Scattering by periodically moving obstacles

B. R. Vainberg

Sbornik. Mathematics, Tome 73 (1992) no. 1, pp. 289-304

Résumé

Suppose $x\in\mathbf R^n$, $L_0(\partial_t,\partial_x)$ is a homogeneous hyperbolic matrix, $U_0(t)$ is the operator taking the Cauchy data for the system $L_0u=0$ for $t=0$ into the corresponding data at time $t$, and $U(t)$ is the analogous operator constructed from the exterior mixed problem for the hyperbolic system $Lu=0$. It is assumed that the boundary of the domain and the coefficients of the operator $L$ are periodic in $t$ with period $T$, $L=L_0$ for $|x|\gg1$, the noncapturing condition is satisfied, the matrix $L_0(0,\partial_x)$ is elliptic, and the energy of solutions of the exterior problem is uniformly bounded for $t\geqslant 0$. Under these conditions it is proved that the space $H$ generated by the eigenfunctions of the monodromy operator $V=U(T)$ with eigenvalues on the unit circle is finite dimensional; for initial data $f$ with compact support the asymptotics of the solution $U(t)f$ of the exterior problem as $t\to\infty$ is obtained; in particular, it is shown that $U(t)f\sim U(t)Pf$, $t\to\infty$, where $P$ is the operator of projection onto $H$; and existence of the wave operators constructed on the basis of $U_0(t)$ and $U(t)$ and of the scattering operator is proved.

Bibliographie

[1] Asymptotic methods in equations of mathematical physics, Gordon and Breach Publ., 1989 | MR | MR | Zbl | Zbl

[2] Vainberg B. R., “Asimptoticheskoe povedenie pri $t\to\infty$ reshenii vneshnikh smeshannykh periodicheskikh po $t$ zadach”, Matem. zametki, 47 (1990), 6–16 | MR | Zbl

[3] Vainberg B. R., “O lokalnoi energii reshenii vneshnikh smeshannykh periodicheskikh po $t$ zadach”, Tr. MMO, 54 (1991)

[4] Vainberg B. R., “Rasseyanie na periodicheski dvizhuschikhsya prepyatstviyakh”, UMN, 45:4 (1990), 131–132

[5] Gelfand I. M., Shilov G. E., Obobschennye funktsii i deistviya nad nimi, Fizmatgiz, M., 1958

[6] Korotyaev E. L., “O rasseyanii vo vneshnem odnorodnom periodicheskom po vremeni magnitnom pole”, Matem. sb., 180 (222) (1984), 491–512 | MR

[7] Yafaev D. R., “Podprostranstva rasseyaniya i asimptoticheskaya polnota dlya nestatsionarnogo uravneniya Shredingera”, Matem. sb., 118 (160) (1982), 262–279 | MR | Zbl

[8] Bachelot A., Petkov V., “Operators d'ondes pour les systemes hyperboliques avec un potential periodique en temps”, Ann. Inst. Henri Poincare (Physique theorique), 47:4 (1987), 383–428 | MR | Zbl

[9] Cooper J., Strauss W., “Energy boundedness and decay of waves reflecting off a moving obstacle”, Indiana Univ. Math. J., 25 (1976), 671–690 | DOI | MR | Zbl

[10] Cooper J., Strauss W., “Scattering of waves by periodically moving bodies”, J. Funct. Anal., 47 (1982), 180–229 | DOI | MR | Zbl

[11] Cooper J., Strauss W., “Abstract scattering theory for time periodic systems with applications to electromagnetism”, Indiana Univ. Math., 34 (1985), 33–83 | DOI | MR | Zbl

[12] Cooper J., Strauss W., “Time periodic scattering of symmetric hyperbolic systems”, J. Math. Anal. Appl., 122:2 (1987), 444–452 | DOI | MR | Zbl

[13] Howland J., “Scattering theory for Hamiltonian periodic in time”, Indiana Univ. Math. J., 28 (1979), 471–494 | DOI | MR | Zbl

[14] Petkov V., Scattering theory for hyperbolic operators, North-Holland, Amsterdam, 1989 | MR | Zbl

[15] Petkov V., Georgiev V., “RAGE theorem for power bounded operators and decay of local energy for moving obstacles”, Ann. Inst. Henry Poincare, 51:2 (1989), 155–185 | MR | Zbl

[16] Schmidt G., “On scattering by time dependent perturbations”, Indiana, Univ. Math. J., 24 (1975), 925–935 | DOI | MR | Zbl

[17] Strauss W., “The existence of the scattering operator for moving obstacles”, J. Funct. Anal., 1979, no. 31, 255–262 | DOI | MR | Zbl

[18] Yajima K., “Scattering theory for Schrödinger equations with potentials periodic in time”, J. Math. Soc. Japan, 29:4 (1977), 729–743 | DOI | MR | Zbl