Reports of the Academy of Sciences of the USSR

1. Volume 158, No. 3

MATHEMATICS

M. V. FEDORYUK

ASYMPTOTICS OF THE DISCRETE SPECTRUM OF THE OPERATOR

\(-w''(x)+\lambda^2 p(x)w(x)\)

(Presented by Academician A. A. Dorodnitsyn, 17 IV 1964)

Consider the eigenvalue problem

\[ w''(x)-\lambda^2 p(x)w(x)=0,\qquad w(\pm\infty)=0, \tag{1} \]

where \(p(z)\) is an entire function of \(z\). We study the asymptotics of the discrete spectrum of problem (1). Our method is as follows. Take a solution \(w(x,\lambda)\) such that \(w(+\infty,\lambda)=0\), continue its asymptotics into the complex \(z\)-plane, find the asymptotics of \(w(x,\lambda)\) as \(x\to-\infty\), and determine the eigenvalues from the equation \(w(-\infty,\lambda)=0\).

1. We first formulate all results for polynomials \(p(z)\). Denote

\[ \alpha_0(z)=-\frac{p'(z)}{4p(z)},\qquad \alpha_{k+1}(z)=-\frac{1}{2\sqrt{p(z)}}\left(\sum_{m=0}^{k}\alpha_m(z)\alpha_{k-m}(z)+\alpha_k'(z)\right). \tag{2} \]

In Theorems 1–3 it is assumed that \(p(z)\) is a polynomial with real coefficients, all real zeros of \(p(z)\) are simple, and \(p(\pm\infty)=+\infty\).

Theorem 1. Suppose that \(p(x)\) has exactly two zeros \(x_1<x_2\). Then as \(n\to\infty\) there is the asymptotic expansion

\[ \lambda_n\int_C \sqrt{p(z)}\,dz \sim 2\pi n i+\pi i+\sum_{k=1}^{\infty}\lambda_n^{-k}\int_C \alpha_k(z)\,dz . \tag{3} \]

Here \(0<\lambda_1<\lambda_2<\cdots\) are the eigenvalues of problem (1), \(C\) is a contour in the \(z\)-plane enclosing the segment \([x_1,x_2]\) and containing no other zeros of \(p(z)\); \(\sqrt{p(x)}>0\) for \(x\in C,\ x>x_2\). The coefficient of \(\lambda_n\) is equal to

\[ 2i\int_{x_1}^{x_2}\sqrt{|p(x)|}\,dx . \]

From formula (3) it is not difficult to obtain an asymptotic series in powers of \(1/n\) for \(\lambda_n\).

Theorem 2. Suppose that \(p(x)\) has \(2k\) zeros \(x_j\), \(x_j<x_{j+1}\). Then for large \(n\) the spectrum of problem (1) consists of \(k\) series of eigenvalues \(\lambda_{nj}\), and as \(n\to\infty\) for \(\lambda_{nj}\) formula (3) holds with \(C\) replaced by \(C_j\), where \(C_j\) is a contour enclosing the segment \([x_{2j-1},x_{2j}]\), \(\sqrt{p(x)}>0\) for \(x\in C_j\), \(x_{2j}<x<x_{2j+1}\).

Corollary. If \(p(x)\) is an even polynomial, then

\[ \lambda_{ni}-\lambda_{n,k-i+1}=O(n^{-\infty}) \]

as \(n\to\infty\).

Theorem 3. If \(p(x)\) is an even polynomial and has 4 zeros \(x_j\), then as \(n\to\infty\)

\[ |\lambda_{n1}-\lambda_{n2}|= \frac{e^{-\lambda_{n1}c}}{a} \left(1+O\left(\frac{1}{n}\right)\right), \]

\[ a=\int_{x_1}^{x_2}\sqrt{|p(x)|}\,dx,\qquad c=\int_{x_2}^{x_3}\sqrt{p(x)}\,dx . \tag{4} \]

Formula (3), with accuracy up to \(O(\lambda_n^{-2})\), was proved by Titchmarsh \((^{1})\) for \(p(z)=z^{2n}-1\). Formula (4) was recently obtained in \((^{2})\).

Let \(p(z)\) be a polynomial, all zeros of \(p(z)\) being simple. Denote
\[ \xi(z,\lambda)=\lambda\int \sqrt{p(z)}\,dz. \]
Suppose that, for all \(\arg\lambda\), the level lines \(\operatorname{Re}\xi(z,\lambda)=\mathrm{const}\) contain no more than two zeros of \(p(z)\), and that for any fixed \(\arg\lambda\) there is no more than one level line containing two zeros of \(p(z)\). In \((^{3})\) the notion was introduced of a complex \(K\) joining \(+\infty\) and \(-\infty\) (a complex of genus II). \(K\) contains exactly two zeros \(z_1,z_2\) of the function \(p(z)\).

Theorem 4. Let \(p(z)\) satisfy the conditions formulated above. Then there exists \(\lambda_0>0\) such that for \(|\lambda|>\lambda_0\) the discrete spectrum of problem (1) either is absent or is situated near some ray \(\arg\lambda=\varphi_0\). The latter occurs only in the case when there exists a complex complex \(K\) joining \(+\infty\) and \(-\infty\), and as \(n\to\infty\) the asymptotics of \(\lambda_n\) is determined by formula (3), where the contour \(C\) encloses the zeros \(p(z)\), \(z_1,z_2\in K\), and contains no other zeros of \(p(z)\).

