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we consider two connected and compact @xmath1-riemannian manifolds @xmath2 and @xmath3 with @xmath0-boundaries @xmath4 and @xmath5 , respectively . we assume @xmath6 and @xmath7 . throughout of the present paper , we assume that    * ( a-1 ) * @xmath8 and @xmath9 have a common boundary @xmath10 . @xmath11 is a disjoint union of a finite number of connected and closed components . the metrics satisfy @xmath12 on @xmath11 . we will add some other assumptions for @xmath13 and @xmath14 in a neighborhood of the boundary later . note that we need our geometric assumptions only in some small neighborhoods of the boundary , in particular , we do not assume that @xmath8 and @xmath9 are diffeomorphic outside of a small neighborhood of the boundary . let @xmath15 , @xmath16 , be the ( negative ) laplace - beltrami operator on each @xmath17 . we consider the following interior transmission eigenvalue ( ite ) problem : @xmath18 where each @xmath19 , @xmath16 , is strictly positive on @xmath17 . we call @xmath20 the index of refraction on @xmath17 . if there exists a pair of non - trivial solutions @xmath21 of ( [ s1_eq_ite1])-([s1_eq_ite3 ] ) , we call corresponding @xmath22 an _ interior transmission eigenvalue_.      ite problems naturally appears in inverse scattering problems for acoustic wave equations on @xmath23 with compactly supported inhomogeneity . in @xmath23 for @xmath24 , time harmonic acoustic waves satisfy the equation @xmath25 where @xmath26 is strictly positive in a bounded domain @xmath27 with a suitable smooth boundary , and @xmath28 . given an incident wave @xmath29 with an incident direction @xmath30 and energy @xmath31 , the scattered wave @xmath32 is described by the difference between the total wave @xmath33 and the incident wave @xmath34 where @xmath33 is the solution of ( [ s1_eq_scatteringeq ] ) satisfying the following asymptotic relation : as @xmath35 @xmath36 here the second term on the right - hand side is the spherical wave scattered to the direction @xmath37 . the function @xmath38 is the scattering amplitude . the _ s - matrix _ is given by @xmath39 where @xmath40 is an integral operator with the kernel @xmath41 . then the s - matrix is unitary operator on @xmath42 . if there exists a non zero function @xmath43 such that @xmath44 i.e. @xmath45 , we call @xmath46 a _ non - scattering energy_. if @xmath47 is a non - scattering energy , we have that @xmath48 vanishes outside of @xmath27 from the rellich type uniqueness theorem ( see @xcite and @xcite ) . hence we can reduce to the ite problem @xmath49 with @xmath50 and @xmath51 . if @xmath52 is a non - scattering energy , @xmath53 is also an ite of the system ( [ s1_eq_ite1euq])-([s1_eq_iteeuq ] ) . ite problems were introduced in @xcite and @xcite in the above view point . for the schrdinger equation @xmath54 with a compactly supported potential @xmath55 which satisfies @xmath56 in @xmath57 , we can state the ite problem similarly . recently , @xcite introduced the ite problems on unbounded domains , considering perturbations which decrease exponentially at infinity . the system ( [ s1_eq_ite1euq])-([s1_eq_iteeuq ] ) is some kind of non - self - adjoint problem . moreover , we can construct a bilinear form associated with this system , but generally this bilinear form is not coercive . note that the @xmath58-coercivity approach is valid for some anisotropic cases i.e. @xmath59 is replaced by @xmath60 where @xmath61 is a strictly positive symmetric matrix valued function and @xmath62 . for the @xmath58-coercivity approach on this case , see @xcite . another common approach is to reduce an ite problem to an equivalent forth - order equation . for ( [ s1_eq_ite1euq])-([s1_eq_iteeuq ] ) , we can reduce to @xmath63 which is formulated as the variational form @xmath64 for any @xmath65 . there are also many works on this approach for acoustic wave equations and schrdinger equations . for more history , technical information and references on ite problems , we recommend the recent survey by cakoni and haddar @xcite .    in this paper , we slightly generalize the  isotropic type " ite problem as a system on two domains @xmath8 and @xmath9 . since we do not assume that @xmath66 and @xmath9 are diffeomorphic , it is difficult to use the forth - order equation approach ( see figure [ s1_fig_manifold ] ) .     