V. E. LYANTSE

ON THE SOLUTION OF CERTAIN BOUNDARY-VALUE PROBLEMS BY THE FOURIER METHOD

(Presented by Academician I. M. Vinogradov, 12 IV 1963)

Let \(l_\theta\) be a (non-self-adjoint) boundary-value problem on the half-axis \(R^+=[0,\infty)\), generated by the differential expression \(l[y]=-y''+p(x)y\) and the boundary condition \(y'(0)-\theta y(0)=0\). To the boundary-value problem \(l_\theta\) there corresponds the so-called \(l_\theta\)-Fourier transform. In our papers \((^1,^2)\), under the assumption that for some \(\varepsilon>0\) the function \(|p(x)|\exp \varepsilon x\) is summable on the half-axis \(R^+\), the \(l_\theta\)-Fourier transform was extended to a certain class of exponentially growing functions. It is of interest to ask whether the theory of the \(l_\theta\)-transform can be used to study the corresponding boundary-value problems by the Fourier method in some class of unbounded functions. Especially interesting is the case where the boundary-value problem \(l_\theta\) has so-called spectral singularities (see \((^2)\)), since in this case the \(l_\theta\)-transform does not admit a unique inversion. From what follows we shall see that even in the presence of spectral singularities, with the aid of the \(l_\theta\)-transform one can obtain fairly concrete results, among them the existence, uniqueness, explicit representation, and also estimates of solutions of the boundary-value problems under consideration. In the present paper we adhere to the same terminology and notation as in \((^1,^2)\).

For each real \(\eta\), by \(L_\eta^2(R^+)\) we denote the Hilbert space corresponding to the norm

\[ \|h\|_\eta=\left\{\int_0^\infty |h(x)e^{\eta x}|^2\,dx\right\}^{1/2}, \tag{1} \]

and put

\[ \Phi=\bigcap_{0<\eta<\varepsilon_0} L_{+\eta}^2(R^+),\qquad F=\bigcup_{0<\eta<\varepsilon_0} L_{-\eta}^2(R^+); \tag{2} \]

here \(\varepsilon_0\) is a positive number depending on the boundary-value problem \(l_\theta\), whose definition is indicated in \((^2)\). We regard the set \(\Phi\) as a countably Hilbert space with the system of norms \(\|\cdot\|_\eta\), \(0<\eta<\varepsilon_0\), and the set \(F\) as the linear space conjugate to the space \(\Phi\): \(F=\Phi'\). Accordingly, we shall write

\[ \int_0^\infty \varphi(x)f(x)\,dx=\langle\varphi,f\rangle,\qquad \varphi\in\Phi,\quad f\in F. \tag{3} \]

One may say that \(\Phi\) is the space of functions decreasing faster than \(\exp(-\varepsilon_0 x)\) as \(x\to\infty\), and \(F\) is the space of functions increasing more slowly than \(\exp \varepsilon_0 x\) as \(x\to\infty\). Let us define the operators generated by the differential expression \(l[y]\) and the boundary condition \(y'(0)=\theta y(0)\) in the spaces under consideration. Denote by \(\mathfrak D_\eta(L)\) the set of functions \(h\) which have a derivative \(h'\) absolutely continuous on every finite interval of the half-axis \(R^+\), satisfy the boundary condition \(h'(0)=\theta h(0)\), and are such that \(h\in L_\eta^2(R^+)\) and \(l[h]\in L_\eta^2(R^+)\). Put

\[ \mathfrak D_\Phi(L)=\bigcap_{0<\eta<\varepsilon_0}\mathfrak D_{+\eta}(L),\qquad \mathfrak D_F(L)=\bigcup_{0<\eta<\varepsilon_0}\mathfrak D_{-\eta}(L). \tag{4} \]

and \(Lh=l[h]\) for \(h\in \mathfrak D_\eta(L)\), \(|\eta|<\varepsilon_0\). Let \(f,g\in F\). It can be shown that \(f\in \mathfrak D_F(L)\) and \(Lf=g\) if and only if for all \(\varphi\in \mathfrak D_\Phi(L)\) the relation \(\langle L\varphi,f\rangle=\langle \varphi,g\rangle\) holds.

