Full Text
Reports of the Academy of Sciences of the USSR
1962. Volume 142, No. 3
MATHEMATICS
L. A. SAKHNOVICH
ON THE DISCRETE SPECTRUM OF THE RADIAL SCHRÖDINGER EQUATION
(Presented by Academician A. N. Kolmogorov on 28 VIII 1961)
1. Consider the Schrödinger equation
\[ -\frac{d^{2}u}{dx^{2}}+ \left[ \frac{\mu(\mu-1)}{x^{2}}-2q(x) \right]u-\lambda u=0, \tag{1} \]
\[ u(0)=0,\qquad 0\le x<\infty,\qquad \mu>1. \]
Many problems of physics lead to equations of this kind. In particular, for \(q(x)=A/x\) \((A>0)\) one obtains the equation for the radial part of the wave function of the hydrogen atom \((^{1})\).
We shall be interested in the behavior of the smallest eigenvalue \(\varphi(\mu)\) of equation (1), as well as in the behavior of the corresponding eigenfunction \(u(x,\mu)\).
In doing so we shall assume that \(q(x)\) is a function continuous for \(x>0\) and, moreover,
\[ q(x)= \begin{cases} \dfrac{\beta}{x}+O(1), & 0\le x\le x_{0},\\[6pt] \dfrac{A_{1}}{x}+\dfrac{A_{2}}{x^{2}}+O\!\left(\dfrac{1}{x^{3}}\right), & x\ge x_{0}, \end{cases} \tag{2} \]
where \(A_{1}>0\).
As is known \((^{2})\), the continuous spectrum of the problem under consideration fills the positive half-axis. On the negative half-axis only a discrete spectrum is possible, bounded from below.
If, for a given \(\mu\), there exist negative eigenvalues of equation (1), then the function \(\varphi(\mu)\) is defined and the inequality \((^{3})\)
\[ \varphi(\mu)\le \int_{0}^{\infty} \left\{ y'(x)^{2}+ \left[ \frac{\mu(\mu-1)}{x^{2}}-2q(x) \right]y^{2}(x) \right\}\,dx, \tag{3} \]
holds, where \(y\) is an absolutely continuous function satisfying the conditions
\[ \int_{0}^{\infty} y^{2}(x)\,dx=1,\qquad y(0)=0. \]
Equality in (3) holds only for the eigenfunction \(u(x,\mu)\) corresponding to the eigenvalue \(\varphi(\mu)\).
We shall set \(\varphi(\mu)=0\) if, for the corresponding \(\mu\), problem (1) has no negative eigenvalues.
Then
\[ \varphi(\mu)= \inf \int_{0}^{\infty} \left\{ y'(x)^{2}+ \left[ \frac{\mu(\mu-1)}{x^{2}}-2q(x) \right]y^{2}(x) \right\}\,dx, \]
where the absolutely continuous function \(y\) satisfies the conditions:
\[ \int_{0}^{\infty} y^{2}(x)\,dx=1,\qquad y(0)=0. \]
We shall call the function \(\varphi(\mu)\) the minimum function of problem (1), and the corresponding eigenfunction the minimizing eigenfunction.
If \(q(x)=A/x\) \((A>0)\), then, as is known,
\[ \varphi(\mu)=-\frac{A^2}{\mu^2}, \qquad u(x,\mu)=x^\mu e^{-\frac{A}{\mu}x}. \]
Theorem 1. The minimum function \(\varphi(\mu)\) has the form
\[ \varphi(\mu)=-\frac{A_1^2}{\mu^2}-\frac{4A_2A_1^2}{\mu^4} +O\left(\frac{1}{\mu^6}\right), \qquad \mu>1. \]
From this theorem and from the monotone increase of the minimum function it follows that
\[ \varphi(\mu)<0, \]
i.e., \(\varphi(\mu)\) is a point of the discrete spectrum, and there always corresponds to it a minimizing eigenfunction \(u(x,\mu)\) (by \(\bar u(x,\mu)\) we shall denote the normalized minimizing eigenfunction: \(\|\bar u\|=1\)).
Thus, the existence of a discrete spectrum for problem (1) depends only on the behavior of \(q(x)\) at infinity \((A_1>0)\).
Theorem 2. The minimizing eigenfunction \(u(x,\mu)\) has the form:
\[ u(x,\mu)=x^\mu\left[1+O\left(\frac{1}{\mu}\right)\right], \qquad 0\le x\le x_0, \]
\[ u(x,\mu)=x^\mu e^{-\frac{A_1}{\mu}x} \left[1+O\left(\frac{1}{\mu^k}\right)\right], \]
where \(0<k<1,\; x_0\le x\le \mu^{\frac{3-k}{2}}\), and, finally,
\[ u(x,\mu)=x^\mu e^{-\frac{A}{\mu}x} \left[ 1+O\left( \frac{1}{\mu^{\frac{1-3k}{2}}} \right) \right], \]
where \(0<k<\frac13,\; \mu^{\frac{3-k}{2}}\le x\le \frac{\mu^2}{A_1}\left[1+\frac{l}{\sqrt{\mu}}\right]\) \((l>0)\).
From Theorem 2 it is easy to obtain:
Corollary. The inequality holds
\[ \int_0^{p_1\mu^2} x^{-m}u^2(x,\mu)\,dx : \int_0^{p_2\mu^2} u^2(x,\mu)\,dx \le O(r^{2\mu}), \]
where \(m\ge 0,\; 0<p_1<p<p_2<\frac{1}{A_1},\; \frac{p_1}{p}e^{A_1(p-p_1)}=r<1\).
