Spectral properties of complex Airy operator on the semi-axis

TL;DR

This paper proves the completeness of root functions for a class of complex Schrödinger operators on the semi-axis, including the Airy operator, under specific conditions on the potential and the argument of the complex parameter.

Contribution

It extends the completeness results for the Airy operator to a broader range of complex parameters using asymptotic analysis of Airy functions.

Findings

Completeness holds for the Schrödinger operator with complex potential under certain growth conditions.

For the Airy operator, completeness is valid if the argument of the complex parameter is less than 5π/6.

The paper introduces a new technique based on asymptotic behavior of Airy functions for proving completeness.

Abstract

We prove the theorem on the completeness of the root functions of the Schroedinger operator $L=-d^2/dx^2+p(x)$ on the semi-axis $\mathbb R_+$ with a complex--valued potential $p(x)$. It is assumed that the potential $p = q \pm ir$ is such that the real functions $q$ and $r$ are subject the conditions $$ q(x) \geqslant c r(x), \quad r(x) \geqslant c_0+ c_1 x^\alpha, \quad \alpha >0, $$ where the constants $c, \ c_0\in \mathbb R$, $c_1>0$ and $\arg(\pm i+c) < 2\alpha\pi/(2+\alpha)$. For the case of the Airy operator $L_c=-d^2/dx^2+cx$, $c=const$, this theorem imply the completeness of the system of the eigenfunctions of this operator if $|\arg c|<2\pi/3$. Using another technique based on the asymptotic behavior of the Airy functions we prove that the completeness theorem for the operator $L_c$ remains valid, provided that $|\arg c|<5\pi/6$.

Equations

$${\mathcal{L}}_c=-\frac{d^2}{d{x}^2}+cx$$

$${\mathcal{L}}_{c,\alpha }=-\frac{d^2}{d{x}^2}+c{x}^{\alpha },x\in [0,\infty ),$$

$$Ly=-y^{\prime \prime }+p(x)y,p(x)=q(x)\pm ir(x),x\in [0,\infty ),$$

$$r(x)\ge M_0,q(x)\ge c_{0}r(x)+M_1,{\underline{\lim }}_{x\to \infty }{x}^{-\alpha }r(x)\ge a>0,\alpha >0,$$

$$l(y)=-y^{\prime \prime }+cxy,x\in [0,+\infty ),$$

$$\mathfrak{D}({\mathcal{L}}_c)=\{y\in L_2({\mathbb{R}}_{+}):y\in W_{2,loc}^2,l(y)\in L_2({\mathbb{R}}_{+}),y(0)=0\}.$$

$$\mathrm{Ai}(0)=\frac{1}{3^{2/3}\Gamma (2/3)},{\mathrm{Ai}}^{\prime }(0)=-\frac{1}{3^{1/3}\Gamma (1/3)},\mathrm{Bi}(0)=\frac{1}{3^{1/6}\Gamma (2/3)},{\mathrm{Bi}}^{\prime }(0)=\frac{3^{1/6}}{\Gamma (1/3)},$$

$$W(\mathrm{Ai},\mathrm{Bi})=\mathrm{Ai}(z){\mathrm{Bi}}^{\prime }(z)-{\mathrm{Ai}}^{\prime }(z)\mathrm{Bi}(z)=1/\pi .$$

$$\mathrm{Ai}(z)=\frac{1}{2\sqrt{\pi }}{z}^{-1/4}{e}^{-\frac{2}{3}{z}^{3/2}}\left(1+O(z^{-3/2})\right),{\mathrm{Ai}}^{\prime }(z)=-\frac{1}{2\sqrt{\pi }}{z}^{1/4}{e}^{-\frac{2}{3}{z}^{3/2}}\left(1+O(z^{-3/2})\right).$$

$$\mathrm{Ai}(-x)=\frac{1}{\sqrt{\pi }}|x{|}^{-1/4}\left(\sin \left(\frac{2}{3}|x{|}^{3/2}+\frac{\pi }{4}\right)+O(|x{|}^{-3/2})\right),|x|\to \infty .$$

$$z_k=-{\left[\frac{3}{2}\pi \left(k-\frac{1}{4}\right)\right]}^{2/3}+O(k^{-4/3}),k=1,2,\dots .$$