Using the theory of the \(l_\theta\)-Fourier transform, one can construct “functions” of the operator \(L\) in the space \(F\). We shall say that a function \(\mathfrak F(\lambda)\) of the complex variable \(\lambda\) belongs to the class \((\mathcal L)\) if it has the following properties: a) the domain of definition of the function \(\mathfrak F(s^2)\) contains the strip \(|\operatorname{Im}s|<\varepsilon_0\); b) the restriction of \(\mathfrak F(s^2)\) to the strip \(|\operatorname{Im}s|<\varepsilon_0\) is a holomorphic function of \(s\), satisfying the condition

\[ |\mathfrak F|_{\eta k}=\sup_{|\operatorname{Im}s|<\eta} \frac{|\mathfrak F(s^2)|}{1+|s|^{2k}}<\infty \tag{5} \]

for some nonnegative integer \(k\) and for all \(\eta\), \(0<\eta<\varepsilon_0\); c) the function \(\mathfrak F(\lambda)\) is holomorphic in a neighborhood of the eigenvalues \(\lambda_1,\ldots,\lambda_r\) of the restriction of the operator \(L\) to the Hilbert space \(L^2(R^+)\).

Theorem. There exists a correspondence \(\mathfrak F\to \mathfrak F(L)\), assigning to each function \(\mathfrak F\in(\mathcal L)\) a linear operator \(\mathfrak F(L)\) acting in the space \(F\), such that the conditions \(1^\circ\)—\(4^\circ\) listed below are satisfied.

\(1^\circ\). If \(|\mathfrak F|_{\eta k}<\infty\), \(0<\eta<\varepsilon_0\), then*

\[ \mathfrak D_{-\eta}(L^k)\subset \mathfrak D_{-\eta}(\mathfrak F(L)), \tag{6} \]

and there exists a constant \(\delta(\eta)\) such that

\[ \|\mathfrak F(L)f\|_{-\eta}\leq \delta(\eta)\|\mathfrak F\|_{\eta k}\cdot \sum_{\nu=0}^{k}\|L^\nu f\|_{-\eta}, \tag{7} \]

where \(\|\mathfrak F\|_{\eta k}\) is the largest of the three numbers**

\[ |\mathfrak F|_{\eta k},\qquad \max_{\substack{j=0,\ldots,\mu_k-1\\ k=1,\ldots,\rho}} |\mathfrak F^{(j)}(\widetilde\lambda_k)|,\qquad \max_{\substack{j=0,\ldots,m_k-1\\ k=1,\ldots,r}} |\mathfrak F^{(j)}(\lambda_k)|. \tag{8} \]

In particular, if \(|\mathfrak F|_{\eta 0}<\infty\), \(0<\eta<\varepsilon_0\), then the operator \(\mathfrak F(L)\) is defined on the whole space \(F\) and is a continuous mapping of each of the Hilbert spaces \(L^2_{-\eta}(R^+)\) into itself.

Relation (7) shows that the operator \(\mathfrak F(L)\) depends on \(\mathfrak F\) continuously in the sense of the norm \(\|\cdot\|_{\eta k}\). The correspondence \(\mathfrak F\to\mathfrak F(L)\) also has the following property of “weak” continuity:

\(2^\circ\). Let \(\{\mathfrak F_t\}\) be a family of functions \(\mathfrak F_t\in(\mathcal L)\) such that: a) \(|\mathfrak F_t|_{\eta k}<C_\eta<\infty\), \(0<\eta<\varepsilon_0\), where \(C_\eta\) is a constant independent of \(t\); b) \(\mathfrak F_t(\lambda)\to0\) as \(t\to0\) uniformly in every compact part of the set \(\{\lambda:\lambda=s^2,\ |\operatorname{Im}s|<\varepsilon_0\}\) and in some neighborhood of the points \(\lambda_1,\ldots,\lambda_r\). Then

\[ \langle \varphi,\mathfrak F_t(L)f\rangle\underset{t\to0}{\longrightarrow}0 \quad\text{for all }\varphi\in\Phi,\ f\in \mathfrak D_F(L^k). \tag{9} \]

\(3^\circ\). If \(\mathfrak F(\lambda)\equiv1\), then \(\mathfrak F(L)\) is the identity operator in the space \(F\); if \(\mathfrak F(\lambda)\equiv\lambda\), then \(\mathfrak F(L)=L\).

