Schrodinger operator: optimal decay of fundamental solutions and boundary value problems

Transcription

1 Schrodinger operator: optimal decay of fundamental solutions and boundary value problems University of Minnesota Department of Mathematics PCMI Research Program Seminar Park City, Utah July 17,

2 The homogeneous second-order elliptic operator Let N N, N 2, and Ω R N open. Write L 0 div A. (1) Here, A = (A ij ) is an N N matrix of complex L coefficients satisfying the following uniform ellipticity condition λ ξ 2 Re A(x)ξ, ξ and A L (Ω) Λ, (2) for some λ > 0, Λ <, and for all ξ C N, x Ω. 2

3 Schrodinger operators Consider L E = diva + V, or L M = ( ia) 2 + V, or more generally L = ( ia) T A( ia) + V. (electric Schrödinger) (magnetic Schrödinger) (generalized magnetic Schrödinger) Example: Schrödinger operator on a manifold. Our drive. Understand the impact of the potential on L and its solutions. Goal 1. Sharp decay estimates on fundamental solutions. Goal 2. Solvability of boundary value problems (BVPs). 3

4 The tug-of-war between The 2nd order term -div A Existence of harmonic measure Necessary conditions on A known Scale-invariance. Homogeneity of equation The potential V Exponential decay of solutions in the presence of confining or disordered (random) potential Introduce different scaling. Exponential decay 4

5 The goals We aim to understand the impact of the potential on L and its solutions. When V is substantial Impact of exponential decay? Estimates on resolvents, fundamental solution. Submitted for publication, [MP]. Long term: Solve BVPs When V is small In which sense is smallness innocent"? Perturbation results for BVPs 5

6 Exponential decay of the fundamental solution Consider L E = div A + V, or L M = ( ia) 2 + V, or more generally L = ( ia) T A( ia) + V. (electric Schrödinger) (magnetic Schrödinger) (generalized magnetic Schrödinger) The fundamental solution Γ, LΓ(x, y) = δ x (y). Question. What is the sharp rate of exponential decay of Γ? Exponential decay of solutions to Schrödinger operators in presence of positive potentials: important in quantum physics. However, establishing a precise rate of decay for complicated potentials is a challenging open problem. (Landis conjecture) 6

7 Exponential decay of the fundamental solution Consider L E = div A + V, or L M = ( ia) 2 + V, or more generally L = ( ia) T A( ia) + V. (electric Schrödinger) (magnetic Schrödinger) (generalized magnetic Schrödinger) The fundamental solution Γ, LΓ(x, y) = δ x (y). Question. What is the sharp rate of exponential decay of Γ? Exponential decay of solutions to Schrödinger operators in presence of positive potentials: important in quantum physics. However, establishing a precise rate of decay for complicated potentials is a challenging open problem. (Landis conjecture) 6

8 Some history First exp. decay in terms of V : Agmon [Agm82], but not sharp. Shen in [She99]: Sharp exp. decay of + V. Polynomial decay also relevant to obtain L p boundedness of associated Riesz transforms: [She95], [She96]. Kurata [Kur00]: non-sharp exp. decay of L E, L M through heat kernel estimates. 7

9 Exponential decay: Main results For Γ the fundamental solution to div A + V, c 1 e ε1d(x,y,v ) x y n 2 Γ(x, y) c 2e ε2d(x,y,v ) x y n 2. (3) Only upper estimate for ( ia) 2 + V. L 2 estimate for ( ia) T A( ia) + V, and resolvents. 8

10 The Reverse Hölder class RH p We say that w L p loc (Rn ), with w > 0 a.e., belongs to the Reverse Hölder class RH p = RH p (R n ) if there exists a constant C so that for any ball B R n, ( ) 1/p w p C B w. (4) B 9

11 The Fefferman-Phong-Shen maximal function m(x, w) Denote and the magnetic field by B, so that D a = ia, B = curl a. (5) For a function w RH p, p n 2, define the maximal function m(x, w) by 1 m(x, w) and the Agmon distance { := sup r : r>0 d(x, y, w) = inf γ ˆ 1 0 ˆ 1 } r n 2 w 1, (6) B(x,r) m(γ(t), w) γ (t) dt, (7) where γ : [0, 1] R n is absolutely continuous and γ(0) = x, γ(1) = y. The Agmon distance characterizes a manifold sensitive to V. 10