1. Let \(p(z)\) be an entire function. A Stokes line is a maximal regular connected component of the line \(\operatorname{Re}\xi(z,\lambda)=\mathrm{const}\), one of whose ends is a zero of \(p(z)\). Denote by \(\Phi(\lambda)\) the collection of all Stokes lines for a given \(\lambda\); by \(R_\varepsilon\) the \(z\)-plane from which nonintersecting neighborhoods of the zeros of \(p(z)\) have been removed, whose images in the \(\xi\)-plane are disks of radius \(\varepsilon\); and by \(\alpha^+(z_0,\lambda)\) the curve joining the points \(z_0,\infty\), along which the function \(\operatorname{Re}\xi(z,\lambda)\) does not decrease and which is mapped by the function \(\xi(z,\lambda)\) onto a broken line. \(\alpha^-(z_0,\lambda)\) is the analogous curve along which \(\operatorname{Re}\xi(z,\lambda)\) does not increase.

Let \(p(z)\) satisfy the following conditions for all \(\lambda\):

1) \(\overline{\Phi}(\lambda)=\Phi(\lambda)\);

2) \(\overline{\Phi}(\lambda)\setminus \overline{S(\lambda)}=\Phi(\lambda)\setminus S(\lambda)\) for any Stokes line \(S(\lambda)\);

3)
\[ \lim_{\substack{z\to\infty\\ z\in l(\lambda)}} \operatorname{Re}\xi(z,\lambda)=\infty \]
for any level line \(l(\lambda):\operatorname{Im}\xi(z,\lambda)=\mathrm{const}\);

4)
\[ \sup_{\lambda;\,z\in R_\varepsilon}\rho(z,\lambda)<\infty,\qquad \lim_{\substack{z\to\infty\\ z\in R_\varepsilon}}\rho(z,\lambda)=0,\qquad \lim_{\substack{z\to\infty\\ z\in R_\varepsilon}}\frac{p'(z)}{[p(z)]^{3/2}}=0; \]

5)
\[ \sup_{\lambda;\,z\in R_\varepsilon}\rho_1(z,\lambda)<\infty,\qquad \lim_{\substack{z\to\infty\\ z\in R_\varepsilon}}\rho_1(z,\lambda)=0; \]

Here
\[ \rho(z,\lambda)=\inf_{\alpha^+(z,\lambda)} \int_{\alpha^+(z,\lambda)}|\delta(z)|\,|dz|, \]
\[ \rho_1(z,\lambda)=\frac{|\delta(z)|}{|\sqrt{p'(z)}|} +\inf_{\alpha^+(z,\lambda)} \int_{\alpha^+(z,\lambda)} \left(\delta(t)\rho(t,\lambda)+ \frac{d}{dt}\left(\frac{\delta(z)}{\sqrt{p(t)}}\right)\right)|dt|, \]
\[ \delta(z)=\frac{[p'(z)]^2}{[p(z)]^{5/2}}+\frac{p''(z)}{[p(z)]^{3/2}}. \]

We note that examples of entire functions \(p(z)\) for which conditions 1) or 2) are not fulfilled are unknown. Condition 3) is approximately equivalent to the following: \(p(z)\) has no asymptotic value 0. Conditions 4), 5) require a certain regularity of the growth of \(p(z)\) as \(z\to\infty\).

Theorem 5. Let \(p(z)\) be an entire function satisfying conditions 1)–5) as \(z\to\infty\). If \(p(z)\) satisfies the conditions of Theorem 1, then, as \(n\to\infty\),
\[ \lambda_n\alpha=n\pi+\frac{\pi}{2} -\frac{i}{64\pi\alpha}\int_C\frac{[p'(z)]^2}{[p(z)]^{5/2}}\,dz +O(n^{-2}); \tag{5} \]
if, however, \(p(z)\) satisfies the conditions of Theorem 2, then for the series \(\lambda_{nj}\) one has

instead of formula (5), with \(a\) replaced by \(a_j\), where

\[ a_j=\int_{x_{2j-1}}^{x_{2j}} \sqrt{|p(x)|}\,dx. \]

If \(p(z)\) satisfies the conditions of Theorem 4 and conditions 1)—5) for all \(\lambda\), then Theorem 4 holds, only for \(\lambda_n\) formula (3) holds with accuracy up to \(O(\lambda_n^{-2})\).

Theorem 3 is also true for entire functions \(p(z)\), but the conditions on \(p(z)\) become very cumbersome.

I express my deep gratitude to M. A. Evgrafov for valuable advice and constant attention to this work.

Moscow Institute of Physics and Technology

Received
13 IV 1964

REFERENCES

¹ E. Titchmarsh, Eigenfunction Expansions, Part 1, IL, 1960. ² L. D. Landau, E. M. Lifshitz, Quantum Mechanics, Moscow, 1963. ³ Yu. N. Dnestrovsky, D. P. Kostomarov, DAN, 152, No. 1 (1963).