\(4^\circ\). If \(\mathfrak F,\mathfrak G\in(\mathcal L)\), and \(\alpha\) and \(\beta\) are arbitrary complex numbers, then

\[ \alpha\mathfrak F(L)+\beta\mathfrak G(L)\subset (\alpha\mathfrak F+\beta\mathfrak G)(L),\quad \mathfrak F(L)\mathfrak G(L)\subset(\mathfrak F\mathfrak G)(L). \tag{10} \]

For lack of space we cannot give the proof of this theorem. Let us only note that, using the same notation as in formula (18) of article \((^2)\), we have

\[ \mathfrak F(L)f(x)=\frac{1}{\pi}\int_{-\infty}^{\infty} s\omega(x,s^2)\mathfrak F(s^2) \frac{1}{A(s)A(-s)}\omega(f,s^2)\,ds+ \]

\[ \text{* } \mathfrak D_{-\eta}(\bullet)\text{ denotes the domain of definition of the restriction of the operator } \bullet \text{ to the space } L^2_{-\eta}(R^+). \]

\[ \text{** } \mu_k \text{ denotes the multiplicity of the spectral singularity } \widetilde\lambda_k,\text{ and } m_k \text{ the multiplicity of the eigenvalue } \lambda_k\ ({}^2). \]

\[ + \sum_{k=1}^{\rho} \sum_{j=0}^{\mu_k-1} C_{kj}\left\{\left(\frac{d}{d\lambda}\right)^j \mathfrak{F}(\lambda)\omega(x,\lambda)\right\}_{\lambda=\tilde{\lambda}_k} + \sum_{k=1}^{r} \left\{\left(\frac{d}{d\lambda}\right)^{m_k-1} M_k(\lambda)\mathfrak{F}(\lambda)\omega_f(\lambda)\omega(x,\lambda)\right\}_{\lambda=\lambda_k}. \tag{11} \]

Accordingly, \(\mathfrak{F}\to \mathfrak{F}(L)\) can be used to construct solutions of various boundary-value problems in the quarter-plane \(x>0,\ t>0\). Consider, for example, the boundary-value problem determined by the relations:

\[ u_t=u_{xx}-p(x)u,\qquad (u_x-\theta u)\big|_{x=0}=0,\qquad u\big|_{t=+0}=f(x). \tag{12} \]

It is not hard to see that the function \(\mathfrak{F}_t(\lambda)=e^{-\lambda t}\) belongs to the class \((\mathscr{L})\) and satisfies condition (5) for \(k=0\) even after multiplication by any power of \(\lambda\). Moreover, \(\mathfrak{F}_t(\lambda)\to 1\) as \(t\to 0\) with observance of the conditions indicated in Proposition \(2^\circ\), and, for \(t>0\),
\([\mathfrak{F}_{t+h}(\lambda)-\mathfrak{F}_t(\lambda)]/h+\lambda\mathfrak{F}_t(\lambda)\to0\), as \(h\to0\), in the sense of the norm \(\|\cdot\|_{\eta,0}\). From this it follows that, for every function \(f\in L^2_{-\eta}(R^+)\), where \(0<\eta<\varepsilon_0\), the function

\[ u=f(x,t)=e^{-tL}f(x) \tag{13} \]

belongs, for \(t>0\), to the domain of definition \(\mathfrak{D}_{-\eta}(L^n)\) of any power \(L^n\) of the operator \(L\) and is (infinitely) differentiable with respect to \(t\) in the sense of the norm \(\|\cdot\|_{-\eta}\). The function (13) satisfies the differential equation and the boundary condition (12). It satisfies the initial condition in the weak sense, i.e. \(\langle \varphi,f(\cdot,+0)\rangle=\langle \varphi,f\rangle\) for all \(\varphi\in\Phi\). If \(f\in\mathfrak{D}_{-\eta}(L)\), then the limit \(f(\cdot,+0)=f\) exists in the sense of the norm \(\|\cdot\|_{-\eta}\). Finally, the function (13) admits an estimate of the form \(\|f(\cdot,t)\|_{-\eta}\le C(\eta)e^{\eta^2t}\|f\|_{-\eta}\).

