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UDC 517.954.4:517.937+513.88
MATHEMATICS
E. A. BEGOVATOV
THE SCHRÖDINGER EQUATION IN HILBERT SPACE
(Presented by Academician A. A. Dorodnitsyn, 8 X 1969)
Let \(H\) be a Hilbert space. Consider the set of all Bochner-integrable functions \(f(x)\) \((-\infty < x < \infty)\) with values in \(H\). This set forms a new Hilbert space \(H_1\), if the inner product in it is defined by the equality
\[ (f,g)_{H_1}=\int_{-\infty}^{\infty}(f(x),g(x))_H\,dx . \]
Consider in the space \(H_1\) the differential operator
\[ L\varphi=[d^2/dx^2-Q(x)]\varphi, \tag{1} \]
defined on the set of functions from \(H_1\) with values in \(D\) (here \(D\) is the domain of definition of the operator \(Q(x)\)) and having a strong second continuous derivative.
In the present note conditions are determined on a family of linear, generally speaking unbounded in \(H\), operators \(Q(x)\), under which the differential operator (1) (more precisely, its closure) is the generator of a strongly continuous contraction semigroup. It will also be shown that the resolvent of this operator is an integral operator with a certain kernel, which by analogy with the scalar case is called the operator Green’s function.
We shall assume that:
\(\alpha\). The closed linear operator \(Q(x)\), for each \(x \in R\), has a dense in \(H\), \(x\)-independent domain of definition \(D\), and for any \(\lambda \geq 0\) the operator \(Q(x)+\lambda I\) has a bounded inverse. The operator \(Q(x)\) is uniformly bounded below:
\[ \operatorname{Re}(Q(x)\varphi,\varphi)_H \geq \delta(\varphi,\varphi)_H,\qquad \varphi \in D . \]
The condition that the domain of definition \(D\) be independent of \(x\), by virtue of the closed graph theorem, entails the boundedness in \(H\) of the operator \(Q(x)\cdot Q^{-1}(y)\), and the norm of this operator, generally speaking, is bounded by a constant depending on \(x\) and \(y\). The following condition imposes a restriction on the growth of this constant.
\(\beta\). There exist constants \(C\) and \(k>0\) such that
\[ \|Q(x)Q^{-1}(y)\|\leq C\exp\{k|x-y|\} \]
and the operator \(Q(x)Q^{-1}(y)\) for \(|x-y|<1\) satisfies the Hölder condition
\[ \|[Q(x)-Q(y)]Q^{-1}(y)\|\leq C|x-y|^\varepsilon,\qquad \varepsilon\in(0,1); \]
moreover,
\[ \|Q^{-1-\delta}(x)[Q(x)-Q(y)]\|\leq C|x-y|^\varepsilon,\qquad \delta<\varepsilon, \]
or
\[ \|[Q^{1/2}(x)-Q^{1/2}(y)]Q^{-1/2}(y)\|\leq C|x-y|^\varepsilon . \]
Let
\[ Q_R=\frac12(Q+Q^*)\quad\text{and}\quad Q_I=\frac{1}{2i}(Q-Q^*) \]
— the real and
imaginary components of the operator \(Q\). We impose additional restrictions on these components.