$$\mathrm{U}(0)=0,{\mathrm{U}}^{\prime }(0)=\frac{2\cdot 3^{1/6}}{\Gamma (1/3)},W(\mathrm{Ai},\mathrm{U})=1/\pi .$$

$$\mathrm{U}(z)=\frac{1}{\sqrt{\pi }}{z}^{-1/4}{e}^{\frac{2}{3}{z}^{3/2}}\left(1+O(z^{-3/2})\right),{\mathrm{U}}^{\prime }(z)=\frac{1}{\sqrt{\pi }}{z}^{1/4}{e}^{\frac{2}{3}{z}^{3/2}}\left(1+O(z^{-3/2})\right).$$

$$y(x)=\pi c^{-1/3}\left[\mathrm{Ai}(c^{1/3}x){\int }_0^x\mathrm{U}(c^{1/3}t)f(t)dt+\mathrm{U}(c^{1/3}x){\int }_x^{\infty }\mathrm{Ai}(c^{1/3}t)f(t)dt\right],$$

$$x^{1/2}y(x)\to 0,y^{\prime }(x)\to 0x\to +\infty ,x^{1/2}y(x),y^{\prime }(x)\in L_2({\mathbb{R}}_{+}).$$

$$y^{\prime }(x)=\pi \left[{\mathrm{Ai}}^{\prime }(c^{1/3}x){\int }_0^x\mathrm{U}(c^{1/3}t)f(t)dt+{\mathrm{U}}^{\prime }(c^{1/3}x){\int }_x^{\infty }\mathrm{Ai}(c^{1/3}t)f(t)dt\right],$$

$$y^{\prime \prime }=\pi \left[{\mathrm{Ai}}^{\prime }(c^{1/3}x)\mathrm{U}(c^{1/3}x)-{\mathrm{U}}^{\prime }(c^{1/3}x)\mathrm{Ai}(c^{1/3}x)\right]f(x)+$$

$$+\pi c^{1/3}\left[{\mathrm{Ai}}^{\prime \prime }(c^{1/3}x){\int }_0^x\mathrm{U}(c^{1/3}t)f(t)dt+{\mathrm{U}}^{\prime \prime }(c^{1/3}x){\int }_x^{\infty }\mathrm{Ai}(c^{1/3}t)f(t)dt\right]=-f(x)+cxy(x),$$

$$\begin{gathered} y=C_1\mathrm{Ai}(c^{1/3}x)+C_2\mathrm{U}(c^{1/3}x)+ \\ +\pi c^{-1/3}\left[\mathrm{Ai}(c^{1/3}x){\int }_0^x\mathrm{U}(c^{1/3}t)f(t)dt+\mathrm{U}(c^{1/3}x){\int }_x^{\infty }\mathrm{Ai}(c^{1/3}t)f(t)dt\right], \end{gathered}$$

$$|\mathrm{U}(c^{1/3}t)|\le M{t}^{-1/4}\exp \left(a{t}^{3/2}\right),a:=\frac{2}{3}|c{|}^{1/2}\cos \frac{\gamma }{2}$$

$$\left|{\int }_0^x\mathrm{U}(c^{1/3}t)f(t)dt\right|\le \left({\int }_0^b+{\int }_b^{x-1}+{\int }_{x-1}^x\right)|\mathrm{U}(c^{1/3}t)||f(t)|dt\le$$

$$\le M_1\Vert f\Vert +M\Vert f\Vert x^{1/4}\exp \left[a(x^{3/2}-x^{1/2})\right]+M{x}^{-1/4}\exp \left(a{x}^{3/2}\right){\left({\int }_{x-1}^x|f(t){|}^{2}dt\right)}^{1/2}.$$

$$\left|\mathrm{Ai}(c^{1/3}x){\int }_0^x\mathrm{U}(c^{1/3}t)f(t)dt\right|\le M{x}^{-1/4}\exp \left(-a{x}^{3/2}\right)\Vert f\Vert +$$

$$+M\exp \left(-a{x}^{1/2}\right)\Vert f\Vert +M{x}^{-1/2}{\left({\int }_{x-1}^x|f(t){|}^{2}dt\right)}^{1/2}.$$

