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MATHEMATICS
V. A. TKACHENKO
ON THE SPECTRAL ANALYSIS OF A ONE-DIMENSIONAL SCHRÖDINGER OPERATOR WITH PERIODIC COMPLEX-VALUED POTENTIAL
(Presented by Academician S. N. Bernstein on 22 XI 1963)
Let \(T\) be a closed differential operator in \(\mathscr L_2(-\infty,\infty)\), generated by the operation
\[ -\frac{d^2}{dx^2}+q(x) \tag{1} \]
with a periodic (of period 1) continuous complex-valued function \(q(x)\). Denote its domain by \(\mathfrak D_T\). The aim of the present note is to study the geometric structure of the operator \(T\). We establish that the spectrum of the operator \(T\) is separable in the sense of the definition of Yu. I. Lyubich and V. I. Macaev \((^1)\), and construct a family of bounded projectors onto spectral subspaces possessing some properties of a resolution of the identity. Under additional assumptions on the spectrum, the operator \(T\) turns out to be unitarily equivalent to the operator of multiplication by a triangular matrix. The work makes essential use of the method proposed by I. M. Gelfand \((^2)\) for the case of real \(q(x)\). A finite-difference analogue of the operator \(T\) was studied by P. B. Naiman \((^3)\).
\(1^\circ\). With the operator \(T\) there is naturally associated a family of operators \(T_t\) \((-\pi \leq t \leq \pi)\) in \(\mathscr L_2(0,1)\), generated by the operation (1) and by the “cyclic” boundary conditions
\(y(1)=e^{it}y(0)\), \(y'(1)=e^{it}y'(0)\). For each \(t\) the operator \(T_t\) is regular, and its spectrum consists of a countable sequence of eigenvalues tending to infinity. Denote by \(\Sigma\) the union of the spectra of the operators \(T_t\) for \(-\pi \leq t \leq \pi\).
Theorem 1. The spectrum of the operator \(T\) coincides \(*\) with \(\Sigma\). For every \(\lambda \notin \Sigma\) the resolvent of the operator \(T\) is an integral operator with kernel
\[ K(x+k,s+n;\lambda)=\frac{1}{2\pi}\int_{-\pi}^{\pi}K_t(x,s;\lambda)e^{it(k-n)}\,dt \]
\[ (0\leq x,s\leq 1;\ k,n=0,\pm1,\pm2,\ldots), \]
where \(K_t(x,s;\lambda)\) is the kernel of the resolvent of the operator \(T_t\).
The set \(\Sigma\), as is known, is the union of the sets of roots of the equations \(A(\lambda)=\cos t\) for \(-\pi \leq t \leq \pi\), where \(A(\lambda)\) is a certain entire function. Therefore \(\Sigma\) consists of a countable system of bounded simple analytic arcs \(\Gamma_1,\Gamma_2,\ldots\), whose endpoints are the roots of the equations
\(A'(\lambda)=0\), \(A(\lambda)=\pm1\). Each of the arcs \(\Gamma_k\) can be given by an equation of the form
\[ \lambda=\omega_k(\cos t), \tag{2} \]
where \(\omega_k\) is an invertible function. Let \(\Delta\) be some closed connected part of one of the arcs \(\Gamma_k\), containing no points at which \(A'(\lambda)=0\), and let
\(*\) This assertion follows from a result obtained recently by F. S. Rofe-Beketov \((^4)\).
\(\delta \subset [-\pi,\pi]\) is its complete preimage by virtue of equation (2). Denote by \(\varphi_t(x)\) and \(\psi_t(x)\) the eigenfunctions of the operators \(T_t\) and \(T_t^*\), corresponding to the eigenvalues \(\lambda(t)=\omega_k(\cos t)\) and \(\overline{\lambda(t)}\) \((t\in\delta)\). We normalize them so that
\[ \int_0^1 |\varphi_t|^2\,dx=\int_0^1 \varphi_t\overline{\psi_t}\,dx=1. \]
We extend each function \(\varphi_t(x)\) and \(\psi_t(x)\) to the whole axis by the equalities
\[ \varphi_t(x+m)=e^{imt}\varphi_t(x),\qquad \psi_t(x+m)=e^{imt}\psi_t(x) \]
\[ (0\leq x\leq 1,\; m=0,\pm1,\pm2,\ldots). \]
Denote by \(\chi_\delta\) the characteristic function of the set \(\delta\), and put, for an arbitrary function \(f(x)\in \mathscr L_2(-\infty,\infty)\),
\[ a(t)=\chi_\delta\int_{-\infty}^{\infty} f(x)\overline{\psi_t(x)}\,dx. \tag{3} \]
Finally, introduce in \(\mathscr L_2(-\infty,\infty)\) the operator \(E(\Delta)\)
\[ E(\Delta)f(x)=\frac{1}{2\pi}\int_{-\pi}^{\pi} a(t)\varphi_t(x)\,dt. \tag{4} \]
Theorem 2. The operator \(E(\Delta)\) is bounded, and
\[
E(\Delta_1)E(\Delta_2)=E(\Delta_1\cap\Delta_2).