12 m(x, w) and the uncertainty principle The function m measures the sum of the contributions of the kinetic energy ReAD a fd a f and potential energy V f 2, and is related to the uncertainty principle through the Fefferman-Phong inequality, its present form obtained by Z. Shen: Suppose that a L 2 loc (Rn ) n, and moreover assume (10) (next slide). Then, for all u Cc 1 (R n ), ˆ m 2 (x, V + B ) u 2 dx C R n ˆ ( D a u 2 + V u 2 ) dx. R n (8) 11

13 L 2 exponential decay Theorem 1 (Mayboroda-P. 2018) For L ( ia) T A( ia) + V and f L 2 c(r n ), constants d, ε, C > 0 such that ˆ 2εd(,supp f,v + B ) { } m (, V + B ) 2 L 1 f 2 e x R n d(x,supp f,v + B ) d ˆ C f 2 1 R m(x, V + B ) 2, (9) n provided i) either a = 0 and V RH n/2, ii) or, more generally, a L 2 loc (Rn ), V > 0 a.e. on R n, and V + B RH n/2, 0 V c m(, V + B ) 2, B c m(, V + B ) 3. (10) 12

14 L 2 exponential decay Theorem 1 (Mayboroda-P. 2018) For L ( ia) T A( ia) + V and f L 2 c(r n ), constants d, ε, C > 0 such that ˆ 2εd(,supp f,v + B ) { } m (, V + B ) 2 L 1 f 2 e x R n d(x,supp f,v + B ) d ˆ C f 2 1 R m(x, V + B ) 2, (9) n provided i) either a = 0 and V RH n/2, ii) or, more generally, a L 2 loc (Rn ), V > 0 a.e. on R n, and V + B RH n/2, 0 V c m(, V + B ) 2, B c m(, V + B ) 3. (10) 12

15 L 2 exponential decay for the resolvent An analogous estimate holds for the resolvent operator (I + t 2 L) 1, t > 0: ˆ ( ) 2 (I { } m, B t + t 2 L) 1 f 2 e x R n d(x,supp f,b t) d where B := V + B + 1 t 2. 2εd(,supp f,bt) ˆ ( ) 2. C f 2 m, B t R n In other words, L 1 f decays as e εd(,supp f,v + B ) away from the 1 εd(,supp f,v + B + support of f and the resolvent decays as e t 2 ). Previous resolvent decay results purely in terms of 1 t. 2 13

16 Upper bound exponential decay Theorem 2 (Mayboroda-P. 2018) If, in addition, solutions have pointwise (Moser) estimates, then Γ(x, y) Ce εd(x,y,v + B ) x y n 2 for all x, y R n. (11) In particular, applies to both electric Schrödinger div A + V, and magnetic Schrödinger ( ia) 2 + V. 14

17 Lower bound exponential decay Theorem 3 (Mayboroda-P. 2018) If, furthermore, an m scale invariant Harnack inequality holds, then Γ(x, y) ce ε2d(x,y,v + B ) x y n 2. (12) If Γ E is the fundamental solution to electric Schrödinger diva + V : c 1 e ε1d(x,y,v ) x y n 2 Γ E (x, y) c 2e ε2d(x,y,v ) x y n 2. (13) 15

18 Properties of the magnetic Schrödinger operator Magnetic Schrödinger exhibits gauge invariance: quantitative assumptions should be put on B rather than a. The diamagnetic inequality u (x) D a u(x). (14) When A I so that L M := L = ( ia) 2 + V, L M is dominated by L E := + V : for each ε > 0, (L M + ε) 1 f ( + ε) 1 f, for each f L 2 (R n ). (15) The above is known as the Kato-Simon inequality. 16