$$\left|{\int }_x^{\infty }\mathrm{Ai}(c^{1/3}t)f(t)dt\right|\le M{\left({\int }_{x+1}^{\infty }{g}^2(t)t^{-3/2}dt\right)}^{1/2}\Vert f\Vert +M{\int }_x^{x+1}g(t)t^{-3/4}|f(t|dt$$

$$\le 2^{1/2}M{x}^{-1/4}g(x+1)\Vert f\Vert +Mg(x)x^{-3/4}{\left({\int }_x^{x+1}|f(t){|}^{2}dt\right)}^{1/2}.$$

$$\mathrm{U}(c^{1/3}x)\le M{x}^{1/4}{g}^{-1}(x),g(x+1)g^{-1}(x)\le M\exp \left(-a{x}^{1/2}\right)$$

$$M\exp \left(-a{x}^{1/2}\right)\Vert f\Vert +M{x}^{-1/2}{\left({\int }_x^{x+1}|f(t){|}^{2}dt\right)}^{1/2}.$$

$$|y(x)|\le M_2\exp \left(-a{x}^{1/2}\right)\Vert f\Vert +M_2{x}^{-1/2}{\left({\int }_{x-1}^{x+1}|f(t){|}^{2}dt\right)}^{1/2},x>b+1,$$

$${\int }_b^{\infty }|x^{1/2}y(x){|}^{2}dx\le M\Vert f{\Vert }^2{\int }_b^{\infty }x\exp \left(-2a{x}^{1/2}\right)dx+$$

$$+M{\int }_b^{\infty }{\int }_{x-1}^{x+1}|f(t){|}^{2}dtdx\le M\Vert f{\Vert }^2+M{\int }_{b-1}^{\infty }|f(t){|}^2{\int }_{t-1}^{t+1}dxdt\le M\Vert f{\Vert }^2.$$

$$({\mathcal{L}}_{c}y,y)=\int_{0}^{\infty}(-y^{\prime\prime}\overline{y}+cxy\overline{y})dx=-\overline{y}(x)y^{\prime}(x)\Big{|}_{0}^{\infty}+\int_{0}^{\infty}|y^{\prime}|^{2}dx+c\int_{0}^{\infty}x|y|^{2}dx.$$

$$({\mathcal{L}}_{c}y,u)={\int }_0^{\infty }\left(-y^{\prime \prime }(x)\overline{u}(x)+cy(x)\overline{u}(x)\right)dx={\int }_0^{\infty }\left(-y(x){\overline{u}}^{\prime \prime }(x)+y(x)\overline{\overline{c}u}(x)\right)dx+$$

Full text

, No 16-01-00706.

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$x\in[0,+\infty)$ . $c$. 1: $L_{2}(\mathbb{R}_{+})$ $|\arg c|<5\pi/6$.

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$\alpha>0,\ \,c\in\mathbb{C}\setminus\mathbb{R}_{-}$.

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$M_{0},\,M_{1},\,c_{0}$ — , . $q$ $r$ . 2: (4) $L_{D}$ (3) , $L_{2}(\mathbb{R}_{+})$ $|\gamma|<2\alpha\pi/(2+\alpha)$, $\gamma=\arg(\pm i+c_{0})\in(0,\pi)$. , – .

, $cx^{\alpha}$ $|c|(\cos\gamma+i\sin\gamma)x^{\alpha}$, (2) $|\arg c|<2\alpha\pi/(2+\alpha)$. , $c=i$ $\alpha>2/3$. 2 , (1) $|\arg c|<2\pi/3$. 2 , [1]. ( . [2, $\S 4$]). [3], , (4) $c_{0}=0$. 1 $|\arg c|\in[2\pi/3,5\pi/6)$ ( 2) . .

1999 ., ( . [4]). , ${\mathcal{L}}_{c}$ . .  .  .  (Y. Almog) [5]: ${\mathcal{L}}_{i,\alpha}=-d^{2}/dx^{2}+ix^{\alpha}$ $\alpha\in(0,2/3]$? , 1. , : $\alpha_{0}<2/3$, , ${\mathcal{L}}_{i,\alpha}$ ${\mathcal{L}}_{2}(\mathbb{R}_{+})$ $\alpha\in(\alpha_{0},2/3]$.

