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UDC 517.948.35
MATHEMATICS
A. A. ARSEN'EV
ON THE ASYMPTOTICS OF THE SPECTRAL FUNCTION OF THE SCHRÖDINGER OPERATOR
(Presented by Academician A. N. Tikhonov on 12 XII 1966)
In the space \(L_2(R_N)\) consider the operator \(L\):
\[ Lu=-\Delta u+V(x)u, \]
where \(\Delta\) is the Laplace operator, and \(V(x)\ge 0\) is a function satisfying the Hölder condition at each point \(x\in R_N\). Let \(\theta(\lambda;x;y)\) be the kernel of the spectral function of the operator \(L\). (It is constructed, for example, in papers \((^{1-3})\).) For the function \(\theta(\lambda;x;y)\) the following asymptotics as \(\lambda\to\infty\) is known:
\[ \theta(\lambda;x;y)= \left(\frac{\sqrt{\lambda}}{2\pi r_{xy}}\right)^{N/2} J_{N/2}\bigl(r_{xy}\sqrt{\lambda}\bigr)+O\bigl(\lambda^{(N-1)/2}\bigr). \tag{1} \]
If \(x\ne y\), then the remainder in formula (1) has a higher order than the first written term. Our aim is to show that in some cases a much more accurate estimate of the function \(\theta(\lambda;x;y)\) is valid. This can be done, however, only by imposing rather strong requirements on the function \(V(x)\). We shall try to formulate these requirements in such a form that the proof of the theorem becomes obvious.
Let \(V(x)\) be a function given on \(R_N\). We shall say that the function \(V(x)\) satisfies, at the points \(x,y\), conditions \(\alpha\), if the following requirements are fulfilled:
1) for any fixed \(\xi\in R_N\), \(\tau\in[0,1]\), the function of the real variable \(t\)
\[ v(t)=V(2\sqrt{t}\,\xi+r(\tau));\qquad r(\tau)=x+(y-x)\tau, \]
admits analytic continuation in \(t\) to the open domain \(D_\eta\):
\[
-\pi/2-\eta<\arg t<\pi/2+\eta,
\]
where \(\eta\) is an arbitrarily small positive number;
2) there exists a number \(\varepsilon>0\) such that for all \(t\in D_\eta\) the estimate
\[ \sup_{\tau\in[0;1]} \left|\exp\left[-tV(2\sqrt{t}\,\xi+r(\tau))\right]\right| \le a(t)\exp\left[2(1-\varepsilon)|\xi|^2\right], \]
holds, where \(a(t)\) is a function finite at each point of the domain \(D_\eta\), and moreover
\[ a(t)/|t|^{N/2+1}\to 0,\qquad |t|\to\infty, \]
uniformly with respect to the argument \(t\), if
\[
-\pi/2-\eta<\arg t<-\pi/2+\eta,\qquad
\pi/2-\eta<\arg t<\pi/2+\eta;
\]
3) there exists a positive number \(\delta>0\) such that for all \(\xi\) satisfying the condition \(|\xi|\le\delta\), and all \(\tau\in[0;1]\), the function \(v(\beta^2)\) is analytic in \(\beta\) in some neighborhood of the point \(\beta=0\).
Potentials satisfying conditions \(\alpha\) exist: for example,
\[ V(x)=\left(\frac{1}{1+x^2}\right)^\nu \]
satisfies conditions \(\alpha\).
Theorem 1. If the potential \(V(x)\) satisfies conditions \(\alpha\), then for the spectral function \(\theta(\lambda;x;y)\) the asymptotic
expansion
\[ \theta(\lambda; x; y) = \left(\frac{\sqrt{\lambda}}{2\pi r_{xy}}\right)^{N/2} \left[ \sum_{n=0}^{M} a_n(x;y) \left(\frac{r_{xy}}{2\sqrt{\lambda}}\right)^n J_{N/2-n}\!\left(r_{xy}\sqrt{\lambda}\right) \right] + \rho_M(\lambda; x; y) \quad (x \ne y), \tag{2} \]
where \(M\) is any positive integer,
\[ |\rho_M(\lambda; x; y)| \le C_M^{(1)}(x;y) \left(\frac{\sqrt{\lambda}}{r_{xy}}\right)^{N/2-1-M} + C_M^{(2)}(x;y) \left(\frac{\sqrt{\lambda}}{r_{xy}}\right)^{N/2} \exp[-\delta' r_{xy}\sqrt{\lambda}], \]
where \(C_M^{(1)}(x;y)\) and \(C_M^{(2)}(x;y)\) are certain constants, finite for any \(x,y\), \(\delta' > 0\).
