1107:
1131:
1188:
341:
168:
511:
423:
209:
996:
579:
197:
659:
822:
1229:
949:
804:
780:
73:
1172:
672:
761:
652:
1031:
676:
530:
431:
52:
44:
55:(1912). The terminology -- both the limiting absorption principle and the limiting amplitude principle -- was introduced by
1222:
827:
525:
48:
883:
1110:
832:
817:
645:
40:
847:
349:
1092:
852:
1268:
1258:
1046:
970:
1087:
1273:
1248:
903:
837:
1263:
1253:
1215:
939:
740:
812:
1165:
1036:
425:. A particular solution to the Helmholtz equation corresponding to outgoing waves is obtained as the limit
1067:
1011:
975:
336:{\displaystyle (\Delta -\partial _{t}^{2})u(x,t)=-F(x)e^{-ikt},\quad t\geq 0,\quad x\in \mathbb {R} ^{3},}
573:
1050:
1016:
954:
668:
36:
1158:
1041:
908:
56:
32:
1021:
28:
1199:
1142:
1026:
944:
913:
893:
878:
873:
868:
705:
606:
176:
888:
842:
790:
785:
756:
637:
610:
24:
715:
1077:
929:
730:
1242:
1082:
1006:
735:
720:
710:
200:
554:
1072:
725:
695:
625:
1138:
1001:
991:
898:
700:
35:. The choice is made by considering a particular time-dependent problem of the
1195:
934:
774:
770:
766:
67:
To find which solution to the
Helmholz equation with nonzero right-hand side
1130:
1187:
163:{\displaystyle \Delta v(x)+k^{2}v(x)=-F(x),\quad x\in \mathbb {R} ^{3},}
39:
due to the action of a periodic force. The principle was introduced by
594:
641:
1203:
1146:
548:
546:
506:{\displaystyle v(x)=\lim _{t\to +\infty }u(x,t)e^{ikt}}
199:, corresponds to the outgoing waves, one considers the
16:
Mathematical concept for solving the
Helmholtz equation
434:
352:
212:
179:
76:
1060:
984:
963:
922:
861:
803:
749:
684:
997:Spectral theory of ordinary differential equations
505:
417:
335:
191:
162:
451:
418:{\displaystyle u(x,0)=0,\,\partial _{t}u(x,0)=0}
31:used for choosing a particular solution to the
1223:
1166:
653:
8:
578:: CS1 maint: multiple names: authors list (
553:Tikhonov, A.N. and Samarskii, A.A. (1948).
1230:
1216:
1173:
1159:
688:
660:
646:
638:
630:. Vol. 4 (6 ed.). Moscow, Nauka.
491:
454:
433:
385:
380:
351:
324:
320:
319:
280:
231:
226:
211:
178:
151:
147:
146:
99:
75:
950:Group algebra of a locally compact group
542:
571:
7:
1184:
1182:
1127:
1125:
1202:. You can help Knowledge (XXG) by
1145:. You can help Knowledge (XXG) by
464:
382:
223:
216:
77:
14:
1186:
1129:
1106:
1105:
1032:Topological quantum field theory
311:
298:
138:
47:. It is closely related to the
531:Sommerfeld radiation condition
484:
472:
458:
444:
438:
406:
394:
368:
356:
273:
267:
255:
243:
237:
213:
132:
126:
114:
108:
89:
83:
53:Sommerfeld radiation condition
45:Alexander Andreevich Samarskii
1:
828:Uniform boundedness principle
526:Limiting absorption principle
49:limiting absorption principle
627:Course in Higher Mathematics
555:"On the radiation principle"
41:Andrey Nikolayevich Tikhonov
21:limiting amplitude principle
1290:
1181:
1124:
971:Invariant subspace problem
599:Doklady Akademii Nauk SSSR
1101:
691:
593:Sveshnikov, A.G. (1950).
940:Spectrum of a C*-algebra
1037:Noncommutative geometry
562:Zh. Eksper. Teoret. Fiz
346:with zero initial data
1198:–related article is a
1141:-related article is a
1093:Tomita–Takesaki theory
1068:Approximation property
1012:Calculus of variations
624:Smirnov, V.I. (1974).
507:
419:
337:
203:with the source term,
193:
192:{\displaystyle k>0}
164:
1088:Banach–Mazur distance
1051:Generalized functions
595:"Radiation principle"
508:
420:
338:
194:
165:
833:Kakutani fixed-point
818:Riesz representation
432:
350:
210:
177:
74:
19:In mathematics, the
1017:Functional calculus
976:Mahler's conjecture
955:Von Neumann algebra
669:Functional analysis
236:
37:forced oscillations
1042:Riemann hypothesis
741:Topological vector
503:
468:
415:
333:
222:
189:
160:
57:Aleksei Sveshnikov
33:Helmholtz equation
23:is a concept from
1269:Mathematics stubs
1259:Scattering theory
1211:
1210:
1154:
1153:
1119:
1118:
1022:Integral operator
799:
798:
601:. Novaya Seriya.