$\mathbb{R}_{+}$, , ( . ). ${\mathcal{L}}_{c}$ $c=i$ ( ., , [6]), $c\in\mathbb{C}\setminus\{0\}$. – ( . [7]– [10] ). ( ., , [6]). , $\varepsilon$- $\sigma_{\varepsilon}:=\{\lambda\in{\mathbb{C}}:\|(L-\lambda I)^{-1}\|\geq\varepsilon^{-1}\}$. , , , , , , , , [11]– [13].

, $p=q+ir$ . , $r$ $q$, . , , , , [14]—[20]. , , . . $q(x)\equiv 0$. , , , , [4], [21]– [26]. $PT$- $p(x)=-\overline{p(-x)}$. , , , , , [27]–[29]. , [30], $c=i$ . , , , , .

. , $y^{\prime}(0)+hy(0)=0,h\in\mathbb{R}$.

1. ${\mathcal{L}}_{c}$

${\mathcal{L}}_{c}$ . , , ${\mathcal{L}}_{c}$ $L_{2}({\mathbb{R}}_{+})$

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$c$ , $c\in{\mathbb{C}}\setminus(-\infty,0]$. , $\gamma:=\arg c\in(-\pi,\pi)$. , ${\mathcal{L}}_{c}$ $(0,+\infty)$, $L_{2}({\mathbb{R}}_{+})$.

— $y^{\prime\prime}=zy$. ( ., , [31, IV, §1] [32, §10.4]), $\mathrm{Ai}(z)$ $\mathrm{Bi}(z)$,

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$3/2$ $2/3$. $|\arg z|<\pi-\varepsilon$, $\varepsilon>0$ , $|z|\to\infty$ $\mathrm{Ai}(z)$ 111 $z^{\alpha}$ $\arg z\in[-\pi,\pi)$.

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(5) . , $\mathrm{Ai}(z)$ $\mathrm{Ai}^{\prime}(z)$ $|z|\to\infty$ $|\arg z|<\pi/3$. , $\arg z=\pi$ $\mathrm{Ai}(-x)$, $x>0$,

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$z_{k}$ $\mathrm{Ai}(z)$ , $\arg z=-\pi$

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$\mathrm{Bi}(z)$ $\mathrm{U}(z):=\mathrm{Bi}(z)-\sqrt{3}\mathrm{Ai}(z)$,

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$|\arg z|<\pi/3$

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, $\mathrm{U}(z)$ $\mathrm{U}^{\prime}(z)$ $|z|\to\infty$ . $\mathrm{U}(z)$ $\mathrm{U}^{\prime}(z)$ , .

1**.**

${\mathfrak{D}}({\mathcal{L}}_{c})$**

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$f(x)$* $L_{2}({\mathbb{R}}_{+})$. $y\in{\mathfrak{D}}({\mathcal{L}}_{c})$:*

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(8) $L_{2}({\mathbb{R}}_{+})$ , ${\mathcal{L}}_{c}$.

Доказательство.

, $c^{1/3}x$, $x>0$, $|\arg z|<\pi/3$ . , $\mathrm{Ai}(c^{1/3}t)$ , (8) . $f\in L_{2}({\mathbb{R}}_{+})$, $y$, (8), $W_{2}^{1}[0,b]$ $b$. ,

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, $y^{\prime}\in W_{2}^{1}[0,b]$ $b$. ,

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. . $y\in W^{2}_{2}[0,b]$ $b>0$ $l(y)=f\in L_{2}(\mathbb{R}_{+})$. $y(0)=0$ $y\in{\mathfrak{D}}({\mathcal{L}}_{c}).$ . , $y\in{\mathfrak{D}}({\mathcal{L}}_{c})$ $l(y)=f\in L_{2}(\mathbb{R}_{+})$. ,

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$C_{1}$, $C_{2}$ — . $y(0)=0$ $C_{1}=0$, $y\in L_{2}({\mathbb{R}}_{+})$ (12), , $C_{2}=0$, . . $y$ (8). , (8) $L_{c}^{-1}$. $|y(x)|\leqslant M\|f\|$ $[0,b]$ (12). $M$ ( $M_{1},M_{2}$) , $\|\cdot\|=\|\cdot\|_{L_{2}({\mathbb{R}}_{+})}$.