and @xmath9 with common boundary @xmath67.,width=377 ]    moreover , in view of assumptions ( a-1 ) and ( a-2 ) which is added in  2.3 , the ite problem is not elliptic , and we can not construct a suitable isomorphism @xmath58 such that the system ( [ s1_eq_ite1])-([s1_eq_ite3 ] ) is @xmath58-coercive . therefore , neither the variational formulation approach nor the @xmath58-coercivity approach are valid for the proof of discreteness of ites in our case . then we adopt arguments by lakshtanov and vainberg @xcite and @xcite in the present paper . they considered isotropic cases in @xcite and anisotropic cases in @xcite in bounded domains in @xmath23 . the approach in @xcite is based on methods of elliptic pseudo differential operators on the boundary and its application to the dirichlet - to - neumann ( d - n ) map . we should also mention about @xcite and @xcite . recently , they proved the weyl s asymptotics including complex ites and evaluated ite - free regions in the complex plane under various conditions . they used the semi - classical analysis for the d - n map associated with an operator of the form @xmath68 where @xmath69 are smooth and positive valued function on a bounded domain @xmath70 .    even for manifolds , these d - n map approaches are valid . more precisely , in this paper , we construct the poisson operator and the associated d - n map as elliptic singular integro - differential operators and we can compute exactly their symbols . using the ellipticity of the d - n map and the analytic fredholm theory , we can prove the discreteness of the set of ites . moreover , we consider a weyl type lower bound of the number of positive ites except for a small neighborhood of the origin . a case which we can use the @xmath58-coercivity approach will be studied in the forthcoming paper @xcite . the plan of the paper is as follows . as has been mentioned above , the main stream of our argument is based on lakshtanov and vainberg @xcite . in  2 , we recall some basic properties of the dirichlet - to - neumann map . for our purpose , we need to study about residues and regular parts of the dirichlet - to - neumann map near its poles . the relation between ites and non - trivial kernels of the difference of dirichlet - to - neumann maps is also introduced here . finally , we construct an approximate solution of the dirichlet boundary value problems as a singular - integro differential operator , and we compute the symbol of the dirichlet - to - neumann map . under the preparation of  2 , we prove our main results in  3 . we use of the analytic fredholm theory , the parameter ellipticity of pseudo differential operators and weyl type asymptotic estimates for the number of dirichlet eigenvalues on compact manifolds . our main results are theorem [ s3_thm_discreteness ] for the discreteness of ites and theorem [ s3_thm_weylnt ] for the lower bound of the number of ites in @xmath71 with sufficiently small @xmath72 . we use the following notations . we put @xmath73 and @xmath74 . for the riemannian metric @xmath75 of @xmath17 , @xmath76 and @xmath77 denote @xmath78 and @xmath79 , respectively . @xmath80 and @xmath81 denote the volume element on @xmath17 and the surface element on @xmath11 induced by @xmath82 , respectively . we often write them as @xmath83 and @xmath84 omitting @xmath85 . letting @xmath86 be a local coordinate of @xmath87 , @xmath88 or @xmath89 denote @xmath90 . for @xmath91 , we use the similar manner . for a multiple index @xmath92 , we write @xmath93 . we often compute some kind of symbols @xmath94 . for short , we use the symbol @xmath95 which means that each @xmath96 replaced by @xmath97 . similarly , when we write @xmath98 , each @xmath99 is replaced by @xmath100 . @xmath101 denotes the outward normal derivative on @xmath11 associated with @xmath87 . for a strictly positive valued function @xmath102 , @xmath103 is the @xmath104 space on @xmath17 with the inner product @xmath105 . here we consider the following dirichlet problems : @xmath106 for @xmath16 . we define the dirichlet - to - neumann ( d - n ) map @xmath107 by @xmath108 where @xmath109 is a solution of ( [ s1_eq_d1 ] ) .    in the following , we call @xmath110 a dirichlet eigenvalue if there exists a non - trivial solution of the equation @xmath111 in fact , ( [ s2_eq_dirichleteigen ] ) is equivalent to @xmath112 which is an eigenvalue problem of the second - order self - adjoint elliptic operator @xmath113 in @xmath114 with the dirichlet boundary condition . then its eigenvalues form an increasing sequence @xmath115 , satisfying the weyl s asymptotics which we derive in  3 . the corresponding eigenfunctions @xmath116 can be chosen so that @xmath117 is an orthonormal basis in @xmath118 . we denote the set of dirichlet eigenvalues by @xmath119 . for @xmath120 , the d - n map @xmath121 is well - defined and extends uniquely as a continuous operator @xmath122 . let @xmath123 such that @xmath124 , and @xmath125 and @xmath126 belong to the same set @xmath127 if and only if @xmath128 . we denote eigenvalues corresponding @xmath127 by @xmath129 . @xmath130 means the set @xmath131 with @xmath132    @xmath133 is meromorphic with respect to @xmath22 and has first order poles at each @xmath134 . moreover , @xmath121 has the following representations : + ( 1 ) for @xmath135 and @xmath136 , we have @xmath137 ( 2 ) in a neighborhood of @xmath138 , we have @xmath139 where @xmath140 is the residue of @xmath141 at @xmath142 given by @xmath143 and @xmath144 is analytic in a neighborhood of @xmath145 .    [ s2_prop_green_meromor ]    proof . we can follow the argument of  4.1.12 in @xcite . let @xmath146 be an extension of @xmath147 into @xmath87 satisfying @xmath148 and @xmath149 for some constants @xmath150 , and @xmath151 is in ( [ s1_eq_d1 ] ) . then we have @xmath152 since @xmath153 is a meromorphic operator valued function with first order poles only at @xmath154 , @xmath155 is also a meromorphic @xmath156-valued function with first order poles only at @xmath154 . next we prove ( [ s2_eq_dnrepresentation ] ) . integrating by parts , we compute the fourier coefficients of @xmath151 with respect to the real - valued eigenfunction @xmath157 : @xmath158 from this formula and the outward normal derivative of @xmath151 , @xmath159 satisfies ( [ s2_eq_dnrepresentation ] ) . finally we verify ( [ s2_eq_resdn ] ) and ( [ s2_eq_pk ] ) . let @xmath160 be the projection to the eigenspace corresponding to @xmath161 i.e. @xmath162 in view of ( [ s2_eq_fouriercoefficients ] ) , we have @xmath163 and this implies ( [ s2_eq_pk ] ) . moreover , @xmath164 is analytic with respect to @xmath53 in a neighborhood of @xmath165 . putting @xmath166 on @xmath11 , we have the proposition . _ the formula ( [ s2_eq_pk ] ) means that the range of @xmath140 is a finite dimensional subspace spanned by @xmath167 for @xmath168 . note that @xmath169 for all @xmath170 are linear independent since @xmath171 are orthogonal basis . hence @xmath172 coincides with the multiplicity of @xmath165 . we can see that the integral kernel of @xmath140 is smooth in @xmath173 by the regularity property of dirichlet eigenfunctions . as has been in propositions [ s2_prop_green_meromor ] , @xmath174 is also meromorphic with respect to @xmath175 and has first order poles at each @xmath176 . in a neighborhood of a pole @xmath177 , we have @xmath178 where @xmath179 and @xmath180 have same properties of @xmath181 and @xmath182 , respectively . in the following , we define the kernel of @xmath183 by @xmath184    let @xmath134 . then the equation ( [ s1_eq_d1 ] ) has a non trivial solution if and only if @xmath185 is orthogonal to @xmath186 in @xmath187 for all @xmath188 . [ s2_lem_singular_sol ]    proof . if @xmath185 is orthogonal to @xmath186 for all @xmath189 , there exist general solutions of the form @xmath190 for any @xmath191 . if @xmath109 is a non trivial solution of ( [ s1_eq_d1 ] ) , we have by green s formula @xmath192 for @xmath193 . since @xmath194 , the left - hand side is equal to zero . then @xmath195 is orthogonal to @xmath196 . the above lemma implies a unique solvability in a subspace as follows . let @xmath197 be a dirichlet eigenvalue . we define @xmath198 as the eigenspace associated by @xmath177 , and @xmath199 as the subspace of @xmath200 spanned by @xmath201 for all @xmath202 . we denote by @xmath203 and @xmath204 their orthogonal complements in @xmath205 and @xmath187 , respectively . for any @xmath206 , there exists a unique solution @xmath207 of ( [ s1_eq_d1 ] ) represented by @xmath208 [ s2_cor_singular_sol2 ]    proof . we have only to check the uniqueness . this is trivial since the equation ( [ s2_eq_dirichleteigen ] ) has only the trivial solution in @xmath209 . now we can state the relation between ites and the d - n map as follows . \(1 ) suppose @xmath210 . then @xmath22 is an ite if and only if @xmath211 . the multiplicity of @xmath53 coincides with @xmath212 . + ( 2 ) suppose @xmath213 . then @xmath22 is an ite if and only if @xmath214 or the ranges of @xmath215 and @xmath216 have a non trivial intersection . the multiplicity of @xmath53 coincides with the sum of @xmath212 and the dimension of the above intersection . [ s2_lem_iteeqcondition ]    proof . we first prove the assertion ( 1 ) . when @xmath217 , this lemma is a direct consequence of the definition of ites . we have only to show for @xmath218 . for @xmath219 , we have @xmath220 . from @xmath221 and ( [ s2_eq_pk ] ) , we have @xmath222 . by lemma [ s2_lem_singular_sol ] and corollary [ s2_cor_singular_sol2 ] , the following equation has a unique non trivial solution : @xmath223 on the other hand , from @xmath224 , we have @xmath225 summarizing ( [ s2_eq_dd1 ] ) and ( [ s2_eq_dd2 ] ) and @xmath226 , @xmath227 is an ite . conversely , if @xmath53 is an ite , from lemma [ s2_lem_singular_sol ] , the equation ( [ s1_eq_d1 ] ) , @xmath228 , with the condition @xmath229 must have a non trivial solution . in view of ( [ s2_eq_pk ] ) , we have @xmath230 , and this implies @xmath231 . this means @xmath232 by using ( [ s2_eq_resdn ] ) . on the other hand , @xmath233 means @xmath234 . therefore , @xmath147 must be in @xmath235 . we have proven the assertion ( 1 ) .    