\(\gamma\). The operators \(Q_R\) and \(Q_I\) are semibounded on a set dense in \(H\), and one of the three self-adjoint extensions in the Friedrichs sense, \(\widetilde Q_R\), \(\widetilde Q_I\), or \(\overline{Q_R+Q_I}\), has a completely continuous resolvent, while the least eigenvalue \(a_1(x)\) satisfies the condition of A. M. Molchanov
\[ \lim_{|x|\to\infty}\int_x^{x+\omega} a_1(x)\,dx=\infty \quad \text{for any } \omega>0 . \]
Condition \(\alpha\) makes it possible to construct the operator
\(\chi(x)=\{Q(x)+\mu^2 I\}^{1/2}\), whose resolvent set is the entire right half-plane \((^1)\). In this case the operator \(\chi\) generates a strongly continuous contraction semigroup
\(U_\mu^x(t)=\exp\{-t\chi(x)\}\), for which
\[ \|\chi^\alpha \exp\{-t\chi\}\|\le C t^{-\alpha}\exp\{\gamma\mu t\},\quad \alpha\ge 0, \tag{2} \]
where \(\gamma>0\) and does not depend on \(\mu>\mu_0\) and \(t>0\). Using, moreover, condition \(\beta\) and (2), one can show that
\[ \|Q^{-\beta}(x)[Q^\alpha(x)-Q^\alpha(y)]\|\le C|x-y|^\varepsilon \quad \text{for } \beta-\delta>\alpha; \tag{3} \]
\[ \|Q^{1/2}(x)[U_\mu^x(t+\Delta t)-U_\mu^x(t)]\| \le C\{\Delta t\}t^{-2}\exp\{-\gamma\mu t\}; \tag{4} \]
\[ \|\chi^n(x)[U_\mu^x(t)-U_\mu^y(t)]\chi^{-k}(y)\| \le C\frac{|x-y|^\varepsilon}{t^{n+\delta-k}}\exp\{-\gamma\mu t\}, \tag{5} \]
where \(n=0,1,2;\ k=0,1;\ n-k\le 1\) and \(\delta<\varepsilon\). From condition \(\beta\) and (2) it follows that for any \(x\ne y\in R\)
\[ \|(Q(x)-Q(y))Q^{-1/2}(y)U_\mu^y(|x-y|)\| \le \frac{C\exp\{-\gamma\mu|x-y|\}}{|x-y|^{1-\varepsilon}}, \]
therefore the integral equation
\[ G(x,y,\mu)=F(x,y,\mu)+\int_{-\infty}^{\infty} G(x,z,\mu)[Q(y)-Q(x)]F(z,y,\mu)\,dz, \]
\[ F(x,y,\mu)=\frac12\chi^{-1}(y)e^{-|x-y|\chi(y)} \tag{6} \]
has a unique solution \(G(x,y,\mu)\), which is determined by the formula
\[ G(x,y,\mu)=F(x,y,\mu)+\int_{-\infty}^{\infty}F(x,z,\mu)\Phi(z,y,\mu)\,dz, \tag{7} \]
where \(\Phi(x,y,\mu)\) is an operator for which the following inequality holds: for any
\(\dfrac{\varepsilon}{1+\varepsilon}<\eta<\varepsilon\),
\[ \|\Phi(x+\Delta x,y,\mu)-\Phi(x,y,\mu)\| \le C|\Delta x|^{\varepsilon-\eta}|x-y|^{\varepsilon-1}, \quad |x-y|<1 . \]
Theorem 1. If the positive number \(\mu\) is sufficiently large and conditions \(\alpha\) and \(\beta\) are satisfied, then equation (6) has a unique solution \(G(x,y,\mu)\), which is the Green’s function for the equation
\[ [L-\mu^2 I]\varphi=-f, \]
i.e., for any function \(f\in H_1\) satisfying the Hölder condition, the integral
\[ g=Af=\int_{-\infty}^{\infty}G(x,y,\mu)f(y)\,dy \tag{8} \]
gives a solution of equation (7).
The proof of the theorem is based on the method of P. E. Sobolevskii \((^2)\), and results \((^1)\) are used here. In proving the existen-
of the second continuous strongly differentiable with respect to \(x\) operator function \(G(x,y,\mu)\), the uniform boundedness with respect to \(x\) and the continuity of the operator
\[ \int_{-\infty}^{\infty} Q(x)F(x,y,\mu)\,dy, \]
which follow from inequalities (2), (3), and (5).
