A new proof of the analyticity of the electronic density of molecules.
|
|
- Camilla Young
- 5 years ago
- Views:
Transcription
1 A new proof of the analyticity of the electronic density of molecules. Thierry Jecko AGM, UMR 8088 du CNRS, Université de Cergy-Pontoise, Département de mathématiques, site de Saint Martin, 2 avenue Adolphe Chauvin, F Cergy-Pontoise cédex, France. thierry.jecko@u-cergy.fr web: Abstract We give a new, short proof of the regularity away from the nuclei of the electronic density of a molecule obtained in [FHHS1, FHHS2]. The new argument is based on the regularity properties of the Coulomb interactions underlined in [KMSW] and on well-known elliptic technics. Keywords: Elliptic regularity, analytic elliptic regularity, molecular Hamiltonian, electronic density, Coulomb potential. 1
2 Analyticity of the electronic density, Introduction. For the quantum description of molecules, it is very useful to study the so-called electronic density and, in particular, its regularity properties. This has be done for molecules with fixed nuclei: see [FHHS1, FHHS2, FHHS3] for details and references. The smoothness and the analyticity of the density away from the nuclei are proved in [FHHS1] and [FHHS2] respectively. In this paper, we propose an alternative proof. Let us recall the framework and the precise results of [FHHS1, FHHS2]. We consider a molecule with N moving electrons N 2) and L fixed nuclei. While the distinct vectors R 1,, R L R 3 denote the positions of the nuclei, the positions of the electrons are given by x 1,, x N R 3. The charges of the nuclei are given by the positive Z 1,, Z L and the electronic charge is 1. In this picture, the Hamiltonian of the system is H := where E 0 = N xj j=1 1 k<k L L k=1 Z k Z k R k R k ) Z k x j R k + 1 j<j N 1 x j x j + E 0 1.1) and xj stands for the Laplacian in the variable x j. Setting := N j=1 xj, we define the potential V of the system as the multiplication operator satifying H = + V. Thanks to Hardy s inequality c > 0 ; f W 1,2 R 3 ), t 1 ft) ) 2 dt c ft) 2 dt, 1.2) R 3 R 3 one can show that V is -bounded with relative bound 0 and that H is self-adjoint on the domain of the Laplacian, namely W 2,2 R 3N ) see Kato s theorem in [RS], p ). Let ψ W 2,2 R 3N ) \ {0} and E R such that Hψ = Eψ. Actually E is smaller than E 0 by [FH]. The electronic density associated to ψ is the following L 1 R 3 )-function ρx) := N j=1 R 3N 1) ψx 1,, x j 1, x, x j,, x N ) 2 dx 1 dx j 1 dx j dx N. Here we used N 2. The regularity result is the following Theorem 1.1. [FHHS1, FHHS2]. The density ρ is real analytic on R 3 \ {R 1,, R L }. Remark 1.2. In [FHHS1], it is proved that ρ is smooth on R 3 \{R 1,, R L }. This result is then used in [FHHS2] to derive the analyticity. Now let us sketch the new proof of Theorem 1.1, the complete proof and the notation used are given in Section 2. We consider the almost everywhere defined L 2 -function ψ : R 3 x ψx,,, ) W 2,2 R 3N 1) ) 1.3) and denote by the L 2 R 3N 1) )-norm. By permutation of the variables, it suffices to show that the map R 3 x ψx) 2 belongs to C ω R 3 \ {R 1,, R L }; R), the space of
3 Analyticity of the electronic density, real analytic functions on R 3 \ {R 1,, R L }. We define the potentials V 0, V 1 by V = V 0 + V 1 with V 0 x) = E 0 L k=1 Z k x R k C ω R 3 \ {R 1,, R L }; R). 1.4) We view the function ψ as a distributional solution in D R 3 ; W 2,2 R 3N 1) )) of x ψ + Qx) ψ = 0, 1.5) where the x-dependent operator Qx) B := LW 2,2 R 3N 1) ); L 2 R 3N 1) )) is given by Qx) = x + V 0 E + V 1 with x = N xj. 1.6) j=2 Considering 1.5) in a small enough neighbourhood Ω of some R 3 \ {R 1,, R L }, we pick from [KMSW] a x-dependent unitary operator U x0 x) on L 2 R 3N 1) ) such that W : Ω x U x0 x)v 1 x)u x0 x) 1 B 1.7) belongs to C ω Ω; B). It turns out that P 0 = U x0 x x )Ux 1 0 is an elliptic differential operator in x with analytic, differential coefficients in B. Applying U x0 to 1.5) and setting ϕ = U x0 ψ, we obtain P0 + W x) + V 0 x) E ) ϕ = ) Since U x0 x) is unitary on L 2 R 3N 1) ), ψx) = ϕx). Thus, it suffices to show that ϕ C ω Ω; L 2 R 3N 1) )). Using 1.8), a parametrix of the operator P 0 + W + V 0, we show by induction that, for all k, ϕ W k,2 Ω; W 2,2 R 3N 1) )). Thus ϕ C Ω; W 2,2 R 3N 1) )). Finally we can adapt the arguments in [H1] p to get ϕ C ω Ω; W 2,2 R 3N 1) )), yielding ϕ C ω Ω; L 2 R 3N 1) )). The main idea in the construction of the unitary operator U x0 is to change, locally in x, the variables x 2,, x N in a x-dependent way such that the x-dependent singularities 1/ x x j becomes locally x-independent see Section 2). In [KMSW], where this clever method was introduced, the nuclei positions play the role of the x variable and the x 2,, x N are the electronic degrees of freedom. The validity of the Born-Oppenheimer approximation is proved there for the computation of the eigenvalues and eigenvectors of the molecule. We point out that this method is the core of a recently introduced, semiclassical pseudodifferential calculus adapted to the treatment of Coulomb singularities in molecular systems, namely the twisted h-pseudodifferential calculus h being the semiclassical parameter). This calculus is due to A. Martinez and V. Sordoni in [MS]. As one can see in [KMSW, MS], the above method works in a larger framework. So do Theorem 1.1 and our proof. For instance, we do not need the positivity of the charges Z k, the fact that E < E 0, and the precise form of the Coulomb interaction. We do not use the self-adjointness or the symmetry) of the operator H. We could replace in 1.1) each xj by i xj + Ax) 2, where A is a suitable, analytic, magnetic vector potential. We could also add a suitable, analytic exterior potential.
