Abstract Generated abstract
The paper studies the long-time behavior of solutions to the Schrödinger equation for a particle initially in a bound state behind an infinite potential barrier that is then replaced by a finite, sufficiently high barrier. Under regularity and compact-support assumptions on the potential, the author defines the wave function of a quasistationary state through an abstract Cauchy problem and analyzes it using resolvent and scattering-operator methods. The main theorem represents the solution as a leading exponentially oscillating and decaying term associated with a pole in the lower half-plane, plus remainder terms bounded in terms of a small parameter measuring the change from the infinite to the finite barrier. The result also gives an approximate formula for the shifted complex energy and notes that the leading spatial factor remains square-integrable, unlike some earlier formulations.
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UDC 517.944/947
MATHEMATICAL PHYSICS
A. A. ARSEN'EV
ON THE ASYMPTOTICS OF THE WAVE FUNCTION OF A QUASISTATIONARY STATE
(Presented by Academician A. N. Tikhonov on 26 II 1969)
I. Let \(R_N\) be an \(N\)-dimensional Euclidean space, \(N \geq 3\), \(x=(x_1,x_2,\ldots,x_N)\), \(\Delta\) the Laplace operator, \(V(x)\) a scalar function (potential), and \(H\) the operator defined on finite functions by the formula
\[ Hu=-\Delta u+V(x)u. \tag{1} \]
Suppose that in \(R_{N+1}\) the section of the surface \(z=V(x)\) by a plane passing through the \(z\)-axis has the form shown in Fig. 1; let \(\psi(x,t)\) be the solution of the Cauchy problem for the Schrödinger equation
\[ \partial\psi/\partial t=-iH\psi,\qquad \psi(x,+0)=\psi_0(x); \]
the support of the function \(\psi_0(x)\) is contained in the ball \(|x|<R\), and
\[ E_0=\langle \psi_0,H\psi_0\rangle \ll V_0. \tag{2} \]
Fig. 1
From physical considerations \((^{1,2})\) it follows that, for a special choice of \(\psi_0(x)\) and under condition (2),
\[ \psi(x,t)\sim \exp(-i\lambda t)A(x)+B(x)t^{-3/2}, \]
\[ t\to +\infty, \]
where the coefficient \(B(x)\) is negligibly small, \(\operatorname{Re}\lambda\) is the quasilevel energy, and \(\operatorname{Im}\lambda\) is its width. Our article is devoted to a rigorous proof of this fact. The main result is formulated by us in Theorem 1.
II. Let \(V(x)\) be a nonnegative function, \(\Omega=\{x,\ V(x)=\infty\}\), \(\Omega_1\) a connected component of the set \(R_N\setminus\Omega\) containing infinitely distant points, and \(\Omega_2=R_N\setminus(\Omega\cup\Omega_1)\). We assume that \(V(x)\) satisfies the conditions \(A(R)\), which consist in the following: 1) for any \(M>0\) the function \(V_M(x)=\min\{V(x),M\}\) locally satisfies the Hölder condition; 2) there exists such an \(R<\infty\) that \(V(x)=0\) for \(|x|>R\), and we assume that \(\operatorname{mes}\Omega_2>0\).
By the symbol \(H\) we shall denote the extension of the operator (1) constructed in § 2 of \((^3)\), and by the symbol \(H_M\) the analogous extension of the operator
\[ H_Mu=-\Delta u+V_M(x)u; \]
the remaining notation is borrowed by us from \((^{3-5})\), and
\[ \theta(M)=\left(\iint |g_M(x,y,\tau)-g(x,y,\tau)|\,dx\,dy\right)^{1/2}. \]
III. Suppose that the operator \(H\) has a nontrivial discrete spectrum \(\{\lambda_j\}\) and that \(\{\psi(x,\lambda_j)\}\) are the eigenfunctions of the discrete spectrum of the operator \(H\).
Definition. The solution of the abstract Cauchy problem
\[ \partial\psi/\partial t=-iH_M\psi;\qquad \psi\subset L^2;\qquad \psi(x,+0)=\psi(x,\lambda_j) \tag{3} \]
we call the wave function of the quasistationary state and denote by the symbol \(\psi_M(x,t,\lambda_j)\).
The solvability of problem (3) is proved by us on the basis of the ideas of work \((^6)\). In the physical formulation, the problem we solve is formulated as follows. Suppose we have an infinite potential barrier (i.e. \(V_0=\infty\)), and suppose that inside it there is a particle in a stationary state. (This means that the wave function of the particle is \(\exp(-i\lambda_j t)\psi(x,\lambda_j)\).) At the time \(t=+0\) we have changed the potential barrier from infinite to finite, but sufficiently large (\(\lambda_j \ll V_0\)). (The inequality \(\lambda_j/d>1\) is not assumed.) The initial state of the particle ceases to be stationary. We compute, to first order in the small* parameter \(\theta(V_0)\), the asymptotics of the wave function of the particle as \(t\to+\infty\).