(9). , (7) $M$, , $t>0$

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( , $\gamma=\arg c$). , $g(t)=t^{-1/4}\exp(at^{3/2})$ $t\geq b=b(a)$. $x>b+1$

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$M_{1}$ — , $b$. , $x$, $(x-1)^{3/2}\leq x^{3/2}-x^{1/2}$, $x$. (5) $x$

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(8). . $t$ $g(t)=t^{1/2}\exp(-at^{3/2})$ , $x$

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$(x+1)^{3/2}\leq x^{3/2}+x^{1/2}$,

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(8)

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C ,

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(9). (9) , (8) (10) , $\mathrm{Ai}^{\prime}$ $\mathrm{U}^{\prime}$ $x^{1/2}$.

, $x^{1/2}y(x)\in L_{2}({\mathbb{R}}_{+})$. (11)

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$y^{\prime}\in L_{2}({\mathbb{R}}_{+})$ , , $|y^{\prime}(x)|$ (11), $x^{1/2}$. . ∎

2**.**

${\mathcal{L}}_{c}$* $S_{\gamma}$ , $\arg\lambda=0$ $\arg\lambda=\gamma$, $\gamma=\arg c$.*

Доказательство.

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, $y(0)=0$ $\overline{y}(x)y^{\prime}(x)\to 0$ $x\to\infty$ (9). ∎

3**.**

${\mathcal{L}}_{c}$* ( ). ${\mathcal{L}}_{c}^{*}$ ${\mathcal{L}}_{\overline{c}}$.*

Доказательство.

1, ${\mathcal{L}}_{c}$, , . , . . $y(x)\in{\mathfrak{D}}(L_{c})$, $u(x)\in{\mathfrak{D}}(L_{\overline{c}})$.

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${\mathcal{L}}_{c}y=z,\ {\mathcal{L}}_{\overline{c}}u=v$.

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, ${\mathcal{L}}_{\overline{c}}^{-1}=\left({\mathcal{L}}_{c}^{-1}\right)^{*}$. ${\mathcal{L}}_{\overline{c}}={\mathcal{L}}_{c}^{*}$. ∎

4**.**

${\mathcal{R}}_{c}(\lambda)=({\mathcal{L}}_{c}-\lambda I)^{-1}$* $\lambda\in{\mathbb{C}}\setminus S_{\gamma}$, $S_{\gamma}$ 2.*

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Доказательство.

$\gamma=0$ ${\mathcal{L}}_{c}$ , $S_{\gamma}={\mathbb{R}}_{+}$ . $\gamma\neq 0$. 2, $T_{1}=e^{i(\pi/2-\gamma)}\operatorname{sign}\gamma\cdot{\mathcal{L}}_{c}$ $T_{2}=-i\operatorname{sign}\gamma\cdot{\mathcal{L}}_{c}$ ( . . $\mathrm{Re}\,(T_{j}y,y)\geq 0$ $y\in{\mathfrak{D}}({\mathcal{L}}_{c})={\mathfrak{D}}(T_{j})$, $j=1,\,2$). 3, , . . .

$z$ $T_{1}-zI$ $T_{2}-zI$ ( ., , [33, .III.10])

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, (13). ∎

2 , ${\mathcal{L}}_{c}$ $S_{\gamma}$. $T=e^{-i\gamma/2}{\mathcal{L}}_{c}$ , 4 , $m$- . $m$- . .

5**.**

${\mathfrak{t}}$* $T=e^{-i\gamma/2}{\mathcal{L}}_{c}$*

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Доказательство.

${\mathfrak{t}}_{0}[u,v]=(Tu,v)$ ${\mathfrak{D}}(T)={\mathfrak{D}}({\mathcal{L}}_{c})$. , 2,

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, ${\mathfrak{D}}({\mathfrak{t}})$ ,

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. . ${\mathfrak{D}}({\mathfrak{t}})$ . , ${\mathfrak{D}}({\mathcal{L}}_{c})$ ${\mathfrak{t}}$ ( . [33, . VI, 2.1]). ${\mathfrak{t}}_{0}$ ${\mathfrak{D}}({\mathfrak{t}})$, ${\mathfrak{t}}$, $T$, (14). ∎

6**.**

$\lambda$* ${\mathcal{R}}_{c}(\lambda)$ . ${\mathcal{L}}_{c}$*

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Доказательство.