for the assertion ( 2 ) , we have only to show the latter case . in fact , if there exists a non trivial solution @xmath236 of @xmath237 with @xmath238 and @xmath239 on @xmath240 , then we have that the ranges of @xmath215 and @xmath216 have a non trivial intersection , recalling @xmath241 for @xmath242 . conversely , if the ranges of @xmath215 and @xmath216 have a non trivial intersection , then there exists a non trivial solution @xmath236 of the above system with the condition @xmath238 and @xmath243 on @xmath240 , since @xmath196 for all @xmath244 are linear independent . then @xmath53 is an ite . _ in @xcite , the authors call @xmath227 _ singular spectral point _ if @xmath53 satisfies the latter condition in the assertion ( 2 ) of lemma [ s2_lem_iteeqcondition ] .      now let us compute the symbol of the d - n map . here we construct the local regularizer for ( [ s1_eq_d1 ] ) . as in @xcite , we follow the argument of  2 in @xcite , slightly modifying it for our case .    in the following , we assume that the equation ( [ s1_eq_d1 ] ) is uniquely solvable in @xmath245 or a suitable subspace of @xmath246 . we take a point @xmath247 and fix it . let @xmath248 be a sufficiently small neighborhood of @xmath249 in @xmath11 . there exist small open domains @xmath250 , @xmath251 , such that @xmath252 and @xmath253 and @xmath254 are diffeomorphic to an open domain @xmath255 . we introduce local coordinates @xmath256 in @xmath257 with the center @xmath258 such that @xmath259 , @xmath27 is given by @xmath260 , @xmath261 for a small @xmath262 , the subset @xmath263 is diffeomorphic to @xmath264 , and @xmath265 is the distance between a point @xmath266 and @xmath267 . then @xmath268 are common local coordinates of @xmath253 and @xmath269 . therefore , we have @xmath270 , \quad y ' = ( y_1 , \cdots , y_{d-1 } ) , \ ] ] in @xmath271 where @xmath272 is a smooth , positive definite and symmetric @xmath273-matrix valued function , and @xmath274 is a @xmath275-dimensional vector valued function . a function @xmath276 with @xmath277 is homogeneous of the generalized degree @xmath278 if @xmath279 satisfies @xmath280 for any @xmath281 . for @xmath282 , we define the homogeneity by the similar manner . taking the @xmath283-coordinate as above , we can rewrite @xmath284 as @xmath285 in @xmath286 with @xmath287 . note that @xmath288 , @xmath289 and @xmath290 are defined by @xmath291 . in view of the assumption ( a-1 ) , we have in @xmath283-coordinates that @xmath292 , @xmath293 . the symbol of @xmath294 is given by @xmath295    in the following , let @xmath296 be a sufficiently large integer . now we take @xmath297 arbitrarily and fix it . using the taylor series of @xmath288 , @xmath298 , @xmath299 and @xmath300 with respect to @xmath283 centered at @xmath301 , we can expand the symbol @xmath302 of @xmath294 as the sum of following terms : @xmath303 @xmath304 and @xmath305 for @xmath306 with the remainder term which has zero of order @xmath307 at @xmath308 or @xmath309 . we rewrite the sum of ( [ s2_eq_expand_symbol1])-([s2_eq_expand_symbol3 ] ) and the remainder term as @xmath310 then each @xmath311 is a homogeneous polynomial in @xmath312 of generalized degree @xmath313 . in particular , @xmath314 is the principal symbol of @xmath294 . @xmath315 vanishes at @xmath316 and the order of the zero is @xmath307 .    in the following arguments , we put @xmath317 we define the following differential operators : @xmath318 @xmath319 and @xmath320    let @xmath321 be a smooth function and homogeneous of the generalized degree @xmath322 with respect to @xmath265 and @xmath323 . then we have that @xmath324 is the homogeneous of the generalized degree @xmath325 with respect to @xmath265 and @xmath323 . [ s2_prop_degree ]    proof . note that @xmath326 . then we can show that @xmath327 and @xmath328 are homogeneous of generalized degree @xmath329 and @xmath330 , respectively . now let us construct an approximate solution of ( [ s1_eq_d1 ] ) . suppose @xmath331 . the system of second order ordinary differential equations @xmath332 has a unique solution @xmath333 such that each @xmath334 converges to