4 Analyticity of the electronic density, Let us now compare our proof with the one in [FHHS1, FHHS2]. Here we only use almost) classical arguments of elliptic regularity. In [FHHS1, FHHS2], the elliptic regularity is essentially replaced by some Hölder continuity regularity result on ψ. The authors introduced an adapted, smartly chosen variable w.r.t. which they can derivate ψ. Here the x-dependent change of variables produces regularity with respect to x. As external tools, we only exploit basic notions of pseudodifferential calculus, the rest being elementary. In [FHHS1, FHHS2], a general, involved regularity result from the literature on PDE is an important ingredient of the arguments. We believe that, in spirit, the two proofs are similar. The shortness and the relative simplicity of the new proof is due to the clever method borrowed from [KMSW], which transforms the singular potential V 1 in an analytic function with values in B. Acknowledgment: The author is supported by the french ANR grant NONAa and by the european GDR DYNQUA. He thanks Vladimir Georgescu, Sylvain Golénia, Hans- Henrik Rugh, and Mathieu Lewin, for stimulating discussions. 2 Details of the proof. Here we complete the proof of Theorem 1.1, sketched in Section 1. Notation and basic facts. For a function f : R d R n x,y) fx,y) R p, let d x f be the total derivative of f w.r.t. x, by x α f with α N d the corresponding partial derivatives. For α N d and x R d, Dx α := i x ) α := i x1 ) α1 i xd ) α d, D x = i x, x α := x α 1 1 x α d d, α := α 1+ +α d, α! := α 1!) α d!), x 2 = x x 2 d, and x := 1 + x 2 ) 1/2. If A is a Banach space and O an open subset of R d, we denote by Cc O; A) resp. Cb O; A), resp. C ω O; A)) the space of functions from O to A which are smooth with compact support resp. smooth with bounded derivatives, resp. analytic). Let D O; A) denotes the topological dual of Cc O; A). We use the traditional notation W k,2 O; A) for the Sobolev spaces of L 2 O; A)-functions with k derivatives in L 2 O; A) when k N and for the dual of W k,2 O; A) when k N. If A is another Banach space, we denote by LA; A ) the space of the continuous linear maps from A to A and set LA) = LA; A). For A LA) with finite dimensional A, A T denotes the transpose of A and DetA its determinant. By the Sobolev injections, W k,2 O; A) C O; A). 2.1) k N Denoting by A the norm of A, it is well-known cf. [H3]) that a function u C O; A) is analytic if and only if, for any compact K O, there exists some C > 0 such that α N d, Dx α u)x) A C α +1 α!). 2.2) sup x K For convenience, we set W k = W k,2 R 3N 1) ), for k N. Recall that B = LW 2 ; W 0 ). Let B = LW 0 ; W 2 ), B 0 = LW 0 ), and B 2 = LW 2 ). Construction of U x0 see [KMSW, MS]). Let τ C c R 3 ; R) such that τ ) = 1 and τ = 0 near R k, for all k {1; ; L}. For x, s R 3, we set fx,s) = s + τs)x ).
5 Analyticity of the electronic density, Notice that x; s) R 3 ) 2, fx, ) = x and fx,s) = s if s supp τ. 2.3) Since d s f)x,s).s = s + τs), s x ), we can choose a small enough, relatively compact neighborhood Ω of such that x Ω, sup d s f)x,s) I 3 LR 3 ) 1/2, 2.4) s I 3 being the identity matrix of LR 3 ). Thus, for x Ω, fx, ) is a C -diffeomorphism on R 3 and we denote by gx, ) its inverse. By 2.4) and a Neumann expansion in LR 3 ), ds f)x,s) ) 1 = I3 + τs), x x0 ) ) n 1 ) τs), x ), n=1 for x, s) Ω R 3. Notice that the power series converges uniformly w.r.t. s. This is still true for the series of the derivatives β s, for β N 3. Since d s g)x, fx,s)) = d s f)x,s) ) 1 and dx g)x, fx,s)) = d s g)x, fx,s)) d x f)x,s), we see by induction that, for α, β N 3, α x β s g ) x, fx,s)) = γ N 3 x ) γ a αβγ s) 2.5) on Ω R 3, with coefficients a αβγ C R 3 ; LR 3 )). For α = β = 0, this follows from gx, fx,s)) = s. Notice that, except for α, β, γ) = 0,0,0) and for β = 1 with α, γ) = 0,0), the coefficients a αβγ are supported in the compact support of τ. For x R 3 and y = y 2,, y N ) R 3N 1), let F x,y) = fx,y 2 ),, fx,y N )). For x Ω, F x, ) is a C -diffeomorphism on R 3N 1) satisfying the following properties: There exists C 0 > 0 such that, for all α N 3, for all x Ω, for all y, y R 3N 1), C0 1 y y F x,y) F x,y ) C 0 y y, 2.6) x α F x,y) x α F x,y ) C 0 y y, 2.7) and, for α 1, α x F x,y) C ) For x Ω, denote by Gx, ) the inverse diffeomorphism of F x, ). By 2.5), the functions Ω R 3N 1) x,y) α x β y G ) x, F x,y)), for α, β) N 3 N 3N 1), are also given by a power series in x with smooth coefficients in y. Given x Ω, let U x0 x) be the unitary operator on L 2 R 3N 1) ) defined by U x0 x)θ)y) = Detd y F )x,y) 1/2 θf x,y)).