IV. Lemma 1. If the potential \(V(x)\) satisfies the conditions \(A(R)\), \(\lambda_j\) is a simple eigenvalue of the discrete spectrum of the operator \(H\), and \(M>M_j\), then the function \(R(1,T_M^+(\lambda))=(E-T_M^+(\lambda))^{-1}\), considered as an element of the space \([\mathfrak A\to\mathfrak A]\), where \(\mathfrak A\) is the Hilbert space with scalar product
\[ \{f_1,f_2\}=\int f_1^*(x)f_2(x)\exp(-|x|)\,dx \]
is holomorphic in \(\lambda\) at all points of some disk \(\{\lambda;\ |\lambda-\lambda_j|<\varepsilon\}\), except for the single point \(\lambda_j^+(M)\), \(\operatorname{Im}\lambda_j^+(M)<0\), and \(\lambda_j^+(M)\) is a pole of first order.
Theorem 1. If the conditions of Lemma 1 are fulfilled and \(M>M_j\), then the function \(\psi_M(x,t,\lambda_j)\) can be represented as the sum of three functions
\[ \psi_M(x,t,\lambda_j)=\exp(-i\lambda_j^+(M)t)A_j(x,M)+B_j(x,M,t)+D_j(x,M,t), \]
where
\[ \begin{aligned} A_j(x,M)=&-(\pi i)[\lambda_j^+(M)]^{N/2-1}(2\pi)^{-N} \int_{|n|=1} dn\,\operatorname{Res}\Bigl\{\bigl(R(1,T_M^-(\lambda))\times \\ &\times T_M^-(\lambda)\exp(-i\sqrt{\lambda}(n,y))\bigr)(x) \int\bigl[\exp(i\sqrt{\lambda}(n,y))+R(1,T_M^+(\lambda))\times \\ &\times \exp(i\sqrt{\lambda}(n,y))\bigr]\psi(y,\lambda_j)\,dy,\quad \lambda=\lambda_j^+(M)\Bigr\}, \end{aligned} \]
the functions \(B_j(x,M,t)\) and \(D_j(x,M,t)\) belong to \(L^2\) and are such that
\[ \|B_j\|_\infty+\|D_j\|_2<C_j\theta(M)^{1/5}, \]
the constant \(C_j\) depends only on \(\lambda_j\).
It can be shown that \(A_j\in L^2\) and
\[ A_j(x,M)=O\bigl(|x|^{-N-2}\bigr),\qquad |x|\to\infty, \]
and the point \(\lambda_j^+(M)\) is a pole of the analytic continuation of the scattering amplitude \(f_M(\sqrt{\lambda},n_{in},n_{out})\) into the lower half-plane.
The number \(\lambda_j^+(M)\) is approximately computed by the formula
\[ \lambda_j^+(M)=\lambda_j+\operatorname{Sp}\bigl(P(T^+-T_M^+)P\bigr)/ \operatorname{Sp}\left(P\frac{\partial T^+}{\partial\lambda}P\right)\Bigg|_{\lambda=\lambda_j} +O(\theta(M)), \]
where
\[ P=(2\pi i)^{-1}\int_{|1-\mu|=\delta} R(\mu,T^+(\lambda_j))\,d\mu, \]
\(\delta\) is sufficiently small, and \(\tau\approx \rho(M,M/2)M^{-1/2}\).**
V. Earlier \((^{2,7})\) an analogous problem was solved in a somewhat different formulation, and the wave function of the quasistationary state obtained was exponentially increasing in modulus as \(|x|\to\infty\).
* Under very broad assumptions \(\theta(M)\sim\exp(-\beta M^{1/2-\delta})\), \(\beta>0\), \(\delta>0\).
** Here \(\tau\) is the quantity which in \((^{3-5})\) we denoted by the symbol \(t\).
But, according to the axioms of quantum mechanics, a function that is not square-integrable cannot be the wave function of a particle localized at the initial moment; therefore the interpretation of the solution obtained is not clear.
VI. The author expresses his gratitude to A. N. Tikhonov and V. A. Il’in for valuable consultations.
Moscow State University
named after M. V. Lomonosov
Received
25 II 1969
References
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- A. I. Baz’, Ya. B. Zel’dovich, A. M. Perelomov, Scattering, Reactions, and Decays in Nonrelativistic Quantum Mechanics, Nauka, 1966.
- A. A. Arsen’ev, Zh. Vychisl. Mat. i Mat. Fiz., 8, No. 3, 544 (1948).
- A. A. Arsen’ev, ibid., 8, No. 6, 1232 (1968).
- A. A. Arsen’ev, ibid., 9, No. 1, p. 137 (1969).
- V. A. Il’in, UMN, 23, issue 2 (140), 61 (1968).
- L. D. Landau, E. M. Lifshitz, Quantum Mechanics. Nonrelativistic Theory, Moscow, 1963.