516:for large times.
450:
29:scattering theory
1281:
1274:Scattering stubs
1249:Linear operators
1232:
1225:
1218:
1190:
1183:
1175:
1168:
1161:
1133:
1126:
1109:
1108:
1027:Jones polynomial
945:Operator algebra
689:
662:
655:
648:
639:
632:
631:
621:
615:
614:
590:
584:
583:
577:
569:
559:
550:
512:
510:
509:
504:
502:
501:
467:
424:
422:
421:
416:
390:
389:
342:
340:
339:
334:
329:
328:
323:
294:
293:
235:
230:
198:
196:
195:
190:
173:with some fixed
169:
167:
166:
161:
156:
155:
150:
104:
103:
1289:
1288:
1284:
1283:
1282:
1280:
1279:
1278:
1264:Spectral theory
1254:Operator theory
1239:
1238:
1237:
1236:
1180:
1179:
1122:
1120:
1115:
1097:
1061:Advanced topics
1056:
980:
959:
918:
884:Hilbert–Schmidt
857:
848:Gelfand–Naimark
795:
745:
680:
666:
636:
635:
623:
622:
618:
592:
591:
587:
570:
557:
552:
551:
544:
539:
522:
487:
430:
429:
381:
348:
347:
318:
276:
208:
207:
175:
174:
145:
95:
72:
71:
65:
51:(1905) and the
25:operator theory
17:
12:
11:
5:
1287:
1285:
1277:
1276:
1271:
1266:
1261:
1256:
1251:
1241:
1240:
1235:
1234:
1227:
1220:
1212:
1209:
1208:
1191:
1178:
1177:
1170:
1163:
1155:
1152:
1151:
1134:
1117:
1116:
1114:
1113:
1102:
1099:
1098:
1096:
1095:
1090:
1085:
1080:
1078:Choquet theory
1075:
1070:
1064:
1062:
1058:
1057:
1055:
1054:
1044:
1039:
1034:
1029:
1024:
1019:
1014:
1009:
1004:
999:
994:
988:
986:
982:
981:
979:
978:
973:
967:
965:
961:
960:
958:
957:
952:
947:
942:
937:
932:
930:Banach algebra
926:
924:
920:
919:
917:
916:
911:
906:
901:
896:
891:
886:
881:
876:
871:
865:
863:
859:
858:
856:
855:
853:Banach–Alaoglu
850:
845:
840:
835:
830:
825:
820:
815:
809:
807:
801:
800:
797:
796:
794:
793:
788:
783:
781:Locally convex
778:
764:
759:
753:
751:
747:
746:
744:
743:
738:
733:
728:
723:
718:
713:
708:
703:
698:
692:
686:
682:
681:
667:
665:
664:
657:
650:
642:
634:
633:
616:
585:
541:
540:
538:
535:
534:
533:
528:
521:
518:
514:
513:
500:
497:
494:
490:
486:
483:
480:
477:
474:
471:
466:
463:
460:
457:
453:
449:
446:
443:
440:
437:
414:
411:
408:
405:
402:
399:
396:
393:
388:
384:
379:
376:
373:
370:
367:
364:
361:
358:
355:
344:
343:
332:
327:
322:
317:
314:
310:
307:
304:
301:
297:
292:
289:
286:
283:
279:
275:
272:
269:
266:
263:
260:
257:
254:
251:
248:
245:
242:
239:
234:
229:
225:
221:
218:
215:
188:
185:
182:
171:
170:
159:
154:
149:
144:
141:
137:
134:
131:
128:
125:
122:
119:
116:
113:
110:
107:
102:
98:
94:
91:
88:
85:
82:
79:
64:
61:
15:
13:
10:
9:
6:
4:
3:
2:
1286:
1275:
1272:
1270:
1267:
1265:
1262:
1260:
1257:
1255:
1252:
1250:
1247:
1246:
1244:
1233:
1228:
1226:
1221:
1219:
1214:
1213:
1207:
1205:
1201:
1197:
1192:
1189:
1185:
1176:
1171:
1169:
1164:
1162:
1157:
1156:
1150:
1148:
1144:
1140:
1135:
1132:
1128:
1123:
1112:
1104:
1103:
1100:
1094:
1091:
1089:
1086:
1084:
1083:Weak topology
1081:
1079:
1076:
1074:
1071:
1069:
1066:
1065:
1063:
1059:
1052:
1048:
1045:
1043:
1040:
1038:
1035:
1033:
1030:
1028:
1025:
1023:
1020:
1018:
1015:
1013:
1010:
1008:
1007:Index theorem
1005:
1003:
1000:
998:
995:
993:
990:
989:
987:
983:
977:
974:
972:
969:
968:
966:
964:Open problems
962:
956:
953:
951:
948:
946:
943:
941:
938:
936:
933:
931:
928:
927:
925:
921:
915:
912:
910:
907:
905:
902:
900:
897:
895:
892:
890:
887:
885:
882:
880:
877:
875:
872:
870:
867:
866:
864:
860:
854:
851:
849:
846:
844:
841:
839:
836:
834:
831:
829:
826:
824:
821:
819:
816:
814:
811:
810:
808:
806:
802:
792:
789:
787:
784:
782:
779:
776:
772:
768:
765:
763:
760:
758:
755:
754:
752:
748:
742:
739:
737:
734:
732:
729:
727:
724:
722:
719:
717:
714:
712:
709:
707:
704:
702:
699:
697:
694:
693:
690:
687:
683:
678:
674:
670:
663:
658:
656:
651:
649:
644:
643:
640:
629:
628:
620:
617:
612:
608:
604:
600:
596:
589:
586:
581:
575:
568:(2): 243–248.