6 Analyticity of the electronic density, Computation of the terms in 1.8) cf. [KMSW, MS]). Consider the functions Ω x J 1 x, ) Cc R 3N 1) ; LR 3N 1) ; R 3 ) ), Ω x J 2 x, ) Cc R 3N 1) ; R 3 ), Ω x J 3 x, ) Cb R 3N 1) ; LR 3N 1) ) ), Ω x J 4 x, ) C c R 3N 1) ; R 3N 1) ), defined by J 1 x,y) = d x Gx,y )) T x, y = F x,y) ), J 2 x,y) = Det dy F x,y) 1/2 Det Dx dy Gx,y ) 1/2 ) y, =F x,y) J 3 x,y) = d y Gx,y )) T x, y = F x,y) ), J 4 x,y) = Det Det d y F x,y) 1/2 D y dy Gx,y ) 1/2) y. =F x,y) Actually, the support of J k x, ), for k 3, is contained in the x-independent, compact support of the function τ cf. 2.3)). So do also the supports of the derivatives α x β y J 3 of J 3, for α + β > 0. Thanks to 2.5), the J k,y) s can also be written as a power series in x with smooth coefficients depending on y. Now U x0 x U 1 = x + J 1 y + J 2 and U x0 x U 1 = J 3 y + J ) In particular, U x0 x) preserves W 2,2 R 3N 1) ), for all x Ω. Furthermore, ) P 0 = U x0 x x U 1 = x + J 1 x; y; D y ) D x + J 2 x; y; D y ), 2.10) where J 2 x; y; D y ) is a scalar differential operator of order 2 and J 1 x; y; D y ) is a column vector of 3 scalar differential operators of order 1. More precisely, the coefficients of J 1 x; y; D y ) and of J 2 x; y; D y ) + y are compactly supported, uniformly w.r.t. x. In particular, these scalar differential operators belong to B. By 2.5), they are given on Ω by a power series of x with coefficients in B and therefore are analytic functions on Ω with values in B cf. [H3]). Next, we look at W defined in 1.7). By 2.3), for j j in {2; ; N}, for k {1; ; L}, and for x Ω, U x0 x) x x j 1) U 1 x) = fx; ) fx; y j ) 1, 2.11) U x0 x) x j R k 1) U 1 x) = fx; y j ) fx; R k ) 1, 2.12) U x0 x) x j x j 1) U 1 x) = fx; y j ) fx; y j ) ) Lemma 2.1. The potential W in 1.7) is an analytic function from Ω to B. Proof: We prove the stronger result: W is analytic from Ω to B := LW 1 ; W 0 ). Notice that W is a sum of terms of the form 2.11), 2.12), and 2.13). We show the regularity of 2.11). Similar arguments apply for the other terms. We first recall the arguments in [KMSW], which proves the C regularity. Using the fact that d x fx, ) fx,y j )) does not depend on x, Dx α fx,x0 ) fx,y j ) 1) = τ ) τy j )) D α α 1 ) fx, ) fx,y j ))
7 Analyticity of the electronic density, for y j. By 2.6) and 2.7), we see that, for all α N 3 and for y j, Dx α fx,x0 ) fx,y j ) 1) 2 α C 0 fx, ) fx,y j ) α D α 1 fx,) fx,y j )) thanks to α N 3, C > 0, y R 3 \ {0}, C 2 α 0 Cα!) fx, ) fx,y j ) 1, D α 1 y) Cα!). 2.14) y α +1 Since x 1 is x -bounded by 1.2) and since U )x) is unitary, fx, ) fx,y j ) 1 is U )x) x U )x)) 1 -bounded with the same bounds. But, by 2.9), U )x) x U )x) 1 y + 1 ) 1/2 is uniformly bounded w.r.t. x. Thus Dx α fx,x0 ) fx,y j ) 1) C B 1C 2 α 0 Cα!), 2.15) uniformly w.r.t. α N 3 and x Ω. Therefore W is a distribution on Ω the derivatives of which belong to L Ω), thus to L 2 Ω). By 2.1), W is smooth. To show the analyticity of W, we just add the following improvement of 2.14), that we prove in appendix below. There exists K > 0 such that α N 3, y R 3 \ {0}, 1 Dα y) K α +1 α!). 2.16) y α +1 Now the l.h.s. of 2.15) is, for α N 3 and x Ω, bounded above by C 1 C 2 α 0 K α +1 α!) K α +1 1 α!), for some K 1 > 0. This yields the result by 2.2). Smoothness. Now we view 1.8) as an elliptic differential equation w.r.t. x with coefficients in B and want to follow usual arguments of elliptic regularity to prove the smoothness of ϕ. In fact, we shall use the basic pseudodifferential calculus in [H2] p ). Since the symbol of P 0 + W + V 0 takes its values in B, which is not an algebra of operators, we verify the validity of the basic calculus in our situation. By 2.10), P 0 + W + V 0 is a differential operator in the x variable the symbol of which px; ξ) = ξ 2 + J 1 x; y; D y ) ξ + J 2 x; y; D y ) + W x) + V 0 x) 2.17) and belongs to the Hörmander class Sm 2, g; B) on Ω R 3 with values in B, where mx; ξ) = ξ ) 1/2 and g = dx 2 + dξ 2 / ξ 2. We can check that the basic calculus of [H2] actually works with the symbols classes Sm, g; B), Sm, g; B ), Sm, g; B 0 ), and Sm, g; B 2 ), for any scalar) order function m on Ω R 3. In particular, we have the following properties: We can give a sense to asymptotic sums of symbols in Sm, g; A), for A {B, B, B 0, B 2 }. The composition of
8 Analyticity of the electronic density, operators a 1 x, D x )a 2 x, D x ) = bx, D x ) with b Sm, g; B 0 ) if a 1, a 2 Sm, g; B 0 ) and if a 1 Sm, g; B) and a 2 Sm, g; B ), also with b Sm, g; B 2 ) if a 1, a 2 Sm, g; B 2 ) and if a 1 Sm, g; B ) and a 2 Sm, g; B). The adjoint of an operator bx, D x ) is b x, D x ) with b Sm, g; B j ) if b Sm, g; B j ), b Sm, g; B ) if b Sm, g; B), and b Sm, g; B) if b Sm, g; B ). For A {B, B, B 0, B 2 }, k Z, and a Sm k, g; A), l Z, ax, D x ) L W l,2 Ω; A); W l k,2 Ω; A) ). 2.18) By the proof of Lemma 2.1, W x) y + 1) 1/2 is uniformly bounded on Ω. By the properties of J 1 and J 2, so are ξ J 1 x; y; D y ) ξ 2 y + 1) 1/2 and J 2 x; y; D y ) + y ) y + 1) 1/2 on Ω R 3. Thus, we can find a 0; 1) and b R such that, for all x, ξ) Ω R 3, for all u W 2, T u W0 a ξ 2 y + 1)u W0 + b u W0 with T = ξ J 1 x; y; D y ) + J 2 x; y; D y ) + y ) + W x) + V 0 x). By Theorem 4.11, p. 291 in [K], there exists C > 0 such that, for all x, ξ) Ω R 3, px, ξ) is bounded below by C + 1. In particular, C + p) 1 is a well-defined symbol in Sm 2, g; B ) on Ω R 3 this is the ellipticity we use). By composition, C + p) 1 x, D x ) p + C)x, D x ) = 1 qx, D x ) with q Sm 1, g; B 2 ). Defining the symbols q k in Sm k, g; B ) by q k x, D x ) = qx, D x ) k C + p) 1 x, D x ) and denoting by q an asymptotic sum of the symbols q k, we have QP 0 + W + V 0 + C) = 1 + R with 2.19) Q := q x, D x ) L W k,2 Ω; B ); W k+2,2 Ω; B ) ), 2.20) R L W k,2 Ω; B 2 ); W k+l,2 Ω; B 2 ) ), 2.21) for all k, l N cf. [H2]). Starting with ϕ W 0,2 Ω; W 2 ) and applying Q to 1.8), we see that ϕ = E +C)Qϕ Rϕ and actually belongs to W 2,2 Ω; W 2 ), by 2.19), 2.20), and 2.21). By induction and by 2.1), we get that ϕ C Ω; W 2 ). We have recovered the result in [FHHS1]. Note that, to get it, we need neither the refined bounds 2.16) nor the power series mentioned above but just use the fact that the functions f, g, F, G are smooth w.r.t. x. Analyticity. To show that ϕ C ω Ω; W 2 ), we adapt the proof of Theorem in [H1] for equation 1.8). So we view the latter as P ϕ = 0 where P = α 2 a α D α x with analytic coefficients a α B cf. Lemma 2.1, 1.4), and 2.10)). Applying D α x to 2.19) with α 2 and using 2.20) and 2.21), we find C > 0 such that, for all v C c Ω; W 2 ) and α N 3, α 2 = D α x v L 2 Ω;W 2 ) C P v L 2 Ω;W 0 ) + C v L 2 Ω;W 2 ). 