Let us also consider the boundary-value problem

\[ u_{tt}=u_{xx}-p(x)u,\qquad (u_x-\theta u)\big|_{x=0}=0,\qquad u\big|_{t=+0}=0,\qquad u_t\big|_{t=0}=f(x). \tag{14} \]

Applying arguments analogous to those used in the preceding example, we arrive at the following conclusions. The operator \(\sin t\sqrt{L}/\sqrt{L}\) is defined on the whole space \(F\), and for every function \(f\in\mathfrak{D}_{-\eta}(L)\) the function

\[ u=f(x,t)=\frac{\sin t\sqrt{L}}{\sqrt{L}}\,f(x) \tag{15} \]

belongs to \(\mathfrak{D}_{-\eta}(L)^*\) and is twice differentiable with respect to \(t\) in the sense of the norm \(\|\cdot\|_{-\eta}\). The function (15) is a solution of the boundary-value problem (14), which satisfies the initial condition in the sense of the norm \(\|\cdot\|_{-\eta}\). The solution (15) also admits an estimate of the form \(\|f(\cdot,t)\|_{-\eta}\le C(\eta)e^{\eta t}\|f\|_{-\eta}\). We note that the solutions we have constructed for the boundary-value problems (12) and (14) preserve the “integral” exponent \(\eta\) of exponential growth of the initial function \(f\).

It is natural also to consider the boundary-value problem for an equation of Schrödinger type \(u_t=i[u_{xx}-p(x)u]\). Here, however, a difficulty is encountered, consisting in the following: the function \(\mathfrak{F}_t(\lambda)=e^{-i\lambda t}\) corresponding to the last equation does not belong to the class \((L)\), since the function \(e^{-is^2t}\) has exponential order of growth with respect to \(s\) in the strip \(|\operatorname{Im}s|<\varepsilon_0\), and consequently \(\mathfrak{F}_t|_{\eta k}=\infty\) for arbitrarily large \(k\) (see relation (5)). We observe, however, that if in our considerations the space \(F\) of functions growing more slowly than \(\exp \varepsilon_0x\) is replaced by the space of functions of finite order of growth, then it will be possible to construct a correspondence \(\mathfrak{F}\to\mathfrak{F}(L)\) in which the role of the strip \(|\operatorname{Im}s|<\varepsilon_0\) is played by the real axis \(\operatorname{Im}s=0\), on which the function \(e^{-is^2t}\) is already bounded. This makes it possible to construct a solution of the boundary-value problem also for the equation \(u_t=i[u_{xx}-p(x)u]\) with an initial sufficiently smooth function \(f\) of finite order of growth. Consequently, there is

* If \(f\in\mathfrak{D}_{-\eta}(L^n)\), then also \(f(\cdot,t)\in\mathfrak{D}_{-\eta}(L^n)\).

the well-known analogy between the boundary-value problems under consideration and the Cauchy problem for the corresponding equations with coefficients independent of \(x\) (see \((^3)\)).

In the case when the boundary-value problem \(l_0\) has no spectral singularities \((^2)\), the uniqueness of the solutions of the boundary-value problems (12) and (14) in the space \(F\) can be proved by the usual Holmgren method (see \((^3)\)). We indicate a device by means of which uniqueness can be proved in the case when there are spectral singularities and, consequently, the \(l_0\)-Fourier transform is not uniquely invertible.

Denote by \(\Phi_0\) the set of functions \(\varphi\in\Phi\) having the property that each of the spectral singularities \(\widetilde\lambda_1,\ldots,\widetilde\lambda_\rho\) of the boundary-value problem \(l_0\) is a zero of multiplicity (at least) \(\mu_1,\ldots,\mu_\rho\), respectively, of the \(l_0\)-Fourier transform \(\omega\varphi(\lambda)\) of the function \(\varphi\).