zero as @xmath335 and satisfies @xmath336 in particular , we have @xmath337 . each solution @xmath338 is smooth and homogeneous with respect to @xmath265 and @xmath339 of generalized degree @xmath340 . ( for @xmath341 , each @xmath334 depends also on @xmath53 . we omit @xmath53 in the notation . ) [ s2_lem_systemode ]    proof . since @xmath342 , we have @xmath337 . obviously , @xmath343 is homogeneous of the generalized degree @xmath344 . let us consider the equation @xmath345 for @xmath346 and @xmath347 with @xmath348 , @xmath349 as @xmath335 . here we assume that @xmath350 decays exponentially as @xmath335 and is homogeneous of the generalized degree @xmath322 . extending @xmath351 and @xmath352 to be zero in @xmath353 , we have @xmath354 then , putting @xmath355 , we have @xmath356 which shows that @xmath351 is homogeneous of the generalized degree @xmath357 with respect to @xmath265 and @xmath323 . in view of proposition [ s2_prop_degree ] , we have @xmath358 is homogeneous of the generalized degree @xmath359 . therefore , we obtain @xmath360 is homogeneous of the generalized degree @xmath361 . repeating the similar argument inductively , we can show that @xmath334 is homogeneous of the generalized degree @xmath340 . let @xmath362 vanish in a neighborhood of @xmath363 , and be equal to one outside a large neighborhood of @xmath364 . taking @xmath365 with a compact support in @xmath366 , we define for @xmath367 @xmath368 and we put @xmath369 letting @xmath370 we have that @xmath371 is a distribution in @xmath372 , and @xmath373 with @xmath374 we represent @xmath294 in the form @xmath375 in the following , we consider @xmath376    let @xmath377 , @xmath378 and @xmath379 be sufficiently large . we have @xmath380 and @xmath381 where @xmath382 and @xmath383 .    [ s2_lem_regurality ] note that @xmath384 and @xmath385 are operators which are given by sums of terms like @xmath386 up to a smooth function with @xmath387 or @xmath388 and @xmath389 . in view of proposition [ s2_prop_degree ] , it is sufficient to show @xmath390 since the derivative @xmath391 is order zero , one or two . now we have @xmath392 since @xmath393 is homogeneous of the generalized degree @xmath394 , using proposition [ s2_prop_degree ] , we have @xmath395 which implies ( [ s2_eq_hlm ] ) . let @xmath396 be sufficiently large . we have that @xmath397 is local regularizer for ( [ s1_eq_d1 ] ) i.e. @xmath398 for @xmath399 which has a compact support in @xmath400 and @xmath401 .    [ s2_thm_localreg ]    proof . note that @xmath402 summing up both sides of ( [ s2_eq_sys0])-([s2_eq_sys2 ] ) , we have @xmath403 in view of lemma [ s2_lem_regurality ] and ( [ s2_eq_ar ] ) , we have that ( [ s2_eq_fourier1 ] ) and ( [ s2_eq_sys33 ] ) imply @xmath404 for @xmath405 . we have that @xmath406 as @xmath407 . since @xmath408 , we have @xmath409 .    _ remark . _ the formal sum @xmath410 is a singular integro - differential operator ( see @xcite ) . in general , a linear operator @xmath411 on a @xmath412-dimensional compact manifold @xmath413 is a singular integro - differential operator of order @xmath414 if there exist homogeneous functions @xmath415 in @xmath91 with homogeneous degree @xmath416 such that for a function @xmath33 with support in a local coordinate neighborhood @xmath417 , @xmath418 where @xmath419 is an arbitrary function which satisfies @xmath420 for @xmath421 and @xmath422 for @xmath423 , and @xmath424 is an operator which increases the smoothness i.e. @xmath425 for any @xmath426 . the principal symbol of @xmath411 is @xmath427 and the full symbol of @xmath411 is the formal sum @xmath428 . then the ellipticity of @xmath411 is defined by @xmath429 for all @xmath430 . here this means that we can construct the parametrix of @xmath411 ( see @xcite ) . therefore , if @xmath411 is an elliptic singular integro - differential operator , @xmath411 is fredholm . since we have @xmath431 in @xmath283-coordinates , we can show the following fact . as a consequence of corollary [ s2_cor_singular_sol2 ] and theorem [ s2_thm_localreg ] . see also lemma 11 and theorem 14 in @xcite . \(1 ) when @xmath53 is not a pole of @xmath141 , @xmath141 is a singular integro - differential operator on @xmath432 with the full symbol given by the following asymptotic series : @xmath433 ( 2 ) when @xmath434 is a pole of @xmath435 , the regular part @xmath182 of @xmath141 at @xmath177 is a singular integro - differential operator on @xmath436 with the full symbol given by ( [ s2_eq_dnsymbol ] ) . [ s2_cor_dnsymbol ]      we compute the principal symbol of @xmath437 . in the following , we denote higher order normal derivatives on @xmath11 associated with @xmath17 by @xmath438 in @xmath283-coordinates , we can locally represent @xmath439 . under the assumption ( a-1 ) , we additionally assume on @xmath240 that    * ( a-2 ) * the metrics @xmath13 , @xmath14 and the indices of refraction @xmath440 , @xmath441 satisfy one of following two cases :    * ( a-2 - 1 ) * for all @xmath135 , @xmath442 for @xmath443 , @xmath444 , and @xmath445 ,    or    * ( a-2 - 2 ) * for all @xmath446 , @xmath447 for @xmath448 , @xmath444 , and @xmath449 , @xmath450 . note that , under the assumptions ( a-1 ) with ( a-2 - 1 ) or ( a-2 - 2 ) , we can see @xmath451 for @xmath452 or @xmath453 , respectively . when @xmath434 is a pole of @xmath454 , we define a subspace @xmath455 of @xmath200 by @xmath456 where @xmath457 if @xmath177 is a dirichlet eigenvalue of @xmath458 , and @xmath459 if otherwise . we denote by @xmath460 the orthogonal complement of @xmath461 in @xmath187 . when @xmath434 is a pole of @xmath174 , we call @xmath174 fredholm if its regular part @xmath462 is fredholm .    in the following , we suppose @xmath463 . + ( 1 ) let @xmath53 be not a pole of @xmath174 . for the case ( a-2 - 1 ) , we have @xmath464 is an elliptic singular integro - differential operator with the principal symbol @xmath465 ( 2 ) let @xmath53 be not a pole of @xmath174 . for the case ( a-2 - 2 ) , we have @xmath466 is an elliptic singular integro - differential operator with the principal symbol @xmath467 ( 3 ) when @xmath53 is a pole of @xmath174 , the regular part of @xmath468 is singular integro - differential operator on @xmath460 with order @xmath361 for ( a-2 - 1 ) or @xmath469 for ( a-2 - 2 ) . its principal symbol is given by ( [ s2_eq_principalsymb1 ] ) or ( [ s2_eq_principalsymb2 ] ) , respectively . + ( 4 ) for both of ( a-2 - 1 ) or ( a-2 - 2 ) , @xmath437 is fredholm for @xmath470 . [ s2_lem_principalsymb ]    proof . let @xmath471 and @xmath441 satisfy ( a-2 - 1 ) . in @xmath283-coordinates , we have @xmath472 , @xmath473 and @xmath474 . then @xmath475 , @xmath476 and @xmath477 a particular solution of this equation is @xmath478 which vanishes at @xmath479 and @xmath335 . then we can take it as @xmath480 , and @xmath481 at @xmath479 is the principal symbol of @xmath437 . in view of the assertion ( 1 ) in corollary [ s2_cor_dnsymbol ] , we have the assertion ( 1 ) . next we assume that @xmath471 and @xmath441 satisfy ( a-2 - 2 ) . as above , we have @xmath482 for @xmath483 , and @xmath484 . then we have @xmath485 hence we obtain the assertion ( 2 ) .    in view of corollary [ s2_cor_singular_sol2 ] and the assertion ( 2 ) in corollary [ s2_cor_dnsymbol ] , we can show the assertion ( 3 ) by the similar way . the ellipticity of @xmath486 implies that @xmath487 is fredholm for @xmath488 . here we prove main results . let us list our assumptions again :                  for the proof of discreteness , we need to use the analytic fredholm theory which was generalized by @xcite . see also appendix a in @xcite . let @xmath490 and @xmath491 are hilbert spaces . we take a connected open domain @xmath492 . an operator valued function @xmath493 for @xmath494 is finitely meromorphic if the principal part of the laurent series at a pole of @xmath495 is a finite rank operator . in particular , @xmath496 is finitely meromorphic in @xmath497 as has been seen in proposition [ s2_prop_green_meromor ] . suppose an operator valued function @xmath493 , @xmath498 , is finitely meromorphic and fredholm . if there exists its bounded inverse @xmath499 at a point @xmath500 , then @xmath501 is finitely meromorphic and fredholm in @xmath502 .      from the above theorem , if there exists a @xmath488 such that @xmath503 is invertible , @xmath503 is invertible in @xmath504 for a discrete subset @xmath505 of @xmath506 . therefore , for the proof of the discreteness , we have only to show that @xmath503 is invertible for some @xmath507 . we expand the symbol of @xmath508 centered at @xmath509 by the same manner in  2.2 . however , here we change the definition of homogeneous functions with generalized degree @xmath322 by @xmath510 for @xmath488 , taking a suitable branch of @xmath511 . we gather terms of the same generalized degree in the sense ( [ s3_def_parameterhomogeneous ] ) , and we denote the symbol in @xmath283-coordinates by @xmath512 up to the remainder term where @xmath513 is homogeneous of degree @xmath313 . in particular , putting @xmath514 , we have @xmath515 where @xmath516 is defined by ( [ s2_def_atilde2 ] ) and @xmath517 we denote by @xmath518 the solution of @xmath519 with the boundary condition @xmath520 , @xmath521 for @xmath522 and @xmath523 as @xmath335 for @xmath524 . let @xmath525 be a pseudo differential operator ( or a singular integro - differential operator ) which is parametrized by @xmath175 such that the full symbol of @xmath526 is a asymptotic series of homogeneous functions in the sense ( [ s3_def_parameterhomogeneous ] ) . if the principal symbol does not vanish when @xmath527 , we call @xmath528 _ parameter elliptic_.    