2.22) For ɛ > 0, let Ω ɛ := {x Ω; dx; R 3 \ Ω) > ɛ} and, for r N, denote the L 2 Ω ɛ ; W r )-norm of v by Nɛ r v). As in [H1] Lemma 7.5.1), we use an appropriate cut-off function, Leibniz formula, and 2.22), to find C > 0 such that, for all v Cc Ω; W 2 ), for all ɛ, ɛ 1 > 0, for all α N 3 such that α 2, ɛ α N 2 ɛ+ɛ 1 D α x v ) Cɛ 2 N 0 ɛ 1 P v ) + C α <2 ɛ α N 2 ɛ 1 D α x v ). 2.23)
9 Analyticity of the electronic density, We used the fact that 2.23) holds true for ɛ large enough since the l.h.s. is zero. Next we show that there exists B > 0 such that, for all ɛ > 0, j N, and α N 3, α < 2 + j = ɛ α N 2 jɛ D α x ϕ ) B α ) This is done by induction on j following the arguments in [H1]. As explained in [H1], ϕ C ω Ω; W 2 ) follows from 2.24) and 2.2). A Appendix Using Cauchy integral formula for analytic functions in several variables cf. [H3]), we prove here the following extension of 2.16). For d N, there exists K > 0 such that α N d, y R d \ {0}, 1 Dα y) K α +1 α!). A.1) y α +1 In dimension d = 1, one can show A.1) with K = 1 by induction. To treat the general case, we use Cauchy inequalities for an appropriate analytic function. Let denote the analytic branch of the square root that is defined on C \ R. Take y R d \ {0}. On the polydisc D = { z = z 1,, z d ) C d ; j, z j < y /4 d) } d, R y j + z j ) 2 7/16) y 2 > 0. j=1 Thus the function u : D C given by uz) = 1 dj=1 y j + z j ) 2 is well defined and analytic. Its modulus is bounded above by y 1 4/ 7. By Cauchy inequalities cf. Theorem 2.2.7, p. 27, in [H3]), α N d, α z u0) 4 y 7 α!) y /4 d) ) α 4 d) α +1 α!) y α +1. A.2) Here zj := 1/2) Rzj + i Izj ) but it can be replaced by Rzj in the formula since u is analytic. Now A.1) follows from A.2) since, for all α, Rzu )0) α = i α D α 1 ) y). References [Fe] H. Federer: Geometric Measure Theory. Springer-Verlag 1969.
10 Analyticity of the electronic density, [FHHS1] S. Fournais, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, T. Østergaard Sørensen: The electron density is smooth away from the nuclei. Comm. Math. Phys. 228, no ), [FHHS2] S. Fournais, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, T. Østergaard Sørensen: Analyticity of the density of electronic wave functions. Ark. Mat. 42, no ), [FHHS3] S. Fournais, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, T. Østergaard Sørensen: Non-isotropic cusp conditions and regularity of the electron density of molecules at the nuclei. Ann. Henri Poincaré ), [FH] R.G. Froese, I. Herbst: Exponential bounds and absence of positive eigenvalues for N-body Schrödinger operators. Comm. Math. Phys. 87, ). [H1] L. Hörmander: Linear partial differential operators. Fourth printing Springer Verlag, [H2] L. Hörmander: The analysis of linear partial differential operators III. Springer Verlag, [H3] L. Hörmander: An introduction to complex analysis in several variables. Elsevier science publishers B.V., [K] T. Kato: Pertubation theory for linear operators. Springer-Verlag [KMSW] M. Klein, A. Martinez, R. Seiler, X.P. Wang: On the Born-Oppenheimer expansion for polyatomic molecules. Comm. Math. Phys. 143, no. 3, ). [MS] A. Martinez, V. Sordoni: Twisted pseudodifferential calculus and application to the quantum evolution of molecules. Preprint 2008, mp arc , to appear in the Memoirs of the AMS. [RS] M. Reed, B. Simon: Methods of Modern Mathematical Physics, Vol. II : Fourier Analysis, Self-adjointness. Academic Press, 1979.
SELF-ADJOINTNESS OF DIRAC OPERATORS VIA HARDY-DIRAC INEQUALITIES
SELF-ADJOINTNESS OF DIRAC OPERATORS VIA HARDY-DIRAC INEQUALITIES MARIA J. ESTEBAN 1 AND MICHAEL LOSS Abstract. Distinguished selfadjoint extension of Dirac operators are constructed for a class of potentials
More information2. Function spaces and approximation
2.1 2. Function spaces and approximation 2.1. The space of test functions. Notation and prerequisites are collected in Appendix A. Let Ω be an open subset of R n. The space C0 (Ω), consisting of the C
More informationDETERMINATION OF THE BLOW-UP RATE FOR THE SEMILINEAR WAVE EQUATION
DETERMINATION OF THE LOW-UP RATE FOR THE SEMILINEAR WAVE EQUATION y FRANK MERLE and HATEM ZAAG Abstract. In this paper, we find the optimal blow-up rate for the semilinear wave equation with a power nonlinearity.
More informationSELF-ADJOINTNESS OF SCHRÖDINGER-TYPE OPERATORS WITH SINGULAR POTENTIALS ON MANIFOLDS OF BOUNDED GEOMETRY
Electronic Journal of Differential Equations, Vol. 2003(2003), No.??, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) SELF-ADJOINTNESS
More informationOn the Virial Theorem in Quantum Mechanics
Commun. Math. Phys. 208, 275 281 (1999) Communications in Mathematical Physics Springer-Verlag 1999 On the Virial Theorem in Quantum Mechanics V. Georgescu 1, C. Gérard 2 1 CNRS, Département de Mathématiques,
More informationWe denote the space of distributions on Ω by D ( Ω) 2.