$T=e^{-i\gamma/2}{\mathcal{L}}_{c}$ $H=\mathrm{Re}\,T$ ( . [33, . VI, 3.3]). $H$ ${\mathfrak{h}}=\mathrm{Re}\,{\mathfrak{t}}$, . .

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$H$ , 5

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$(H-\lambda)^{-1}$ ( ., , [34, . VII, §24]). , ${\mathcal{R}}_{c}(\lambda)$ , $\sigma({\mathcal{L}}_{c})$ .

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$t=-\lambda c^{-2/3}+xc^{1/3}$. $-y^{\prime\prime}_{tt}+ty=0$, . .

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(5) (7), $C_{2}=0$, $y(0)=0$, $\mathrm{Ai}(-\lambda c^{-2/3})=0$. , $\mathrm{Ai}(z)$ , (15) (6). (18) $\lambda=\lambda_{n}$, (16). ∎

${\mathcal{L}}_{c}$

1**.**

${\mathcal{L}}_{c}$* $L_{2}({\mathbb{R}}_{+})$ $|\arg c|<(5\pi)/{6}$.*

Доказательство.

$\{y_{n}\}$ ${\mathcal{L}}_{c}$

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$\{z_{k}\}$ — ${\mathcal{L}}_{\overline{c}}$. ( . [35, 4]), $\mathrm{Ai}({\mathrm{w}}+c^{1/3}x)$, $y_{n}$ ${\mathrm{w}}=-t_{n}$. $\mathrm{Im\,}c\geq 0$, . . $\gamma\in[0,\pi)$, $\gamma=\arg c$. $\gamma\in(-\pi,0]$ ( , ).

. $f\in L_{2}(\mathbb{R}_{+})$ ${\mathcal{L}}_{c}$.

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, $F$ $\rho\leq 3/2$ $\rho=3/2$. , $F$

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, $\alpha_{0}\in(0,\gamma/3)$, , (20) $S^{\prime}=S\bigcup S_{0}$,

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$\alpha_{0}$ ( ), , $\mathbb{C}\setminus S^{\prime}$ $2\pi/3$, $\gamma<5\pi/6$. (20) . $F({\mathrm{w}})\equiv 0$, $f(x)\equiv 0$. .

1. , $F_{0}$ (19) ${\mathrm{w}}\in{\mathbb{C}}$. $\delta\in(0,\pi/4]$, , $\gamma/2+\delta<\pi/2$. ${\mathrm{w}}$ $|{\mathrm{w}}|\leq R$. $F_{0}$

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, , ${\mathrm{w}}$ $|{\mathrm{w}}|\leq R$, , $x>x_{0}(R)$

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$x>x_{0}(R)$ $|{\mathrm{w}}|\leq R$

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$a_{1}:=2/3\cos\left(\gamma/{2}+\delta\right)$. , (5) (23) $x>x_{0}(R)$

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$\mathrm{Ai}^{\prime}({\mathrm{w}}+c^{1/3}x)^{3/2}$, $|{\mathrm{w}}+c^{1/3}x|^{1/4}$. , $F_{2}({\mathrm{w}})$ ${\mathrm{w}}$ $|{\mathrm{w}}|\leq R$ ${\mathrm{w}}$, . . $F_{2}$ $|{\mathrm{w}}|\leq R$. $R$ , $F_{0}=F_{1}+F_{2}$ . , $F({\mathrm{w}})$, ${\mathrm{w}}=-t_{k}$ ( , $F_{0}({\mathrm{w}})$ , $\mathrm{Ai}({\mathrm{w}})$ ).

$F_{0}({\mathrm{w}})$ $|{\mathrm{w}}|\to\infty$. $F_{1}$ $R$,

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$v(x)=\exp\left\{-a_{1}(|c|^{1/3}x-R)^{3/2}+x\right\}$, $a_{1}$ (23). $x>x_{0}$, $x_{0}=x_{0}(R)$ . (24)

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, $|F_{2}({\mathrm{w}})|\to 0$ ${\mathrm{w}}\to\infty$. (25) , $F_{0}=F_{1}+F_{2}$ — $3/2$ $3/2$. $F({\mathrm{w}})$, $F_{0}({\mathrm{w}})$ $\mathrm{Ai}({\mathrm{w}})$, , ( ., , [35, . I, §9]), . . $F$ $\rho\leq 3/2$, $\rho=3/2$, .

