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}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|\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}
|\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(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}^{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|=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}}(\lambda)-\mathfrak{Ft(\lambda)]/h+\lambda\mathfrak{F}_t(\lambda)\to0), as (h\to0), in the sense of the norm (|\cdot|(R^+)), where (0<\eta<\varepsilon_0), the function}). From this it follows that, for every function (f\in L^2_{-\eta

[
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|}). 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|). Finally, the function (13) admits an estimate of the form (|f(\cdot,t)|{-\eta}\le C(\eta)e^{\eta^2t}|f|).

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|=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|). 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)|). 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).}\le C(\eta)e^{\eta t}|f|_{-\eta

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|-p(x)u]) with an initial sufficiently smooth function (f) of finite order of growth. Consequently, there is}=\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

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