let @xmath529 . + ( 1 ) we assume that ( a-2 - 1 ) holds . then @xmath530 is parameter elliptic . its principal symbol is @xmath531 ( 2 ) we assume that ( a-2 - 2 ) holds . then @xmath530 is parameter elliptic . its principal symbol is @xmath532 where @xmath533 and @xmath534 . we fix an arbitrary point @xmath535 . suppose that ( a-2 - 1 ) holds . obviously we have @xmath536 under the assumption , we also have @xmath537 so that @xmath538 . then the principal symbol @xmath539 of @xmath540 is given by ( [ s3_eq_symbola31 ] ) . let us consider the case ( a-2 - 2 ) . in view of @xmath541 on @xmath240 , we have @xmath542 so that @xmath543 since we have assumed ( a-1 ) and ( a-2 - 2 ) , we have @xmath544 then @xmath545 satisfies the equation @xmath546 precisely , we obtain @xmath547 then the principal symbol @xmath548 of @xmath540 is given by ( [ s3_eq_symbola32 ] ) .    in view of lemma [ s3_lem_parametersymbol ] , we can obtain a uniform estimate in @xmath53 of @xmath549 and its inverse . in the following , we define the hilbert space @xmath550 for @xmath551 by the norm @xmath552 we take an arbitrary closed sector @xmath553 centered at the origin such that @xmath554 . we put @xmath555 . for the proof of the next result , see theorem 4.4.6 in @xcite and see also @xcite . we assume ( a-1 ) and one of ( a-2 - 1 ) and ( a-2 - 2 ) . the set of ites consists of a discrete subset of @xmath563 with the only possible accumulation points at @xmath344 and infinity . there exist at most finitely many ites in @xmath564 . [ s3_thm_discreteness ]            in view of ( a-1 ) , there exists a positive integer @xmath573 such that @xmath574 where each @xmath575 is a closed and connected component of @xmath11 . taking an arbitrary point @xmath576 , we take a small neighborhood @xmath577 of @xmath249 and a sufficiently small open domain @xmath27 which is diffeomorphic to @xmath578 such that @xmath579 as has been defend in the beginning of  2.2 . then , identifying @xmath580 with the corresponding point on @xmath267 , the following function @xmath581 for @xmath582 is well - defined constant function @xmath583 or @xmath584 in @xmath264 . obviously , the function @xmath585 or @xmath584 can be naturally extended on @xmath575 . we define the function @xmath586 on @xmath587 by @xmath588    in the following , we use an auxiliary operator defined by @xmath589 where @xmath590 for ( a-2 - 1 ) or @xmath562 for ( a-2 - 2 ) . here @xmath591 is given by @xmath592 where @xmath593 is the laplace - beltrami operator on @xmath11 . then @xmath594 is a first order singular integro - differential operator when @xmath53 is not a pole of @xmath595 . \(1 ) suppose @xmath210 . then @xmath22 is an ite if and only if @xmath596 . the multiplicity of @xmath53 coincides with @xmath597 . + ( 2 ) suppose @xmath213 . then @xmath22 is an ite if and only if @xmath598 or the range of @xmath599 and @xmath600 have a non trivial intersection . the multiplicity of @xmath53 coincides with the sum of @xmath597 and the dimension of the above intersection . [ s2_lem_iteeqconditionb ]      let @xmath602 be not a pole of @xmath603 . + ( 1 ) then @xmath603 is a first order , symmetric and elliptic singular integro - differential operator . its principal symbol is @xmath604 for ( a-2 - 1 ) , or @xmath605 for ( a-2 - 2 ) , where @xmath606 . + ( 2 ) for @xmath607 , the spectrum of @xmath603 is discrete and consists of the set of real eigenvalues @xmath608 . [ s3_lem_symbb ]      since @xmath603 has a positive principal symbol and @xmath603 is meromorphic with respect to @xmath53 , we also have the following lemma . the proof is completely same of lemma 2.4 and 2.5 in @xcite . note that , in view of ( [ s2_eq_resdn ] ) , we define the residue @xmath609 of @xmath610 at a pole @xmath177 by the expansion @xmath611 where @xmath612 is analytic in a small neighborhood of @xmath177 . \(1 ) for each compact interval @xmath613 such that any pole of @xmath603 are not included in @xmath614 , there exists a constant @xmath615 such that @xmath616 for @xmath617 , @xmath618 . + ( 2 ) if @xmath603 is analytic in a neighborhood of @xmath177 , all eigenvalues @xmath610 are analytic in this neighborhood . if @xmath177 is a pole of @xmath603 and @xmath619 is the rank of the residue of @xmath603 at @xmath177 , @xmath352 eigenvalues @xmath610 and its eigenfunctions have a pole at @xmath177 . moreover , @xmath620 are eigenvalues of @xmath621 . now we consider the relation between @xmath626 and @xmath608 for @xmath627 . roughly speaking , we can evaluate @xmath628 by the number of the singular spectral points and the number of @xmath227 satisfying @xmath629 . we put @xmath630 assume that @xmath631 moves from @xmath632 to @xmath633 . since @xmath634 is meromorphic with respect to @xmath631 , @xmath635 changes only when some @xmath636 pass through @xmath344 or @xmath631 passes through a pole of @xmath637 . when @xmath631 moves from @xmath632 to @xmath638 , we denote by @xmath639 the change of @xmath640 due to the first case , and @xmath641 as the change due to the second case i.e. @xmath642 for a pole @xmath643 of @xmath603 , we put @xmath644 for any @xmath645 .        proof . in view of lemma [ s3_lem_positiveb ] , some eigenvalues @xmath610 have a pole at @xmath177 . if @xmath649 , we have @xmath650 as @xmath651 and @xmath652 as @xmath653 , respectively . then the number of negative eigenvalues decreases for @xmath654 and increases for @xmath655 when @xmath53 passes through @xmath177 from @xmath632 . this implies the lemma . we denote by @xmath662 a union of connected components @xmath575 such that @xmath663 , respectively . let @xmath664 . we put @xmath665 and @xmath666 , and we split @xmath667 and @xmath668 with @xmath669 and @xmath670 for a pole @xmath671 of @xmath603 .        proof . first we prove the assertion ( 1 ) . suppose @xmath677 . we can expand @xmath603 in a small neighborhood of @xmath177 as @xmath678 where @xmath679 is analytic . from lemma [ s3_lem_residuepositive ] , we have @xmath680 and also @xmath681 so that @xmath682 has exactly @xmath683 strictly negative eigenvalues . then , in view of the assertion ( 2 ) in lemma [ s3_lem_positiveb ] and lemma [ s3_lem_jump ] , we have @xmath684 , so @xmath685 with @xmath686 . for the case @xmath687 , we can see @xmath688 with @xmath689 by the similar way . plugging these two cases , we obtain the assertion ( 1 ) . let us prove the assertion ( 2 ) . suppose @xmath690 for @xmath691 and @xmath692 . then we have the following representation in a small neighborhood of @xmath177 @xmath693 with @xmath694 . we see that @xmath695 on @xmath696 and @xmath697 on @xmath698 . then we have @xmath699 and @xmath700 . these inequalities and lemma [ s3_lem_jump ] imply the assertion ( 2 ) . we assume ( a-1 ) and one of ( a-2 - 1 ) and ( a-2 - 2 ) . for large @xmath701 , we have @xmath702 where the summation is taken over poles @xmath703 $ ] of @xmath530 . moreover , if @xmath704 or @xmath705 and @xmath706 or @xmath707 respectively where @xmath708 are defined in ( [ s3_eq_weyldirichlet ] ) , @xmath628 satisfies asymptotically as @xmath709 @xmath710      proof . we prove for the case @xmath711 . for @xmath712 , the proof is similar and can be slightly simplified . letting us recall that we call @xmath227 is a singular spectral point when @xmath227 satisfies the latter condition of the assertion ( 2 ) of lemma [ s2_lem_iteeqcondition ] , we put @xmath713 \subset { \bf r } _ { > 0 } \ } .\ ] ] here @xmath714 counts the number of singular spectral points with multiplicities taken into account . note that @xmath715 by the definition of @xmath716 and lemma [ s2_lem_iteeqconditionb ] . we take the summation of @xmath717 over all poles @xmath718 of @xmath719 in @xmath720 $ ] . then we have @xmath721 see also remark of proposition [ s2_prop_green_meromor ] . plugging this inequality and ( [ s3_eq_diffnb ] ) , we have @xmath722 since @xmath723 , we obtain ( [ s3_eq_lower1 ] ) . if @xmath724 , we see @xmath725 and @xmath726 for any poles @xmath718 , respectively , so that we have from ( [ s3_eq_lower1 ] ) @xmath727 the asymptotic estimate ( [ s3_eq_lower2 ] ) is a direct consequence of ( [ s3_eq_lower1 ] ) and theorem [ s3_thm_weyl ] . p. m. blekher , _ operators that depend meromorphically on a parameter _ , vestnik moskov . mech . , * 24 * ( 1969 ) , 30 - 36 . ( in russian ) ; english transl . : moscow univ . math . bull . , * 24 * ( 1969 ) , 21 - 26 .
in this paper , we consider an interior transmission eigenvalue ( ite ) problem on some compact @xmath0-riemannian manifolds with smooth boundary . 
 in particular , we do not assume that two domains are diffeomorphic , but we impose some conditions of riemannian metrics and indices of refraction on the boundary . 
 then we prove the discreteness of the set of ites , the existence of infinitely many ites , and its weyl type lower bound . 
 our problem is non - coercive . 
 however , the argument by lakshtanov and vainberg @xcite which reduces the problem to the elliptic one using the dirichlet - to - neumann map , is valid for manifolds .