2.

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$\varepsilon\in(0,\pi/3)$ . (26) (5),

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$\varphi=\arg{\mathrm{w}}$. $S_{1}\bigcup S_{2}$ , (27), $\cos(3\varphi/2)\leq\sin(3\varepsilon/2)$. $\varepsilon>0$ ,

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$R\to\infty$,

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, (5),

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, $\xi^{\prime}(x)=-\mathrm{Re}\,\left(c^{1/3}({\mathrm{w}}+c^{1/3}x)^{1/2}\right)$, , $\xi^{\prime}(x)\leq 0$,

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$\xi(x)$ , $\xi(0)=0$, $\xi(x)\leq 0$. (32) (30)

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, (32) $S_{1}\bigcup S_{2}$ $x\geq 0$. . $\varphi\in[-\pi+\gamma/3,-\pi/3+\varepsilon]$, ${\mathrm{w}}$ $c^{1/3}x$ $-\pi+\gamma/3\leq\arg z\leq\gamma/3$, . $\varphi\in[\pi/3-\varepsilon,\pi-2\gamma/3]$, ${\mathrm{w}}$ $c^{1/3}x$ $0\leq\arg z\leq\pi-2\gamma/3$ $\leq\pi$, . (32) , $S_{1}\bigcup S_{2}$ , (27), (33). (29), , $S_{1}\bigcup S_{2}$

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$S$ (21), $S\setminus(S_{1}\bigcup S_{2})$ $2\pi/3$. – $\widetilde{F}({\mathrm{w}})=F({\mathrm{w}})({\mathrm{w}}+1)^{-1/2}$, , (34) $S$.

3. $F$ $S^{\prime\prime}=\mathbb{C}\setminus S$. $S^{\prime\prime}$ $\gamma$; $F$ – , $\gamma<2\pi/3$. $\gamma\in[2\pi/3,5\pi/6)$, , .

$-\pi<\varphi<-\pi+\gamma/3$. $|F_{2}({\mathrm{w}})|/|\mathrm{Ai}({\mathrm{w}})|$ ( - ), $\cos(3\varphi/2)<0$ (28). (33) $S_{0}$, (22), $\alpha_{0}<\gamma/3$. $\xi(x)$, (31). , $z^{3/2}$ $\arg z\in[-\pi,\pi)$. , $\varphi\in(-\pi,-\pi+\gamma/3)$ ${\mathrm{w}}+c^{1/3}x$, $x>0$, $(-\infty,0)$, $({\mathrm{w}}+c^{1/3}x)^{3/2}$, $x\in[0,+\infty)$, , $x_{1}$. , , $x_{1}$ — , $\xi(x)$ . $\xi$ $x\in(0,x_{1})$ $x\in(x_{1},+\infty)$. $\xi^{\prime}(x)<0$, ${\mathrm{w}}+c^{1/3}x$ $\varphi$ ( $x=0$) $-\pi$ ( $x=x_{1}$), . . (32). ${\mathrm{w}}+c^{1/3}x$ $\pi$ ( $x=x_{1}+0$) $\gamma/3$ ( $x\to+\infty$), , $x_{2}$, $\arg({\mathrm{w}}+c^{1/3}x)=\pi-2\gamma/3$. $x_{2}$ $\xi^{\prime}$ , . . $x_{2}$ — $\xi$. , (33), , (30) $\xi(x)\leq 0$ $x\in[0,+\infty)$, , $\xi(x_{2})\leq 0$. $\xi(x_{2})$ [math], ${\mathrm{w}}$ ${\mathrm{w}}+c^{1/3}x_{2}$.

$\alpha+2\gamma/3$, $\gamma/3-\alpha$ $\pi-\gamma$ ( $\alpha=\pi+\varphi\in(0,\gamma/3)$), $\varphi=\arg{\mathrm{w}}$.

