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Index

absorbing medium 214 acoustic obstacle scattering 46 acoustic waves 13 addition theorem 27, 30, 67

vector 185 adiabatic hypothesis 14 analytic 235 analytic Riesz-F'redholm theory 236 approximation of scattered field 134, 203 associated Legendre equation 24 associated Legendre functions 24 asymptotics of Hankel functions 30, 66

Bessel differential equation 64 spherical 28

Bessel functions 64 spherical 28

Bojarski identity 114 Born approximation 223, 273 boundary element method 78

combined double- and single-layer po-tential48

compact imbedding 40 complete set 55 completely continuous operator 86 completeness of far fields 56, 188, 191,

226,258 conductor 251 coupled finite element and boundary el­

ement 246

dielectric 251 Dirichlet eigenvalue 56 Dirichlet problem

exterior 46 interior 109

Dirichlet to Neumann map 49

discrepancy principle 89, 95, 99 domain of class C k 16 double-layer potential 39 dual space method 146, 206, 280, 290

modified 285, 296

eigenvalue Dirichlet 56 Maxwell 175 transmission 226

electric dipole 163 electric far field pattern 164 electric to magnetic boundary map 175 electromagnetic Herglotz pair 187 electromagnetic obstacle scattering 172 entire solutions 20, 163 equation of continuity 13 Euler's equation 13 exterior Dirichlet Problem 46 exterior impedance problem 285

electromagnetic 265 exterior Maxwell Problem 172 exterior Neumann problem 50

far field equation 315 far field operator

acoustic 57, 229 electric 190

far field pattern 20, 223 electric 164 magnetic 164

function spaces Ck(D) 16 Ck(tJ) 17 Co,Q(G) 40 C1,Q(G) 40 Ck,Q(G) 214 T2(fl) 177

Td(aD) 169 T~,a(aD) 169 T~,a(aD) 170

fundamental solution 16, 66 Funk-Heeke formula 32

graded mesh 72 Green's formula 17 Green's theorem 17

vector 155

Hankel function 65 spherical 28

Helmholtz equation 15 vector 160

Helmholtz representation 18 Herglotz pair 187 Herglotz wave function 55 Holder continuous 40 Holder space 20, 214 holomorphic

strongly 233 weakly 233

Huygen's principle 53, 176

ill-posed problem 86 mildly 92 severely 92 nonlinear 102

impedance boundary condition 15, 155 impedance problem

exterior 265, 285 interior 287, 296

improperly posed problem 86 integral operators

F 57, 135, 145, 190, 204, 207, 229 H 147, 209 K 41 K'42 M 167 M'171 M 204 N 170 S 41, 135 T 42 Te 256 Tm 218, 312 TO" 292

interior Dirichlet problem 109

Index 333

interior impedance problem 287 electromagnetic 296

interior transmission problem 225 electromagnetic 257 weak solution 292, 312

interior Maxwell problem 208 inverse crimes 133, 304 inverse medium problem 272

electromagnetic 290

Jacobi-Anger expansion 32, 67 jump relations 39

in L2 45,172 vector 165

Karp's theorem 112, 197 Kirchhoff approximation 53

Lax's theorem 43 Legendre differential equation 23

associated 23 Legendre polynomials 23 Lidski's theorem 229 limited-aperture problem 141 Lippmann-Schwinger equation 216 Lyapunov boundaries 52

magnetic dipole 163 magnetic far field pattern 164 mapping properties in Holder spaces 42 mapping properties in Sobolev spaces 44 Maxwell eigenvalue 175 Maxwell equations 154 Maxwell problem

exterior 172 interior 208

monotonicity of eigenvalues 108

near field 140 Neumann function 65

spherical 28 Neumann problem 50 Newton's method 131 Nystrom method 69

in lR3 79 on a graded mesh 76

penalty method 98 perfect conductor 155 physical optics approximation 53, 113

334 Index

Picard's theorem 92 potential

double-layer 39 single-layer 39 vector 165 volume 214

projection theorem 311 properly posed problem 86

quasi-solution 100 nonlinear 104

radiating solutions 19, 160 radiation condition

Silver-Miiller 160 Sommerfeld 16, 19, 66

reciprocity relation acoustic 54, 223, 285 electromagnetic 186, 257, 269

refractive index 214, 251 regularization methods 87 regularization parameter 87 regularization scheme 87 regularization strategy 89

regular 89 Rellich's lemma 32 resonance region 105 resonance states 4

scattering amplitude 20 scattering operator 233 Silver-Miiller radiation condition 160 single-layer potential 39 singular system 91 singular values 90

decomposition 91 Sommerfeld radiation condition 16, 19,

66 sound-hard obstacle 15 sound-soft obstacle 15 spectral cut-off 94 spectral decomposition 90 speed of sound 14 spherical Bessel differential equation 28

spherical Bessel functions 28 spherical Hankel functions 28 spherical harmonics 21 spherical Neumann functions 28 spherical vector wave functions 180 spherical wave functions 30, 67 spherically stratified medium 226, 259 starlike surface 115 state equation 14 Stratton-Chu formula 156, 158, 160 strongly holomorphic 233 superposition of incident fields 146, 206 surface divergence theorem 168 surface divergence 168 surface gradient 167 surface of class C k 16

Tikhonov functional, 98 Tikhonov regularization 97

nonlinear 103 trace class operator 229 transmission conditions 15 transmission eigenvalue 226 transmission problem 15, 155

interior 225, 312 interior electromagnetic 257

trial and error 89

unique continuation principle 219, 255

vector addition theorem 185 vector Green's theorem 155 vector Helmholtz equation 160 vector jump relations 165 vector potential 165 vector spherical harmonics 177 volume potential 214

wave equation 14 wave number 15, 154 weakly holomorphic 233 well-posed problem 86 Wronskian 29,65