Lemma. In order that a function \(f\in F\) satisfy the relation

\[ \langle \varphi, f\rangle = 0 \quad \text{for all } \varphi\in\Phi_0, \tag{16} \]

it is necessary and sufficient that \(*\)

\[ f(x)=\sum_{k=1}^{\rho}\sum_{j=0}^{\mu_k-1} C_{kj}\omega^{(j)}(x,\widetilde\lambda_k), \tag{17} \]

where \(C_{kj}\) are certain constants.

Let now \(f(x,t)\) be a solution of the boundary-value problem (12) corresponding to the initial function \(f(x)\equiv 0\). We must show that if \(f(\cdot,t)\in F\), then \(f(x,t)\equiv 0\)**. It can be proved that for an arbitrary function \(\varphi\in\Phi_0\) the boundary-value problem (12) with initial function \(f=\varphi\) has a solution \(\varphi(x,t)\), which for all \(t\ge 0\) also belongs to the set \(\Phi_0\). Consider an arbitrary \(t>0\) and, for \(\tau\le t\), put \(\alpha(\tau)=\langle \varphi(\cdot,t-\tau), f(\cdot,\tau)\rangle\). We have \(\alpha(0)=0\) and, as is easy to see, \(\alpha'(\tau)=0\). Consequently, \(\alpha(\tau)\equiv 0\) and, in particular, \(\langle \varphi, f(\cdot,t)\rangle=0\). On the basis of the preceding lemma we conclude from this that

\[ f(x,t)=\sum_{k=1}^{\rho}\sum_{j=0}^{\mu_k-1} C_{kj}(t)\omega^{(j)}(x,\widetilde\lambda_k). \tag{17′} \]

Taking into account that the function (17′) satisfies the differential equation \(u_t+l[u]=0\), that \(l[\omega^{(j)}(\cdot,\widetilde\lambda_k)]=\widetilde\lambda_k\omega^{(j)}(\cdot,\widetilde\lambda_k)+j\omega^{(j-1)}(\cdot,\widetilde\lambda_k)\), and that the functions \(\omega^{(j)}(\cdot,\widetilde\lambda_k)\) are linearly independent, we arrive at the conclusion that the functions \(C_{kj}(t)\) satisfy a certain linear homogeneous system of (independent) differential equations. Since, moreover, \(C_{kj}(0)=0\) (in view of the initial condition \(f(x,+0)=0\)), it follows that \(C_{kj}(t)\equiv 0\), as was required to prove. An analogous device can be applied to prove uniqueness in the space \(F\) of the solution of the boundary-value problem (14). In this case it is expedient to define the function \(\alpha(t)\) by the formula \(\alpha(\tau)=\langle \varphi_t(\cdot,t-\tau), f(\cdot,\tau)\rangle+\langle \varphi(\cdot,t-\tau), f_t(\cdot,\tau)\rangle\), where \(\varphi(x,t)\) is the solution of the boundary-value problem (14) corresponding to the initial function \(f=\varphi\in\Phi_0\), such that \(\varphi(\cdot,t)\in\Phi_0\).

Lviv Polytechnic Institute

Received
4 IV 1963

REFERENCES

\(^{1}\) V. E. Lyantse, DAN, 149, No. 2 (1963).
\(^{2}\) V. E. Lyantse, DAN, 150, No. 5 (1963).
\(^{3}\) I. M. Gelfand, G. E. Shilov, Generalized Functions, vol. 3, Moscow, 1958.

\(*\) \(\omega(x,\lambda)\) denotes the solution of the equation \(l[y]=\lambda y\) with initial values \(\omega(0,\lambda)=1,\ \omega_x'(0,\lambda)=0\); \(\omega^{(j)}\) is the derivative of order \(j\) of \(\omega\) with respect to the variable \(\lambda\).

\(**\) A. G. Kostyuchenko pointed out to the author that in fact uniqueness here holds in the class of functions growing as \(x\to+\infty\) like \(\exp(x^2)\).