19 About L 2 decay L 2 Germinet and Klein [GK03]: resolvent exp. decay purely in terms of 1 t 2. Also Combes-Thomas estimates. The estimate (9) (for the operator L 1 ) is entirely new and is a consequence of the decay afforded by our assumptions on V and B. Resolvent estimate = estimate on f(l) for any holomorphic f. Our results are in the nature of best possible. 17

20 Some challenges Existence of fundamental solution. Some authors (B. Ben Ali [Ben10], Kurata and Sugano [KS00]) took ad-hoc assumptions on a, V. We prove existence of a fund. solution in the natural context, by a smooth approximation technique. A variable, non self-adjoint. When A is not self-adjoint: difficulty in dealing with the non-symmetric part. Dealing with complex coefficients, possible lack of Harnack or even of Moser. 18

21 Open problems For eigenfunctions, the decay is governed by the uncertainty principle - see Arnold, David, Filoche, Jerison and Mayboroda Localization of eigenfunctions via an effective potential [ADFJM]. Obtain exponential decay in terms of the landscape potential 1/u, Lu = 1. Use sharper uncertainty principle of [ADFJM]. Solve boundary value problems, Neumann problem in particular. 19

22 V is small Let us turn our attention to the general second-order elliptic operators, and describe some history. 20

23 The general second-order elliptic operator Let N N, N 2, and Ω R N open. Write L = div(a + b 1 ) + b 2 + V. (16) Here, A is an N N matrix satisfying λ > 0, Λ <, ξ C N, x Ω, λ ξ 2 Re A(x)ξ, ξ and A L (Ω) Λ. The terms b 1, b 2 are called drift terms, while V is called the potential. In particular, electric and magnetic Schrödinger. 21

24 Non-tangential maximal operators In this generality, Lipschitz domain half-space. Given q R n, write the non-tangential approach region { Γ(q) := (x, t) R n+1 + : x q < t}. (17) If u L loc (Rn+1 + ), write If u L 2 loc (Rn+1 + ), write N u(q) := Ñ u(q) := sup u(x, t). (18) (x,t) Γ(q) ( sup u 2) 1 2. (19) (x,t) Γ(q) x y <t t s < t 2 We use the notation u f non-tangentially (or u f n.t.) to mean that for almost every q R n, we have lim u(x) = f(q). (20) Γ(q) x q 22

25 The Dirichlet problem (D p ) The Dirichlet problem with data f in L p, p (1, ) is div(a u + b 1 u) + b 2 u + V u = 0 in R n+1 u f non-tangentially, N u Lp (R n ) f Lp (R n ). +, 23

26 The Neumann problem (N p ) and Regularity problem (R p ) div(a u + b 1 u) + b 2 u + V u = 0 in R n+1 u Neumann (N p ) : ν = g Lp (R n ), Ñ ( u) Lp (R n ) g Lp (R n ). +, div(a u + b 1 u) + b 2 u + V u = 0 in R n+1 Regularity (R p ) : u f n.t., Ñ ( u) L p (R n ) f L p (R n ). +, 24

27 BVPs: Early results L = on Lipschitz Ω: Dalhberg [Dah77] solved (D 2 ). Jerison and Kenig [JK81b] (N 2 ), (R 2 ). Verchota [Ver84] layer potentials. L = div A, how general can A be? Caffarelli, Fabes and Kenig [CFK81]: some regularity in transverse direction is necessary for (D 2 ). Thus it is natural to consider t independent A. 25

28 BVPs: t independent A A real, symmetric: Jerison and Kenig [JK81a] (D 2 ), Kenig and Pipher [KP93] (N 2 ), (R 2 ). A real, non-symmetric: Kenig, Koch, Pipher and Toro [KKPT00], Kenig and Rule [KR09], Barton [Bar13], Hofmann, Kenig, Mayboroda and Pipher [HKMP15a]-[HKMP15b]. A complex block: (N 2 ), (R 2 ) equivalent to solving a Kato square root problem. Full solution in [AHLMT02] Auscher, Hofmann, Lacey, McIntosh, Tchamitchian. 26

