UDC 517.948.35

MATHEMATICS

A. A. ARSEN’EV

ON EXPANSION IN EIGENFUNCTIONS OF THE SCHRÖDINGER OPERATOR WITH A STRONGLY SINGULAR POTENTIAL

(Presented by Academician A. N. Tikhonov, 3 I 1968)

Consider the behavior of a quantum particle in the potential field shown in Fig. 1. It is clear that if the height of the barrier (V_0) is sufficiently large, then the wave function will differ little from the wave function of a particle in the potential field shown by the dotted line in Fig. 1. Our article is devoted to a rigorous proof of this fact. The usual methods of perturbation theory are not suitable here, since the perturbation of the potential in the present case is infinite. A. A. Samarskii ((^1)), using the example of the Helmholtz equation, showed that in such problems it is necessary to introduce a certain nontrivial metric for the perturbation. We shall proceed in the same way.

Fig. 1

Let (R_N) be an (N)-dimensional Euclidean space ((N \ge 3)), (\Delta) the Laplace operator, (V(x)) a scalar function (potential), (\Omega={x;\, V(x)=+\infty}). Below we assume that the function (V(x)) satisfies two conditions:
1) each function (V_M(x)=\min{V(x),M}), for (M \ge 0), is nonnegative and locally satisfies the Hölder condition;
2) there exist constants (\alpha>0), (C<\infty), and (R<\infty), independent of (M), such that for all (x) lying outside the ball of radius (R), the estimate
[
V(x)<C|x|^{-N-\alpha}
]
holds.

Let (G_M(t)) be the Green function of the Cauchy problem
[
u(x,0)=u_0(x);\quad \partial u/\partial t=\Delta u-V_M(x)u,\quad t>0,\quad x\in R_N;\quad u(x,t)\in L^\infty .
]

Lemma. The semigroup of operators (G_M(t)), as (M\to\infty), converges in the uniform operator topology of the space ([L^p\to L^p,\ 1<p<\infty]) to the operator (G(t)); the operators (G(t)) form a semigroup in (t) in (L^p), (1\le p\le \infty).

It can be verified that in (L^p(R_N\setminus\Omega)) the semigroup (G(t)) is a semigroup of class (C_0). Let (A) be the infinitesimal operator of the semigroup (G(t)), (A_M) the infinitesimal operator of the semigroup (G_M(t)), the operator (P) be defined by the formula ((Pu)(x)=u(x)), (x\in R_N\setminus\Omega), ((Pu)(x)=0), (x\in\Omega), and the operators (H) and (H_M) be defined by the formulas (Hu=-APu), (H_Mu=-A_Mu). It is proved that (H) and (H_M) are self-adjoint operators; by (E(\lambda,H)) and (E(\lambda,H_M)) we denote the spectral functions of the operators (H) and (H_M).

The spectral function (E(\lambda,H)) is constructed with the aid of the solution of the scattering problem (u(x,k)), i.e. the solution of the equation (Hu=\lambda u), (u\in L^\infty), (x\in R_N), which can be represented as the sum of two functions
[
u(x,k)=\exp(ikx)+\varphi(x,k),\qquad k^2=\lambda,
]
where the function (\varphi(x,k)) satisfies the radiation conditions:
[
\varphi(x,k)=O\bigl(|x|^{(1-N)/2}\bigr),\qquad
(\partial/\partial |x|-i|k|)\varphi(x,k)=o\bigl(|x|^{(1-N)/2}\bigr),
]
[
|x|\to\infty .
]

(The functions (u_M(x,k)) and (\varphi_M(x,k)) are defined analogously with the aid of the operator (H_M).)

Theorem 1. 1) Let (f(x)) be an arbitrary function from (L^{2}), and let (f_n(x)\in L^{1}\cap L^{2}) be a sequence converging in (L^{2}(R_N\setminus\Omega)) to the function (f(x)). To each function (f_n(x)) we associate the function
[
(\widetilde f_n)(k)=\int u(x,k)f_n(x)\,dx .
]
It is asserted that the sequence ((\widetilde f_n)(k)) converges in (L^{2}) to a certain function ((\widetilde f)(k)), and the function ((\widetilde f)(k)) does not depend on the choice of the sequence (f_n(x)), but is determined only by the function (f(x)).