The first four coefficients \(a_n(x;y)\) are given by the following formulas:
\[ a_0(x;y)=1; \]
\[ a_1(x;y) = -\int_0^1 V(x+(y-x)\tau)\,d\tau; \]
\[ a_2(x;y) = -0.5\left(\int_0^1 V(x+(y-x)\tau)\,d\tau\right)^2 - \int_0^1 \tau(1-\tau)\Delta V(x+(y-x)\tau)\,d\tau^*, \]
\[ \begin{aligned} a_3(x;y) ={}& -0.5\int_0^1 \tau^2(1-\tau)^2 \Delta\Delta V(x+(y-x)\tau)\,d\tau \\ &+ \left(\int_0^1 V(x+(y-x)\tau)\,d\tau\right) \cdot \int_0^1 \tau(1-\tau)\Delta V(x+(y-x)\tau)\,d\tau \\ &+ \int_0^1 \left[ \int_\tau^1 \nabla V(x+(y-x)\xi)\,d\xi \right]^2 d\tau - \left( \int_0^1 \int_\tau^1 \nabla V(x+(y-x)\xi)\,d\xi\,d\tau \right)^2 \\ &- \frac{1}{6} \left(\int_0^1 V(x+(y-x)\tau)\,d\tau\right)^2 . \end{aligned} \]
Let us dwell on the proof of this theorem. Consider the Cauchy problem for the equation:
\[ \partial u/\partial t=-Lu;\qquad u(t;x)\in L_2(R_N);\qquad u|_{t=0}=u_0(x). \tag{3} \]
Lemma 1. The Green function of problem (3) is given by the formula
\[ G(x;y;t)= \frac{\exp[-r_{xy}^2/4t]}{(4\pi t)^{N/2}} \times \]
\[ \times\, \mathcal{E}\left\{ \exp\left[ -t\int_0^1 V\bigl(2\sqrt{t}(x(\tau)-\tau x(1))+x+(y-x)\tau\bigr) \,d\tau \right] \right\}, \tag{4} \]
where the symbol \(\mathcal{E}\{F(x(\tau))\}\) denotes the integral with respect to Wiener measure of the functional \(F(x(\tau))\); \(\tau\in[0;1]\); \(x(\tau)\in C_{[0;1]}(R_N)\); \(x(0)=0\).
The proof of this lemma is simplest to obtain by showing, by direct calculation according to the scheme of work (5), that the function on the right-hand side of formula (4), for any \(t>0\), satisfies equation (3); it is obvious that it has the required singularity as \(t\to0\). Next we show that the integral on the right-hand side of equality (4) converges for \(t\in D_\eta\) and is the analytic continuation into the domain \(D_\eta\) of the Green function \(G(x;y;t)\), which, as is known, can be computed by the formula:
\[ G(x;y;t) = \int_0^\infty e^{-\lambda t}\,d_\lambda \theta(\lambda; x; y). \]
\[ \text{* In note (}^1\text{) a misprint has been made in this formula.} \]
The conditions on \(\alpha\) make it possible, upon inverting the Laplace transform, to move the contour of integration into the left half-plane; the computation of the remaining integral is elementary.
We note that from the asymptotic formula (2) there follows quite simply the consequence: the function \(f(x)\) may be arbitrarily smooth in a neighborhood of the given point \(x_0\) (equal to \(0\)), but its Fourier expansion in the eigenfunctions of the operator \(L\) may diverge at the point \(x_0\). This fact for the eigenfunctions of the Laplace operator was first proved by V. A. Il’in.
The author expresses deep gratitude to V. A. Il’in for numerous consultations.
Moscow State University
named after M. V. Lomonosov
Received
30 XI 1966
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