29 BVPs: Perturbation results L perturbations of t independent A: Fabes, Jerison and Kenig [FJK84], Alfonseca, Auscher, Axelsson, Hofmann and Kim [AAAHK11], Auscher, Axelsson and Hofmann [AAH08], Auscher, Axelsson and McIntosh [AAM10]... Carleson-norm perturbations: Fefferman, Kenig and Pipher [FKP91], Kenig and Pipher [KP93]-[KP95], Auscher and Axelsson [AA11], Hofmann, Mayboroda and Mourgoglou [HMM15]... Perturbation results for t independent matrices Perturbation results under small Carleson-norm assumption 27

30 BVPs: With lower order terms Very little is known. Three results: Kenig and Pipher [KP01], Dindos, Petermichl and Pipher [DPP07]: b 1 V 0 and some regularity on b 2 = (D p ). Used results of Hofmann and Lewis [HL01] Shen [She94]: A I, b 1 b 2 0, V RH = (N p ), (R p ). More recently: The Ph.D thesis [Sak17] of G. Sakellaris. Solve (D 2 ), (R 2 ) for L = diva + b 2 and its adjoint, A Lipschitz, Ω bounded Lipschitz, b 2 essentially bounded and integrability condition on div b 2. 28

31 A perturbation result through layer potentials Theorem 4 (Bortz-Hofmann-Luna García-Mayboroda-P. 2018) Let Ω = R n+1 +, n 3. Suppose that A is a t independent, complex, elliptic, bounded matrix, the coefficients b 1, b 2, V are t independent, and { } max b 1 L n (R n ), b 2 L n (R n ), V n L 2 (Rn ε ) 0 1. If (D 2 ) s, (N 2 ) s, (R 2 ) s are well-posed for L 0 := div A and its adjoint L 0, then the problems (D 2 ) s, (N 2 ) s, (R 2 ) s have solutions for L, where div(a u + b 1 u) + b 2 u + V u in R n+1 +, (D 2 ) s : u(, t) f strongly in L 2 (R n ) as t 0, sup t>0 u(, t) L 2 (R n ) f L 2 (R n ). (N 2 ) s : div(a u + b 1 u) + b 2 u + V u in R n+1 u ν = g L2 (R n ), sup t>0 u(, t) L2 (R n ) g L 2 (R n ). +, 29

32 A perturbation result through layer potentials Theorem 4 (Bortz-Hofmann-Luna García-Mayboroda-P. 2018) Let Ω = R n+1 +, n 3. Suppose that A is a t independent, complex, elliptic, bounded matrix, the coefficients b 1, b 2, V are t independent, and { } max b 1 L n (R n ), b 2 L n (R n ), V n L 2 (R ε n ) 0 1. (21) If (D 2 ) s, (N 2 ) s, (R 2 ) s are well-posed for L 0 := div A and its adjoint L 0, then the problems (D 2 ) s, (N 2 ) s, (R 2 ) s have solutions for L. Gesztesy, Hofmann and Nichols [GHN16]: Solution to a Kato problem for operators with lower order terms. Hypotheses hold for A real and symmetric, in particular. L p spaces in (21) are scale-invariant with respect to the second-order term diva. 30

33 Some main obstacles: no essential boundedness Our first strategy was to use the methods by Alfonseca, Auscher, Axelsson, Hofmann and Kim [AAAHK11]. But [AAAHK11] relies heavily on DeGiorgi-Nash-Moser bounds, which guarantee good decay properties of the fundamental solution Γ. In our setting, the DeGNM bounds do not hold, and neither do good decay properties of the fundamental solution, if it even exists. In [AAAHK11], they obtain the non-tangential maximal function control through a Cotlar-type argument using the fine properties of t Γ. This argument is not available to us. 31

34 On the failure of the DeGNM theory at the critical L p spaces for b 1, V The DeGiorgi-Nash-Moser theory gives Hölder-regularity for positive solutions to ( + V )u = 0 when V L n 2 +ε, for ε > 0. What happens when L + V, V L n 2 (R n )? Lemma 5 Let n 3, B a ball, and V L n 2 (R n ) with arbitrarily small norm. There exists u W 1,2 loc (B) a local weak solution to Lu = 0 on B, such that u > 0, u / L loc (B). Kim and Sakellaris [KS]: shown similar result. 32