From this corollary, in particular, it follows that
\[ \int_0^{p_1\mu^2} \bar u^2(x,\mu)\,dx \le O(r^{2\mu}), \]
where \(0<p_1<p<\frac{1}{A_1},\; \frac{p_1}{p}e^{A_1(p-p_1)}=r<1\).
Theorem 2 describes the behavior of \(u(x,\mu)\) for large \(\mu\) only on a certain interval of the positive half-axis.
Using the usual methods \((^2)\) for estimating the solution of a differential equation for large values of \(x\), it is easy to characterize the behavior of \(u(x,\mu)\) for \(x\gg \mu^2\).
The following two theorems describe the behavior of the minimizing eigenfunction on the entire half-axis.
Theorem 3. The minimizing normalized eigenfunction \(\bar u(x,\mu)\) satisfies the relation
\[ \left\|\bar u(x,\mu)-c(\mu_1)e^{-\frac{A_1}{\mu_1}x}\right\|^2 \leqslant O\left(\frac{1}{\mu^3}\right), \]
where
\[ \mu_1=\frac{1+\sqrt{(2\mu-1)^2-8A_2}}{2},\qquad c(\mu_1)=\left[\int_0^\infty x^{2\mu_1}e^{-\frac{2A_1}{\mu_1}x}\,dx\right]^{-\frac12}. \]
Theorem 4. The minimizing normalized eigenfunction \(\bar u(x,\mu)\) satisfies the relation
\[ \left\|u(x,\mu)-c(\mu)x^\mu e^{-\frac{A_1}{\mu}x}\right\|^2 \leqslant O\left(\frac{1}{\mu}\right), \]
where
\[ c(\mu)=\left[\int_0^\infty x^{2\mu}e^{-\frac{2A_1}{\mu}x}\,dx\right]^{-\frac12}. \]
Let us note that Theorems 3 and 4 are obtained from Theorem 1 and the following lemma:
Lemma. Let \(A\) be a self-adjoint operator such that the leftmost point of its spectrum, \(\lambda_1\), is an isolated point of multiplicity 1. Then, for a function \(y\) \((\|y\|=1)\) belonging to the domain of definition of \(A\), the inequality
\[ \left\|y-\alpha_1 y_1\right\|\leqslant \frac{(Ay,y)-\lambda_1}{m},\qquad \|y_1\|=1, \]
holds, where \(y_1\) is the normalized eigenfunction corresponding to the number \(\lambda_1\); \(m\) is the distance from the point \(\lambda_1\) to the remaining part of the spectrum; \(\alpha_1=(y,y_1)\) satisfies the inequalities
\[ 1\geqslant |\alpha_1|\geqslant 1-\frac{(Ay,y)-\lambda_1}{m}. \]
The validity of this lemma is easily established if one uses the spectral decomposition of the operator \(A\).
Theorem 5. Let \(\varphi_1(\mu)\) and \(\varphi_2(\mu)\) be the minimum functions of two boundary-value problems, and suppose that
\[ q_1(x)-q_2(x)=\frac{A_m}{x^m}+O\left(\frac{1}{x^{m+1}}\right),\qquad x\geqslant x_0,\quad m>1; \]
then the equality
\[ \varphi_1(\mu)=\varphi_2(\mu)-\frac{2A_m A_1^m}{\mu^{2m}}+O\left(\frac{1}{\mu^{2m+\frac12}}\right) \]
holds.
Using Theorem 5, one can refine Theorem 1 in the following way:
Theorem 6. If the potential \(q(x)\) has the form:
\[ q(x)=\frac{A_1}{x}+\frac{A_2}{x^2}+\frac{A_3}{x^3}+O\left(\frac{1}{x^4}\right),\qquad x\geqslant x_0, \]
then the equality holds
\[ \varphi(\mu)=-\frac{A_1^2}{\mu^2}-\frac{4A_2A_1^2}{\mu^4} -\frac{4A_2^2A_1^2+2A_3A_1^3}{\mu^6} +O\left(\frac{1}{\mu^{13/2}}\right). \]
- The following problem seems important to us:
to determine whether the minimum function \(\varphi(\mu)\) uniquely determines the potential \(q(x)\), and to give a method for constructing \(q(x)\) from \(\varphi(\mu)\).
Theorems 1–6 allow one to draw certain, although only preliminary, conclusions.
Theorem 7. Let the minimum functions \(\varphi_1(\mu)\) and \(\varphi_2(\mu)\) be such that
\[ \varphi_1(\mu)=\varphi_2(\mu)+O\left(\frac{1}{\mu^m}\right), \]
where \(m=1,2,\ldots\).
If the corresponding potentials \(q_1(x)\) and \(q_2(x)\) are analytic functions, regular at infinity, then
\[ \varphi_1(\mu)=\varphi_2(\mu),\qquad q_1(x)=q_2(x). \]
Let us note that the theorem remains valid also in the case when the equality
\[ \varphi_1(\mu)=\varphi_2(\mu)+o\left(\frac{1}{\mu^m}\right),\qquad m=1,2,\ldots, \]
holds only for some sequence of points \(\mu_1,\mu_2,\mu_3,\ldots\to\infty\), and this sequence may depend on \(m\).
Thus, analytic potentials are determined uniquely by means of \(\varphi(\mu)\).
From Theorem 6 it is clear how to find the first three coefficients \((A_1,A_2,A_3)\) of the expansion of \(q(x)\) at infinity, if the function \(\varphi(\mu)\) is known.
Odessa Institute
of the Food and Refrigeration Industry
Received
1 VII 1961
REFERENCES
- L. Schiff, Quantum Mechanics, IL, 1959.
- M. A. Naimark, Linear Differential Operators, 1954.
- S. G. Mikhlin, Variational Methods in Mathematical Physics, 1957.