2) Let ({\lambda_i}) be the set of eigenvalues of the discrete spectrum of the operator (H), and let (\psi(x,\lambda_i)) be the corresponding eigenfunctions. If (\lambda\notin{\lambda_i}), then the operator (E(\lambda,H)) can be computed by the formulas
[
\langle f_1,E(\lambda,H)f_2\rangle
=(2\pi)^{-N}\int_{|k|^2\le \lambda}(\widetilde f_1)^{}(k)(\widetilde f_2)(k)\,dk
+\sum_{\lambda_i<\lambda}(f_1)_i^{}(f_2)i,
]
[
(E(\lambda,H)f)(x)
=(2\pi)^{-N}\intu^{}(x,k)(\widetilde f)(k)\,dk
+\sum_{\lambda_i<\lambda} f_i\psi(x,\lambda_i),
\tag{1}
]
where (f_i=\langle\psi(x,\lambda_i),f\rangle) is the scalar product of the functions (\psi(x,\lambda_i)) and (f(x)) in (L^{2}). The integral over (k) in formula (1) is understood in the following sense: let (\widetilde f_n(k)) be a sequence of functions converging in (L^{2}) to ((\widetilde f)(k)), with each of the functions (\widetilde f_n(k)) equal to zero in a certain neighborhood, depending on (n), of the set ({k,\ k^2\in{\lambda_i}}). For each (n) the integral
[
I_n(x)=\int_{|k|^2\le\lambda}u^{}(x,k)\widetilde f_n(k)\,dR
]
exists as a Lebesgue integral. It is asserted that the sequence (I_n(x)) converges in (L^{2}), and its limit depends only on ((\widetilde f)(k)).

3) The transformation
[
f\to Uf=\bigl[(2\pi)^{-N/2}(\widetilde f)(k);\ {f_i}\bigr],\qquad
f_i=\langle\psi(x,\lambda_i),f\rangle
]
is a one-to-one unitary transformation of (L^{2}(R_N\setminus\Omega)) onto the orthogonal sum of the spaces (L^{2}(R_N)\oplus l_m^{2}), where (m) is the cardinality of the set ({\lambda_i}). The inverse transformation is given by the formula
[
f(x)=U^{-1}\bigl[(2\pi)^{-N/2}(\widetilde f)(k);\ {f_i}\bigr]
=\lim_{\lambda\to\infty}\left((2\pi)^{-N}
\int_{|k|^2\le\lambda}u^{*}(x,k)(\widetilde f)(k)\,dk
+\sum_{\lambda_i<\lambda} f_i\psi(x,\lambda_i)\right).
\tag{2}
]
In formula (2) the integral over (k) is understood in the sense indicated above, and the limit with respect to (\lambda) is in the (L^{2}) metric.

The spectral expansion of the operator (H_M) is constructed in a completely analogous way. The existence and uniqueness of the functions (u(x,k)) were proved in work (2); there we also showed that the functions (\varphi(x,k)) and (\varphi_M(x,k)), which are solutions of the equations
[
\varphi(x,k)=T^{+}(\lambda)(\exp(iky)+\varphi(y,k)),
]
[
\varphi_M(x,k)=T_M^{+}(\lambda)(\exp(iky)+\varphi_M(y,k)),\qquad \lambda=k^2,
]
where the operators (T^{+}(\lambda)) and (T_M^{+}(\lambda)) are integral operators completely continuous in the metric ([L^{p}\to L^{p},\ 2N/(N-1)<p<\infty]), and that the function (\psi(x,\lambda_i)\in L^{2}) is an eigenfunction of the operator (H) if and only if it satisfies the equation
[
T^{+}(\lambda_i)\psi=\psi .
]

The fact that, in the uniform operator topology of the space
([L^{p}\to L^{p},\ 2N/(N-1)<p<\infty]), the equality
[
T^{+}(\lambda_j)=\lim_{\rho_j(\lambda,M)\to 0}T_M^{+}(\lambda),
\qquad
\text{where }\rho_j(\lambda,M)=\sqrt{|\lambda-\lambda_j|^2+M^{-2}},
]
holds allows us to prove the theorem:

Theorem 2. There exists a (\delta_j>0) such that for all (\lambda,M) such that
[
0<\rho_j(\lambda,M)<\delta_j,
]
the function (u_M(x,k)) can be represented in the form