567:
563:
556:
549:
547:
543:
536:
532:
529:
527:
524:
523:
519:
517:
498:
495:
492:
488:
481:
478:
475:
469:
461:
455:
447:
441:
435:
428:
427:
426:
412:
409:
403:
400:
397:
391:
386:
377:
374:
371:
365:
362:
359:
353:
330:
325:
315:
312:
308:
305:
302:
299:
295:
290:
287:
284:
281:
277:
270:
264:
261:
258:
252:
249:
246:
240:
232:
227:
219:
206:
205:
204:
202:
201:wave equation
186:
183:
180:
157:
152:
142:
139:
135:
129:
123:
120:
117:
111:
105:
100:
96:
92:
86:
80:
70:
69:
68:
62:
60:
58:
54:
50:
46:
42:
38:
34:
30:
26:
22:
1204:expanding it
1193:
1147:expanding it
1136:
1121:
1073:Balanced set
1047:Distribution
985:Applications
838:Krein–Milman
823:Closed graph
626:
619:
602:
598:
588:
574:cite journal
565:
561:
515:
345:
172:
66:
20:
18:
1139:mathematics
1002:Heat kernel
992:Hardy space
899:Trace class
813:Hahn–Banach
775:Topological
605:: 917–920.
63:Formulation
1243:Categories
1196:scattering
935:C*-algebra
750:Properties
611:0040.41903
537:References
909:Unbounded
904:Transpose
862:Operators
791:Separable
786:Reflexive
771:Algebraic
757:Barrelled
465:∞
459:→
383:∂
316:∈
303:≥
282:−
262:−
224:∂
220:−
217:Δ
143:∈
121:−
78:Δ
1111:Category
923:Algebras
805:Theorems
762:Complete
731:Schwartz
677:glossary
520:See also
914:Unitary
894:Nuclear
879:Compact
874:Bounded
869:Adjoint
843:Min–max
736:Sobolev
721:Nuclear
711:Hilbert
706:Fréchet
671: (
889:Normal
726:Orlicz
716:Hölder
696:Banach
685:Spaces
673:topics
609:
1194:This
1137:This
701:Besov
558:(PDF)
1200:stub
1143:stub
1049:(or
767:Dual
580:link
184:>
43:and
27:and
607:Zbl
452:lim
1245::
675:–
597:.
576:}}
572:{{
566:18
564:.
560:.
545:^
59:.
1231:e
1224:t
1217:v
1206:.
1174:e
1167:t
1160:v
1149:.
1053:)
777:)
773:/
769:(
679:)
661:e
654:t
647:v
613:.
603:5
582:)
499:t
496:k
493:i
489:e
485:)
482:t
479:,
476:x
473:(
470:u
462:+
456:t
448:=
445:)
442:x
439:(
436:v
413:0
410:=
407:)
404:0
401:,
398:x
395:(
392:u
387:t
378:,
375:0
372:=
369:)
366:0
363:,
360:x
357:(
354:u
331:,
326:3
321:R
313:x
309:,
306:0
300:t
296:,
291:t
288:k
285:i
278:e
274:)
271:x
268:(
265:F
259:=
256:)
253:t
250:,
247:x
244:(
241:u
238:)
233:2
228:t
214:(
187:0
181:k
158:,
153:3
148:R
140:x
136:,
133:)
130:x
127:(
124:F
118:=
115:)
112:x
109:(
106:v
101:2
97:k
93:+
90:)
87:x
84:(
81:v
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.