\]
The operator \(E^*(\Delta)\), adjoint to \(E(\Delta)\), is defined by the equalities
\[ a(t)=\chi_\delta\int_{-\infty}^{\infty} f(x)\overline{\varphi_t(x)}\,dx, \]
\[ E^*(\Delta)f(x)=\frac{1}{2\pi}\int_{-\pi}^{\pi} a(t)\psi_t(x)\,dt. \]
If in Theorem 2 we put \(\Delta_1=\Delta_2=\Delta\), then we obtain \(E^2(\Delta)=E(\Delta)\). Thus \(E(\Delta)\) turns out to be a bounded projector onto the subspace \(\mathscr L_\Delta\) of functions of the form (4). Equality (4) defines the operator \(V_\Delta\), which maps \(\mathscr L_\Delta\) into the subspace \(Q_\delta\) of functions from \(\mathscr L_2(-\pi,\pi)\) that are equal to zero outside \(\delta\). At the same time the operator \(V_\Delta^{-1}\) exists and is defined on all of \(Q_\delta\).
Theorem 3. The operator \(V_\Delta\) is isometric, and for every function \(h\in\mathscr L_\Delta\) one has
\[ h\in\mathfrak D_T,\qquad Th=V_\Delta^{-1}\Lambda_\Delta V_\Delta h, \]
where \(\Lambda_\Delta\) is the operator of multiplication by \(\lambda(t)\) in \(Q_\delta\).
The following theorem establishes the completeness of the system of projectors \(E(\Delta)\).
Theorem 4. Let \(E(\Delta)f=0\) for some function \(f\in\mathscr L_2(-\infty,\infty)\) for all \(\Delta\). Then \(f=0\).
Next denote by \(\mathfrak M\) the linear span of all possible functions of the form (4). From Theorems 2 and 4 it is not difficult to obtain the following proposition.
Theorem 5. The linear manifold \(\mathfrak M\) is dense in \(\mathscr L_2(-\infty,\infty)\).
Operation (1) induces on \(\mathfrak M\) a certain operator \(S\). Since \(S\subset T\), \(S\) admits a closure, and moreover \(\overline S\subseteq T\). Furthermore, the following holds:
Theorem 6. The operator \(T\) coincides with the closure of the operator \(S\).
Let \(P\) be a bounded projector, not necessarily orthogonal. We shall say that it reduces the operator \(T\) if its range is contained in \(\mathfrak D_T\) and is invariant with respect to \(T\), together with the range of the operator \(I-P\). It is obvious that if \(P\) reduces the operator \(T\), then \(P\) and \(T\) commute.
Theorem 7. The projector \(E(\Delta)\) reduces the operator \(T\) for any \(\Delta\).
By the preceding theorems, in particular, it has been established that:
1) \(\mathscr{L}_{\Delta} \subset \mathfrak{D}_{T}\), \(T\mathscr{L}_{\Delta} \subset \mathscr{L}_{\Delta}\), and 2) the spectrum of the part of the operator \(T\) in \(\mathscr{L}_{\Delta}\) coincides with \(\Delta\).
Theorem 8. If the subspace \(R_{\Delta}\) has properties 1) and 2), then \(R_{\Delta} \subseteq \mathscr{L}_{\Delta}\).
\(2^\circ\). Suppose now that \(A'(\lambda) \ne 0\) for \(\lambda \in \Sigma\). In this case the \(\Gamma_k\) have no common points, and each of them is described by the equation \(\lambda=\lambda_k(t)\), in which \(\{\lambda_k(t)\}_{k=1}^{\infty}\), for fixed \(t\), is the sequence of eigenvalues of the operator \(T_t\). The corresponding sequence of eigenfunctions can, for each \(t\), be transformed into an orthonormal basis \(\{u_{n,t}(x)\}_{n=1}^{\infty}\) in \(\mathscr{L}_2(0,1)\). In this basis we shall have
\[ T_t u_{n,t}(x)=\sum_{i=1}^{n}\lambda_{in}(t)u_{i,t}(x), \qquad \lambda_{ii}(t)=\lambda_i(t), \]
where \(\lambda_{in}(t)\) are continuous functions of \(t\) on the interval \([-\pi,\pi]\). Denote by \(\Lambda\) the right triangular matrix-function with elements \(\lambda_{ik}(t)\) for \(i\leq k\). We next introduce into consideration the Hilbert space \(\overrightarrow{\mathscr{L}}_2(-\pi,\pi)\) of vector-functions \(\alpha=\{a_k(t)\}_{k=1}^{\infty}\), for which
\[ \sum_{k=1}^{\infty}\int_{-\pi}^{\pi}|a_k(t)|^2\,dt<\infty, \]
with scalar product
\[ (\alpha,\beta)=\sum_{k=1}^{\infty}\frac{1}{2\pi}\int_{-\pi}^{\pi}a_k(t)\overline{b_k(t)}\,dt. \]
Define on finite vectors from \(\overrightarrow{\mathscr{L}}_2(-\pi,\pi)\) the operator \(\Lambda_0\) of multiplication by the matrix \(\Lambda\). It can be shown that \(\Lambda_0\) admits a closure. The following theorem shows that \(\Lambda_0\) is a triangular model of the operator \(T\) in \(\overrightarrow{\mathscr{L}}_2(-\pi,\pi)\).
Theorem 9. The operator \(T\) is unitarily equivalent to the closure of the operator \(\Lambda_0\).
The author expresses his gratitude to I. M. Glazman for posing the problem and to Yu. I. Lyubich for discussion of the results.
Kharkov Polytechnic Institute
named after V. I. Lenin
Received
22 XI 1963
REFERENCES
- Yu. I. Lyubich, V. I. Macaev, Mat. sbornik, 56 (98), 4 (1962).
- I. M. Gel'fand, DAN, 73, No. 6 (1950).
- P. B. Nayman, Dokl. Akad. Nauk SSSR, No. 10 (1963).
- F. S. Rofe-Beketov, DAN, 152, No. 6 (1963).