Since the set of functions \(f(x)\) satisfying the Hölder condition is dense in \(H_1\), it follows from the theorem of B. M. Lidskii ((3), p. 110) and B. M. Levitan and T. A. Suvorchenkova ((4), p. 56) that the operator (8) is completely continuous in \(H_1\), if the family of operators \(Q(x)\) satisfies, in addition, condition \(\gamma\). From condition \(\alpha\) it is easy to obtain that in \(H_1\) the inequality
\[ 0 \leqslant \operatorname{Re}(Af,f)_{H_1} \leqslant \frac{(f,f)_{H_1}}{\delta+\mu} \]
holds.
Therefore, by what was said above, \(\mu\) is a regular point of the operator \(A\) ((5), p. 302), and the integral equation
\[ G(x,y)=G(x,y,\mu)+\int_{-\infty}^{\infty}G(x,z,\mu)G(z,y)\,dz \tag{9} \]
has a unique solution.
Splitting the integral operator in (9) into a finite-dimensional operator and an integral operator whose norm is sufficiently small, one can show that the operator \(G(x,y)\) is uniformly bounded in \(H\). By virtue of the properties of the operator Green’s function \(G(x,y,\mu)\), one can prove the following theorem.
Theorem 2. If conditions \(\alpha,\beta,\gamma\) are satisfied, then there exists a unique Green’s function \(G(x,y)\) for the problem \(L\varphi=-f\).
The extension by continuity of the integral operator generated by the operator Green’s function \(G(x,y)\) to all of \(H_1\) leads to a certain closed extension of the operator \(L\), whose range is all of \(H_1\). If one uses condition \(\alpha\) and the Hille–Yosida theorem ((1), p. 109), one can obtain
Corollary 1. Under the assumptions of Theorem 2, the closure of the operator \(L\) is the infinitesimal generator of a strongly continuous contraction semigroup \(T_t=\exp\{-t\bar L\}\); moreover,
\[ dT_t v/dt=d^2T_t v/dx^2-Q(x)T_t v \]
for any \(v\in D(L)\) and \(t>0\).
If the semigroups generated by the operators \(d^2/dx^2\) and \(Q(x)\) are denoted by \(R_t\) and \(Q_t\), respectively, then, applying Trotter’s theorem (6), one can obtain a representation of the semigroup \(T_t\) in terms of \(R_t\) and \(Q_t\).
Corollary 2. Under the assumptions of Theorem 2, the equality
\[ T_t v=\lim (R_hQ_h)^{[t/h]}v \]
holds for any \(v\in H_1\).
Remark 1. If one uses the result of article (7), condition \(\gamma\) can be somewhat weakened.
Remark 2. In the case of a self-adjoint operator \(Q(x)\), results analogous to Theorem 1 were obtained in (8), and in the case of a finite interval results analogous to Theorems 1 and 2 were obtained in (9).
In conclusion, I take the opportunity to express my gratitude to B. M. Levitan, conversations with whom substantially contributed to the writing of this paper.
Kazan State University
named after V. I. Ulyanov-Lenin
Received
3 X 1969
References
- S. G. Krein, Linear differential equations in Banach spaces, Moscow, 1967.
- P. E. Sobolevskii, Trudy Moskov. Mat. Obshch., 10, 297 (1961).
- B. M. Lidskii, Trudy Moskov. Mat. Obshch., 8, 84 (1959).
- B. M. Levitan, T. A. Suvorchenkova, Functional analysis and its applications, 2, No. 2 (1968).
- M. Sh. Gokhberg, M. G. Krein, Introduction to the theory of nonselfadjoint operators, Moscow, 1965.
- H. F. Trotter, Proc. Am. Math. Soc., 10, No. 4 (1959).
- V. P. Maslov, Functional analysis and its applications, 2, No. 2 (1968).
- B. M. Levitan, Mat. Sbornik, 76 (118), 2 (1968).
- G. I. Laptev, Litovsk. Mat. Sbornik, 8, No. 1 (1968).