Sep. 1 0, 008 Distributions Distributions are generalized functions. Some familiarity with the theory of distributions helps understanding of various function spaces which play important roles in the study
More informationMATH 205C: STATIONARY PHASE LEMMA
MATH 205C: STATIONARY PHASE LEMMA For ω, consider an integral of the form I(ω) = e iωf(x) u(x) dx, where u Cc (R n ) complex valued, with support in a compact set K, and f C (R n ) real valued. Thus, I(ω)
More informationEuler Equations: local existence
Euler Equations: local existence Mat 529, Lesson 2. 1 Active scalars formulation We start with a lemma. Lemma 1. Assume that w is a magnetization variable, i.e. t w + u w + ( u) w = 0. If u = Pw then u
More informationA new class of pseudodifferential operators with mixed homogenities
A new class of pseudodifferential operators with mixed homogenities Po-Lam Yung University of Oxford Jan 20, 2014 Introduction Given a smooth distribution of hyperplanes on R N (or more generally on a
More informationProceedings of the 5th International Conference on Inverse Problems in Engineering: Theory and Practice, Cambridge, UK, 11-15th July 2005
Proceedings of the 5th International Conference on Inverse Problems in Engineering: Theory and Practice, Cambridge, UK, 11-15th July 2005 SOME INVERSE SCATTERING PROBLEMS FOR TWO-DIMENSIONAL SCHRÖDINGER
More informationhal , version 1-22 Nov 2009
Author manuscript, published in "Kinet. Relat. Models 1, 3 8) 355-368" PROPAGATION OF GEVREY REGULARITY FOR SOLUTIONS OF LANDAU EQUATIONS HUA CHEN, WEI-XI LI AND CHAO-JIANG XU Abstract. By using the energy-type
More informationPOINTWISE BOUNDS ON QUASIMODES OF SEMICLASSICAL SCHRÖDINGER OPERATORS IN DIMENSION TWO
POINTWISE BOUNDS ON QUASIMODES OF SEMICLASSICAL SCHRÖDINGER OPERATORS IN DIMENSION TWO HART F. SMITH AND MACIEJ ZWORSKI Abstract. We prove optimal pointwise bounds on quasimodes of semiclassical Schrödinger
More informationWEYL S LEMMA, ONE OF MANY. Daniel W. Stroock
WEYL S LEMMA, ONE OF MANY Daniel W Stroock Abstract This note is a brief, and somewhat biased, account of the evolution of what people working in PDE s call Weyl s Lemma about the regularity of solutions
More informationTHE FORM SUM AND THE FRIEDRICHS EXTENSION OF SCHRÖDINGER-TYPE OPERATORS ON RIEMANNIAN MANIFOLDS
THE FORM SUM AND THE FRIEDRICHS EXTENSION OF SCHRÖDINGER-TYPE OPERATORS ON RIEMANNIAN MANIFOLDS OGNJEN MILATOVIC Abstract. We consider H V = M +V, where (M, g) is a Riemannian manifold (not necessarily
More informationAn inverse scattering problem for short-range systems in a time-periodic electric field. François Nicoleau
An inverse scattering problem for short-range systems in a time-periodic electric field. François Nicoleau Laboratoire Jean Leray UMR CNRS-UN 6629 Département de Mathématiques 2, rue de la Houssinière
More informationSharp Gårding inequality on compact Lie groups.
15-19.10.2012, ESI, Wien, Phase space methods for pseudo-differential operators Ville Turunen, Aalto University, Finland (ville.turunen@aalto.fi) M. Ruzhansky, V. Turunen: Sharp Gårding inequality on compact
More informationi=1 α i. Given an m-times continuously
1 Fundamentals 1.1 Classification and characteristics Let Ω R d, d N, d 2, be an open set and α = (α 1,, α d ) T N d 0, N 0 := N {0}, a multiindex with α := d i=1 α i. Given an m-times continuously differentiable
More informationLow frequency resolvent estimates for long range perturbations of the Euclidean Laplacian
Low frequency resolvent estimates for long range perturbations of the Euclidean Laplacian Jean-Francois Bony, Dietrich Häfner To cite this version: Jean-Francois Bony, Dietrich Häfner. Low frequency resolvent
More informationTitle: Localized self-adjointness of Schrödinger-type operators on Riemannian manifolds. Proposed running head: Schrödinger-type operators on
Title: Localized self-adjointness of Schrödinger-type operators on Riemannian manifolds. Proposed running head: Schrödinger-type operators on manifolds. Author: Ognjen Milatovic Department Address: Department
More informationAN EIGENVALUE PROBLEM FOR THE SCHRÖDINGER MAXWELL EQUATIONS. Vieri Benci Donato Fortunato. 1. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume, 998, 83 93 AN EIGENVALUE PROBLEM FOR THE SCHRÖDINGER MAXWELL EQUATIONS Vieri Benci Donato Fortunato Dedicated to
More informationMicrolocal Analysis : a short introduction
Microlocal Analysis : a short introduction Plamen Stefanov Purdue University Mini Course, Fields Institute, 2012 Plamen Stefanov (Purdue University ) Microlocal Analysis : a short introduction 1 / 25 Introduction
More informationTHE L 2 -HODGE THEORY AND REPRESENTATION ON R n
THE L 2 -HODGE THEORY AND REPRESENTATION ON R n BAISHENG YAN Abstract. We present an elementary L 2 -Hodge theory on whole R n based on the minimization principle of the calculus of variations and some
More informationNotes for Elliptic operators
Notes for 18.117 Elliptic operators 1 Differential operators on R n Let U be an open subset of R n and let D k be the differential operator, 1 1 x k. For every multi-index, α = α 1,...,α n, we define A
More informationSébastien Chaumont a a Institut Élie Cartan, Université Henri Poincaré Nancy I, B. P. 239, Vandoeuvre-lès-Nancy Cedex, France. 1.
A strong comparison result for viscosity solutions to Hamilton-Jacobi-Bellman equations with Dirichlet condition on a non-smooth boundary and application to parabolic problems Sébastien Chaumont a a Institut
More informationAsymptotic behavior of infinity harmonic functions near an isolated singularity
Asymptotic behavior of infinity harmonic functions near an isolated singularity Ovidiu Savin, Changyou Wang, Yifeng Yu Abstract In this paper, we prove if n 2 x 0 is an isolated singularity of a nonegative
More informationP(E t, Ω)dt, (2) 4t has an advantage with respect. to the compactly supported mollifiers, i.e., the function W (t)f satisfies a semigroup law:
Introduction Functions of bounded variation, usually denoted by BV, have had and have an important role in several problems of calculus of variations. The main features that make BV functions suitable
More informationPseudo-Poincaré Inequalities and Applications to Sobolev Inequalities
Pseudo-Poincaré Inequalities and Applications to Sobolev Inequalities Laurent Saloff-Coste Abstract Most smoothing procedures are via averaging. Pseudo-Poincaré inequalities give a basic L p -norm control
More informationA Walking Tour of Microlocal Analysis
A Walking Tour of Microlocal Analysis Jeff Schonert August 10, 2006 Abstract We summarize some of the basic principles of microlocal analysis and their applications. After reviewing distributions, we then
More informationIntroduction to Spectral Theory
P.D. Hislop I.M. Sigal Introduction to Spectral Theory With Applications to Schrodinger Operators Springer Introduction and Overview 1 1 The Spectrum of Linear Operators and Hilbert Spaces 9 1.1 TheSpectrum
More informationNon Adiabatic Transitions in a Simple Born Oppenheimer Scattering System