[
u_M(x,k)=\exp(ikx)+\mu_j(\lambda,M)(1-\mu_j(\lambda,M))^{-1}\psi(x,\lambda_j,\lambda,M)\times
]
[
\times(\psi^*)_0(k,\lambda_j,\lambda,M)+x_j(x,k,M), \qquad \lambda=k^2,
]

where the function (\psi(x,\lambda_j,\lambda,M)) is an eigenfunction of the equation
[
T_M^+(\lambda)\psi=\mu_j(\lambda,M)\psi,
]
((\psi^)_0(k,\lambda_j,\lambda,M)) is the Fourier transform of the eigenfunction of the equation
[
T_M^+(\lambda)^\psi^=\mu_j\psi^,
]
and
[
\lim_{\rho_j(\lambda,M)\to 0}\mu_j(\lambda,M)=1,\qquad
\lim_{\rho_j(\lambda,M)\to 0}|\psi(x,\lambda_j,\lambda,M)-\psi(x,\lambda_j)|_{L^p}=0,
]
[
\frac{2N}{N-1}<p<\infty,
]
and the function (x_j(x,k,M)) is bounded in the metric (L^p), (2N/(N-1)<p<\infty), as (\rho_j(\lambda,M)\to 0).

We now formulate the main result of our work, which is a consequence of Theorems 1 and 2. Suppose that (F(\lambda)) is a continuous function bounded on ([0,\infty]), (D(M)=F(H)-F(H_M)). The function (D(M)f) can be represented as the sum of two terms
[
(D(M)f)(x)=\int_0^\lambda F(\lambda')\,d{E(\lambda',H)-E(\lambda',H_M)}f+\int_\lambda^\infty F(\lambda')\,d{E(\lambda',H)-
]
[
-E(\lambda',H_M)}f=D_1(M,\lambda)f+D_2(M,\lambda)f,\qquad \lambda\notin{\lambda_i}.
]

Theorem 3. If (f(x)\in \dot W_2^1(R_N\setminus\Omega)\cap L^1\cap L^\infty), and all eigenvalues ({\lambda_i}) are simple, then there exists a constant (C_0(f)), independent of (M,F), and (\lambda), such that
[
|D_2(M,\lambda)f|2^2\lambda}|F(\xi)|^2,
]
and the function (D_1(M,\lambda)f) satisfies the equality
[
\text{s.l. }\limD_1(M,\lambda)f=0.
]
Moreover, whatever bounded closed set (S) may be,
[
\lim_{M\to\infty}|D_1(M,\lambda)f|_{L^1(S)}=0.
]

A consequence of Theorem 3 is the equality
[
\text{s.l. }\lim_{M\to\infty}F(H_M)f=F(H)f.
]

Let (\Omega_1) be the largest open connected set contained in (R_N\setminus\Omega) and containing the points ({x,\ |x|>R}), and let
[
\Omega_2=R_N\setminus(\Omega_1\cup\Omega).
]

Theorem 4. Suppose that the function (V(x)) satisfies the following conditions: (\operatorname{mes}\Omega_2>0), (\rho(\Omega_1,\Omega_2)>0), the function (F(\lambda)) is measurable and bounded on ([0,\infty)), and the function (f(x)) is equal to zero outside the set (\Omega_2) and
[
\int f(x)(Hf)(x)\,dx=E_0<\infty.
]
Let (\varepsilon) be an arbitrary positive number, let the number (\lambda(\varepsilon)\notin{\lambda_j}) satisfy the inequality
[
\lambda(\varepsilon)>4\varepsilon^{-2}E_0\max |F(\xi)|^2.
]
Then there exists a number (M(\varepsilon)<\infty) and a function (\sigma(M)\to+0) as (M\to\infty), such that for all (M\ge M(\varepsilon)) the inequality
[
\left|F(H_M)f-\sum_{\lambda_j<\lambda(\varepsilon)}(2\pi)^{-N}
\int_{\lambda_j-\sigma(M)\le k^2\le \lambda_j+\sigma(M)}
u_M^*(x,k)(\tilde f)_M(k)F(k^2)\,dk\right|_2<\varepsilon
]
holds.

The author expresses his gratitude to A. N. Tikhonov for valuable remarks.

Moscow State University
named after M. V. Lomonosov

Received
20 XII 1967

CITED LITERATURE

1. A. A. Samarskii, DAN, 63, No. 6, 631 (1948).
2. A. A. Arsen'ev, DAN, 178, No. 6 (1968).