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, $\xi(x_{2})\leq 0$

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— $\alpha\in[0,\gamma/3]$ $\gamma\in[2\pi/3,\pi)$ $\eta(\alpha)$. , $\eta(\alpha)$ , $\eta(0)>0$ $\eta(\gamma/3)<0$. , (35) $\alpha\in[\alpha_{0},\gamma/3]$ $\alpha_{0}\in(0,\gamma/3)$. , (33), (34), $\arg{\mathrm{w}}=\varphi\in[-\pi+\alpha_{0},\pi-2\gamma/3]$. – , , $\alpha_{0}+2\gamma/3$ $2\pi/3$. $\alpha_{0}=\alpha_{0}(\gamma)$ $\eta(\alpha)=0$. , , $\eta$

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4. , $\gamma<5\pi/6$ $F({\mathrm{w}})$ (34) , , . , . , $F({\mathrm{w}})\to 0$ ${\mathrm{w}}\to\infty$ . 2, ${\mathrm{w}}=Re^{-i\pi/2}$ , $S_{1}$, (29). (33). $x\in[0,x_{0}]$ $x_{3}(R)=R^{-\theta}$, $\theta\in(0,1/2)$ .

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$\xi(x)$ (31) $[0,+\infty)$ $\xi(0)=0$ $-\infty$,

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, $R\to\infty$

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$R^{1/2}e^{\xi(x_{3}(R))}=o(1)$. , $F({\mathrm{w}})\equiv 0.$

5. , $F_{0}({\mathrm{w}})=\int_{0}^{\infty}\mathrm{Ai}({\mathrm{w}}+c^{1/3}x)\,d\mu\equiv 0$ ( $d\mu=f(x)dx$). , $\mathrm{Ai}^{\prime\prime}(t)=t\mathrm{Ai}(t)$,

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$P_{n}$ $Q_{n}$ — .

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$\deg P_{n}(t)=p_{n}$, $\deg Q_{n}(t)=q_{n}$. $n\geq 5$

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, $n\geq 2$

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${\mathrm{w}}$,

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$j\geq 0$. ,

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,

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— $n=0$ $n=1$. . (38) $n<N$, $N\geq 2$, , (37) $j=2N$ (36),

[TABLE]

, ,

[TABLE]

(37) $j=2N+1$ (36),

[TABLE]

, ,

[TABLE]

(38) .

(38) $f(x)\equiv 0$.

[TABLE]

(5) $\lambda$. ,

[TABLE]

$h(\lambda)\equiv 0$. $\mathrm{Ai}(c^{1/3}x)f(x)$, $f(x)\equiv 0$. 1. ∎

$L$

2**.**

$L$* (3)*

[TABLE]

$AC_{loc}$* — , (4).*

[TABLE]

L $L_{2}(\mathbb{R}_{+})$. , $\beta$ $\beta\in\left(\frac{2+\alpha}{2\alpha},\frac{\pi}{\gamma},\right).$

Доказательство.

, . , ( . [2, $\S 5$]).

[TABLE]

$\{y_{k}\}$ — $L$, $\lambda_{k}$, $\{z_{k}\}$ — . , $\forall f\in L_{2}(\mathbb{R}_{+})$ $S(t,f)$ $\forall t>0$ $L_{2}({\mathbb{R}}_{+})$ $S(t,f)\to f$ $t\to+0$. , $\{y_{k}\}$ , , , $f$ .

, $M_{0},\,M_{1}$ (4) . ${\mathfrak{D}}_{0}(L)$ ${\mathfrak{D}}(L)$, . [3], , , . . $l(y)=0$ $L_{2}({\mathbb{R}}_{+})$. $L$ . , ${\mathfrak{D}}_{0}(L)$ $L$, . . $L$ ${\mathfrak{D}}_{0}(L)$ $L$. $f\in{\mathfrak{D}}_{0}(L)$ $(Lf,f)$ , $\arg\gamma$ $\mathbb{R}_{+}$. (4) . $L$ , $T=e^{-i\gamma/2}L$ $m$- $\gamma$. $H=\mathrm{Re}\,T$ ${\mathfrak{D}}(H)={\mathfrak{D}}(L)$

[TABLE]

(4) $\widetilde{r}(x)\geq Mr(x)\geq Max^{\alpha}$. , $s_{n}$ $H$

[TABLE]

. . $H$ $L$ $2\alpha/(2+\alpha)$. – , . [2, 5.1]. ∎

1**.**

, . , .

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