1 Non Adiabatic Transitions in a Simple Born Oppenheimer Scattering System George A. Hagedorn Department of Mathematics and Center for Statistical Mechanics, Mathematical Physics, and Theoretical Chemistry
More informationA SHARP STABILITY ESTIMATE IN TENSOR TOMOGRAPHY
A SHARP STABILITY ESTIMATE IN TENSOR TOMOGRAPHY PLAMEN STEFANOV 1. Introduction Let (M, g) be a compact Riemannian manifold with boundary. The geodesic ray transform I of symmetric 2-tensor fields f is
More informationEigenfunctions for smooth expanding circle maps
December 3, 25 1 Eigenfunctions for smooth expanding circle maps Gerhard Keller and Hans-Henrik Rugh December 3, 25 Abstract We construct a real-analytic circle map for which the corresponding Perron-
More informationNew Proof of Hörmander multiplier Theorem on compact manifolds without boundary
New Proof of Hörmander multiplier Theorem on compact manifolds without boundary Xiangjin Xu Department of athematics Johns Hopkins University Baltimore, D, 21218, USA xxu@math.jhu.edu Abstract On compact
More informationSobolev Spaces. Chapter 10
Chapter 1 Sobolev Spaces We now define spaces H 1,p (R n ), known as Sobolev spaces. For u to belong to H 1,p (R n ), we require that u L p (R n ) and that u have weak derivatives of first order in L p
More informationOn a class of pseudodifferential operators with mixed homogeneities
On a class of pseudodifferential operators with mixed homogeneities Po-Lam Yung University of Oxford July 25, 2014 Introduction Joint work with E. Stein (and an outgrowth of work of Nagel-Ricci-Stein-Wainger,
More informationAgmon-Type Exponential Decay Estimates for Pseudodifferential Operators
J. Math. Sci. Univ. Tokyo 5 (1998), 693 712. Agmon-Type Exponential Decay Estimates for Pseudodifferential Operators By Shu Nakamura Abstract. We study generalizations of Agmon-type estimates on eigenfunctions
More informationEXPOSITORY NOTES ON DISTRIBUTION THEORY, FALL 2018
EXPOSITORY NOTES ON DISTRIBUTION THEORY, FALL 2018 While these notes are under construction, I expect there will be many typos. The main reference for this is volume 1 of Hörmander, The analysis of liner
More informationFrom the N-body problem to the cubic NLS equation
From the N-body problem to the cubic NLS equation François Golse Université Paris 7 & Laboratoire J.-L. Lions golse@math.jussieu.fr Los Alamos CNLS, January 26th, 2005 Formal derivation by N.N. Bogolyubov
More informationContinuous Functions on Metric Spaces
Continuous Functions on Metric Spaces Math 201A, Fall 2016 1 Continuous functions Definition 1. Let (X, d X ) and (Y, d Y ) be metric spaces. A function f : X Y is continuous at a X if for every ɛ > 0
More informationMICROLOCAL ANALYSIS METHODS
MICROLOCAL ANALYSIS METHODS PLAMEN STEFANOV One of the fundamental ideas of classical analysis is a thorough study of functions near a point, i.e., locally. Microlocal analysis, loosely speaking, is analysis
More informationClass Meeting # 1: Introduction to PDEs
MATH 18.152 COURSE NOTES - CLASS MEETING # 1 18.152 Introduction to PDEs, Spring 2017 Professor: Jared Speck Class Meeting # 1: Introduction to PDEs 1. What is a PDE? We will be studying functions u =
More informationProperties of the Scattering Transform on the Real Line
Journal of Mathematical Analysis and Applications 58, 3 43 (001 doi:10.1006/jmaa.000.7375, available online at http://www.idealibrary.com on Properties of the Scattering Transform on the Real Line Michael
More informationAALBORG UNIVERSITY. Third derivative of the one-electron density at the nucleus. S. Fournais, M. Hoffmann-Ostenhof and T. Østergaard Sørensem
AALBORG UNIVERSITY Third derivative of the one-electron density at the nucleus by S. Fournais, M. Hoffmann-Ostenhof and T. Østergaard Sørensem R-2006-26 July 2006 Department of Mathematical Sciences Aalborg
More informationFourier Transform & Sobolev Spaces
Fourier Transform & Sobolev Spaces Michael Reiter, Arthur Schuster Summer Term 2008 Abstract We introduce the concept of weak derivative that allows us to define new interesting Hilbert spaces the Sobolev
More informationOn stable inversion of the attenuated Radon transform with half data Jan Boman. We shall consider weighted Radon transforms of the form
On stable inversion of the attenuated Radon transform with half data Jan Boman We shall consider weighted Radon transforms of the form R ρ f(l) = f(x)ρ(x, L)ds, L where ρ is a given smooth, positive weight
More informationSobolev spaces. May 18
Sobolev spaces May 18 2015 1 Weak derivatives The purpose of these notes is to give a very basic introduction to Sobolev spaces. More extensive treatments can e.g. be found in the classical references
More informationLECTURE 5: THE METHOD OF STATIONARY PHASE
LECTURE 5: THE METHOD OF STATIONARY PHASE Some notions.. A crash course on Fourier transform For j =,, n, j = x j. D j = i j. For any multi-index α = (α,, α n ) N n. α = α + + α n. α! = α! α n!. x α =
More informationFACTORIZATION OF SECOND-ORDER STRICTLY HYPERBOLIC OPERATORS WITH LOGARITHMIC SLOW SCALE COEFFICIENTS AND GENERALIZED MICROLOCAL APPROXIMATIONS
Electronic Journal of Differential Equations, Vol. 28 28, No. 42, pp. 49. ISSN: 72-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu FACTORIZATION OF SECOND-ORDER STRICTLY HYPERBOLIC OPERATORS
More informationPartial Differential Equations, 2nd Edition, L.C.Evans Chapter 5 Sobolev Spaces
Partial Differential Equations, nd Edition, L.C.Evans Chapter 5 Sobolev Spaces Shih-Hsin Chen, Yung-Hsiang Huang 7.8.3 Abstract In these exercises always denote an open set of with smooth boundary. As
More informationEXISTENCE RESULTS FOR OPERATOR EQUATIONS INVOLVING DUALITY MAPPINGS VIA THE MOUNTAIN PASS THEOREM
EXISTENCE RESULTS FOR OPERATOR EQUATIONS INVOLVING DUALITY MAPPINGS VIA THE MOUNTAIN PASS THEOREM JENICĂ CRÎNGANU We derive existence results for operator equations having the form J ϕu = N f u, by using
More informationAN INVERSE PROBLEM FOR THE WAVE EQUATION WITH A TIME DEPENDENT COEFFICIENT
AN INVERSE PROBLEM FOR THE WAVE EQUATION WITH A TIME DEPENDENT COEFFICIENT Rakesh Department of Mathematics University of Delaware Newark, DE 19716 A.G.Ramm Department of Mathematics Kansas State University
More informationOn m-accretive Schrödinger operators in L p -spaces on manifolds of bounded geometry
On m-accretive Schrödinger operators in L p -spaces on manifolds of bounded geometry Ognjen Milatovic Department of Mathematics and Statistics University of North Florida Jacksonville, FL 32224 USA. Abstract
More informationPartial Differential Equations
Part II Partial Differential Equations Year 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2015 Paper 4, Section II 29E Partial Differential Equations 72 (a) Show that the Cauchy problem for u(x,
More informationChapter One. The Calderón-Zygmund Theory I: Ellipticity
Chapter One The Calderón-Zygmund Theory I: Ellipticity Our story begins with a classical situation: convolution with homogeneous, Calderón- Zygmund ( kernels on R n. Let S n 1 R n denote the unit sphere
More information2) Let X be a compact space. Prove that the space C(X) of continuous real-valued functions is a complete metric space.
