discrete part of its spectrum. Considering, on the other hand, the operator (L) as the result of perturbing the operator (\widetilde L) by the function (p_2(x)), we can draw the converse conclusion.

On the basis of the corollary formulated, one may, for example, assert that the spectrum of the equation

[
-\Delta u+ i p(x)u=\lambda u,
]

where (p(x)) is any continuous complex function tending to zero as (|x|\to 0), consists of all points of the positive half-plane ((\lambda)) and, possibly, of a countable set forming the discrete part of the spectrum.

Moscow Institute of Physics and Technology

Received
5 IX 1956

REFERENCES

1. I. M. Gel'fand, Uspekhi Mat. Nauk, 7, no. 6 (1952).
2. A. M. Molchanov, Trudy Moskov. Mat. Obshch., 2 (1953).