University of Bergen General Functional Analysis Problems with solutions 6 ) Prove that is unique in any normed space. Solution of ) Let us suppose that there are 2 zeros and 2. Then = + 2 = 2 + = 2. 2)
More informationu xx + u yy = 0. (5.1)
Chapter 5 Laplace Equation The following equation is called Laplace equation in two independent variables x, y: The non-homogeneous problem u xx + u yy =. (5.1) u xx + u yy = F, (5.) where F is a function
More informationBLOWUP THEORY FOR THE CRITICAL NONLINEAR SCHRÖDINGER EQUATIONS REVISITED
BLOWUP THEORY FOR THE CRITICAL NONLINEAR SCHRÖDINGER EQUATIONS REVISITED TAOUFIK HMIDI AND SAHBI KERAANI Abstract. In this note we prove a refined version of compactness lemma adapted to the blowup analysis
More informationDEGREE AND SOBOLEV SPACES. Haïm Brezis Yanyan Li Petru Mironescu Louis Nirenberg. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 13, 1999, 181 190 DEGREE AND SOBOLEV SPACES Haïm Brezis Yanyan Li Petru Mironescu Louis Nirenberg Dedicated to Jürgen
More informationThe oblique derivative problem for general elliptic systems in Lipschitz domains
M. MITREA The oblique derivative problem for general elliptic systems in Lipschitz domains Let M be a smooth, oriented, connected, compact, boundaryless manifold of real dimension m, and let T M and T
More informationRegularity for Poisson Equation
Regularity for Poisson Equation OcMountain Daylight Time. 4, 20 Intuitively, the solution u to the Poisson equation u= f () should have better regularity than the right hand side f. In particular one expects
More informationThe Calderon-Vaillancourt Theorem
The Calderon-Vaillancourt Theorem What follows is a completely self contained proof of the Calderon-Vaillancourt Theorem on the L 2 boundedness of pseudo-differential operators. 1 The result Definition
More informationSecond Order Elliptic PDE
Second Order Elliptic PDE T. Muthukumar tmk@iitk.ac.in December 16, 2014 Contents 1 A Quick Introduction to PDE 1 2 Classification of Second Order PDE 3 3 Linear Second Order Elliptic Operators 4 4 Periodic
More informationEnergy method for wave equations
Energy method for wave equations Willie Wong Based on commit 5dfb7e5 of 2017-11-06 13:29 Abstract We give an elementary discussion of the energy method (and particularly the vector field method) in the
More informationNATIONAL BOARD FOR HIGHER MATHEMATICS. Research Scholarships Screening Test. Saturday, January 20, Time Allowed: 150 Minutes Maximum Marks: 40
NATIONAL BOARD FOR HIGHER MATHEMATICS Research Scholarships Screening Test Saturday, January 2, 218 Time Allowed: 15 Minutes Maximum Marks: 4 Please read, carefully, the instructions that follow. INSTRUCTIONS
More informationChap. 1. Some Differential Geometric Tools
Chap. 1. Some Differential Geometric Tools 1. Manifold, Diffeomorphism 1.1. The Implicit Function Theorem ϕ : U R n R n p (0 p < n), of class C k (k 1) x 0 U such that ϕ(x 0 ) = 0 rank Dϕ(x) = n p x U
More informationNOTES FOR CARDIFF LECTURES ON MICROLOCAL ANALYSIS
NOTES FOR CARDIFF LECTURES ON MICROLOCAL ANALYSIS JARED WUNSCH Note that these lectures overlap with Alex s to a degree, to ensure a smooth handoff between lecturers! Our notation is mostly, but not completely,
More informationA proof of the abstract limiting absorption principle by energy estimates
A proof of the abstract limiting absorption principle by energy estimates Christian Gérard Laboratoire de Mathématiques, Université de Paris-Sud 91405 Orsay Cedex France Abstract We give a proof in an
More informationON EFFECTIVE HAMILTONIANS FOR ADIABATIC PERTURBATIONS. Mouez Dimassi Jean-Claude Guillot James Ralston
ON EFFECTIVE HAMILTONIANS FOR ADIABATIC PERTURBATIONS Mouez Dimassi Jean-Claude Guillot James Ralston Abstract We construct almost invariant subspaces and the corresponding effective Hamiltonian for magnetic
More informationGreen s Functions and Distributions
CHAPTER 9 Green s Functions and Distributions 9.1. Boundary Value Problems We would like to study, and solve if possible, boundary value problems such as the following: (1.1) u = f in U u = g on U, where
More informationRigidity and Non-rigidity Results on the Sphere
Rigidity and Non-rigidity Results on the Sphere Fengbo Hang Xiaodong Wang Department of Mathematics Michigan State University Oct., 00 1 Introduction It is a simple consequence of the maximum principle
More informationOn Schrödinger equations with inverse-square singular potentials
On Schrödinger equations with inverse-square singular potentials Veronica Felli Dipartimento di Statistica University of Milano Bicocca veronica.felli@unimib.it joint work with Elsa M. Marchini and Susanna
More informationMixed exterior Laplace s problem
Mixed exterior Laplace s problem Chérif Amrouche, Florian Bonzom Laboratoire de mathématiques appliquées, CNRS UMR 5142, Université de Pau et des Pays de l Adour, IPRA, Avenue de l Université, 64000 Pau
More informationComplex geometrical optics solutions for Lipschitz conductivities
Rev. Mat. Iberoamericana 19 (2003), 57 72 Complex geometrical optics solutions for Lipschitz conductivities Lassi Päivärinta, Alexander Panchenko and Gunther Uhlmann Abstract We prove the existence of
More informationFunctional Analysis HW #3
Functional Analysis HW #3 Sangchul Lee October 26, 2015 1 Solutions Exercise 2.1. Let D = { f C([0, 1]) : f C([0, 1])} and define f d = f + f. Show that D is a Banach algebra and that the Gelfand transform
More informationA Perron-type theorem on the principal eigenvalue of nonsymmetric elliptic operators
A Perron-type theorem on the principal eigenvalue of nonsymmetric elliptic operators Lei Ni And I cherish more than anything else the Analogies, my most trustworthy masters. They know all the secrets of
More informationTHE GROUP OF AUTOMORPHISMS OF A REAL
THE GROUP OF AUTOMORPHISMS OF A REAL RATIONAL SURFACE IS n-transitive JOHANNES HUISMAN AND FRÉDÉRIC MANGOLTE To Joost van Hamel in memoriam Abstract. Let X be a rational nonsingular compact connected real
More informationThe following definition is fundamental.
1. Some Basics from Linear Algebra With these notes, I will try and clarify certain topics that I only quickly mention in class. First and foremost, I will assume that you are familiar with many basic
More informationExponentially Accurate Semiclassical Tunneling Wave Functions in One Dimension
Exponentially Accurate Semiclassical Tunneling Wave Functions in One Dimension Vasile Gradinaru Seminar for Applied Mathematics ETH Zürich CH 8092 Zürich, Switzerland, George A. Hagedorn Department of
More information2 A Model, Harmonic Map, Problem
ELLIPTIC SYSTEMS JOHN E. HUTCHINSON Department of Mathematics School of Mathematical Sciences, A.N.U. 1 Introduction Elliptic equations model the behaviour of scalar quantities u, such as temperature or
More information1.4 The Jacobian of a map
1.4 The Jacobian of a map Derivative of a differentiable map Let F : M n N m be a differentiable map between two C 1 manifolds. Given a point p M we define the derivative of F at p by df p df (p) : T p
More informationThe Lusin Theorem and Horizontal Graphs in the Heisenberg Group
Analysis and Geometry in Metric Spaces Research Article DOI: 10.2478/agms-2013-0008 AGMS 2013 295-301 The Lusin Theorem and Horizontal Graphs in the Heisenberg Group Abstract In this paper we prove that
More informationExercise Solutions to Functional Analysis
Exercise Solutions to Functional Analysis Note: References refer to M. Schechter, Principles of Functional Analysis Exersize that. Let φ,..., φ n be an orthonormal set in a Hilbert space H. Show n f n
More informationMath 209B Homework 2
Math 29B Homework 2 Edward Burkard Note: All vector spaces are over the field F = R or C 4.6. Two Compactness Theorems. 4. Point Set Topology Exercise 6 The product of countably many sequentally compact
More informationCOMPLEX SPHERICAL WAVES AND INVERSE PROBLEMS IN UNBOUNDED DOMAINS
COMPLEX SPHERICAL WAVES AND INVERSE PROBLEMS IN UNBOUNDED DOMAINS MIKKO SALO AND JENN-NAN WANG Abstract. This work is motivated by the inverse conductivity problem of identifying an embedded object in
More informationApplied Math Qualifying Exam 11 October Instructions: Work 2 out of 3 problems in each of the 3 parts for a total of 6 problems.
Printed Name: Signature: Applied Math Qualifying Exam 11 October 2014 Instructions: Work 2 out of 3 problems in each of the 3 parts for a total of 6 problems. 2 Part 1 (1) Let Ω be an open subset of R
More informationTopics in Harmonic Analysis Lecture 6: Pseudodifferential calculus and almost orthogonality
Topics in Harmonic Analysis Lecture 6: Pseudodifferential calculus and almost orthogonality Po-Lam Yung The Chinese University of Hong Kong Introduction While multiplier operators are very useful in studying
More informationASYMPTOTIC BEHAVIOR OF GENERALIZED EIGENFUNCTIONS IN N-BODY SCATTERING
ASYMPTOTIC BEHAVIOR OF GENERALIZED EIGENFUNCTIONS IN N-BODY SCATTERING ANDRAS VASY Abstract. In this paper an asymptotic expansion is proved for locally (at infinity) outgoing functions on asymptotically
More informationLax Solution Part 4. October 27, 2016
Lax Solution Part 4 www.mathtuition88.com October 27, 2016 Textbook: Functional Analysis by Peter D. Lax Exercises: Ch 16: Q2 4. Ch 21: Q1, 2, 9, 10. Ch 28: 1, 5, 9, 10. 1 Chapter 16 Exercise 2 Let h =
More informationPseudodifferential calculus and Hadamard states
Pseudodifferential calculus and Hadamard states Local Quantum Physics and beyond - in memoriam Rudolf Haag Hamburg, Sept. 26-27 2016 Christian Gérard Département of Mathématiques Université Paris-Sud 1
More informationSpectral theory for magnetic Schrödinger operators and applicatio. (after Bauman-Calderer-Liu-Phillips, Pan, Helffer-Pan)
Spectral theory for magnetic Schrödinger operators and applications to liquid crystals (after Bauman-Calderer-Liu-Phillips, Pan, Helffer-Pan) Ryukoku (June 2008) In [P2], based on the de Gennes analogy
More informationNotes on Distributions
Notes on Distributions Functional Analysis 1 Locally Convex Spaces Definition 1. A vector space (over R or C) is said to be a topological vector space (TVS) if it is a Hausdorff topological space and the
More informationTransformations from R m to R n.
Transformations from R m to R n 1 Differentiablity First of all because of an unfortunate combination of traditions (the fact that we read from left to right and the way we define matrix multiplication
More informationarxiv: v1 [math.ap] 18 May 2017
Littlewood-Paley-Stein functions for Schrödinger operators arxiv:175.6794v1 [math.ap] 18 May 217 El Maati Ouhabaz Dedicated to the memory of Abdelghani Bellouquid (2/2/1966 8/31/215) Abstract We study
More informationOptimal elliptic estimates for solutions of Ginzburg-Landau systems revisited
Optimal elliptic estimates for solutions of Ginzburg-Landau systems revisited Bernard Helffer Mathématiques - Univ Paris Sud- UMR CNRS 8628 (After S. Fournais and B. Helffer) International Conference in
More informationSobolev Spaces. Chapter Hölder spaces
Chapter 2 Sobolev Spaces Sobolev spaces turn out often to be the proper setting in which to apply ideas of functional analysis to get information concerning partial differential equations. Here, we collect
More informationTD 1: Hilbert Spaces and Applications
Université Paris-Dauphine Functional Analysis and PDEs Master MMD-MA 2017/2018 Generalities TD 1: Hilbert Spaces and Applications Exercise 1 (Generalized Parallelogram law). Let (H,, ) be a Hilbert space.
More informationOn the mathematical treatment of the Born-Oppenheimer approximation.
On the mathematical treatment of the Born-Oppenheimer approximation. Thierry Jecko AGM, UMR 8088 du CNRS, Université de Cergy-Pontoise, Département de mathématiques, site de Saint Martin, 2 avenue Adolphe
More informationFRAMES AND TIME-FREQUENCY ANALYSIS
FRAMES AND TIME-FREQUENCY ANALYSIS LECTURE 5: MODULATION SPACES AND APPLICATIONS Christopher Heil Georgia Tech heil@math.gatech.edu http://www.math.gatech.edu/ heil READING For background on Banach spaces,
More informationBASIC MATRIX PERTURBATION THEORY
BASIC MATRIX PERTURBATION THEORY BENJAMIN TEXIER Abstract. In this expository note, we give the proofs of several results in finitedimensional matrix perturbation theory: continuity of the spectrum, regularity
More informationCONTINUITY WITH RESPECT TO DISORDER OF THE INTEGRATED DENSITY OF STATES
Illinois Journal of Mathematics Volume 49, Number 3, Fall 2005, Pages 893 904 S 0019-2082 CONTINUITY WITH RESPECT TO DISORDER OF THE INTEGRATED DENSITY OF STATES PETER D. HISLOP, FRÉDÉRIC KLOPP, AND JEFFREY
More information