35 On the failure of the DeGNM theory at the critical L p spaces for b 1, V The DeGiorgi-Nash-Moser theory gives Hölder-regularity for positive solutions to ( + V )u = 0 when V L n 2 +ε, for ε > 0. What happens when L + V, V L n 2 (R n )? Lemma 5 Let n 3, B a ball, and V L n 2 (R n ) with arbitrarily small norm. There exists u W 1,2 loc (B) a local weak solution to Lu = 0 on B, such that u > 0, u / L loc (B). Kim and Sakellaris [KS]: shown similar result. 32

36 Some main obstacles: constants are not solutions If L 0 div A and u solves L 0 u = 0 on Ω, then for any c C, u c also solves L 0 u = 0 on Ω. The above property does not hold for the operators with non-zero terms b 1, V. Linked to the failure of the DeGNM bounds. 33

37 Some definitions Denote 2 = 2(n+1) n 1 Y 1,2 (R n+1 + ) =, define the space { u L 2 (R n+1 + ) u L 2 (R n+1 + ) Following A. Barton [Bar17], define the single-layer potential, S L : (H (R n )) Y 1,2 (R n+1 ), by ( S L := Tr 0 (L 1 ) ). (22) The above definition coincides with the classical single-layer potential for the operator : ˆ S L f(x, t) = Γ(x, t, y, 0)f(y) dy, (x, t) R n+1. R n The square function S : ( S u = u := R n dx dt u(x, t) t ) 1 2. }, 34

38 Some definitions Denote 2 = 2(n+1) n 1 Y 1,2 (R n+1 + ) =, define the space { u L 2 (R n+1 + ) u L 2 (R n+1 + ) Following A. Barton [Bar17], define the single-layer potential, S L : (H (R n )) Y 1,2 (R n+1 ), by ( S L := Tr 0 (L 1 ) ). (22) The above definition coincides with the classical single-layer potential for the operator : ˆ S L f(x, t) = Γ(x, t, y, 0)f(y) dy, (x, t) R n+1. R n The square function S : ( S u = u := R n dx dt u(x, t) t ) 1 2. }, 34

39 The Neumann problem As a main step towards solvability of (N 2 ) s, we want to show the estimate sup S L f(, t) L 2 (R n ) f L 2 (R n ). (23) t>0 This is the sup-on-slices estimate. To obtain it, we want to show the following estimates, which together imply it. Tr < S : S L f(, t) L2 (R n ) t 2 2 t S L f + t 2 t S L f, The square function estimates: for each m 1, t m t m+1 S L f f L 2 (R n ). 35

40 The Neumann problem As a main step towards solvability of (N 2 ) s, we want to show the estimate sup S L f(, t) L 2 (R n ) f L 2 (R n ). (23) t>0 This is the sup-on-slices estimate. To obtain it, we want to show the following estimates, which together imply it. Tr < S : S L f(, t) L2 (R n ) t 2 2 t S L f + t 2 t S L f, The square function estimates: for each m 1, t m t m+1 S L f f L 2 (R n ). 35

41 The square function estimate Apply a vector-valued T b theorem (see Grau de la Herrán and Hofmann [GH16]). {Θ t } vector-valued family, has uniform L 2 bounds, off-diagonal decay in L 2, and quasi-orthogonality on a subspace H of L 2. {b Q } family of matrix-valued functions with components in H, indexed in dyadic cubes Q, satisfying ˆ R n b Q (x) 2 dx C 0 Q, ˆ l(q) 0 ˆ Q ( δ ξ 2 Re ξ Θ t b Q (x) 2 dx dt t C 0 Q, ) b Q dxξ, ξ C m. Q Then [GH16] Θ t f f L 2 (R n ), f H. 36

42 Tr< S S L f(, t) L 2 (R n ) t 2 2 t S L f + t 2 t S L f. Use integration by parts, Green s formula, adjoint relations of the layer potentials, and the square function estimates. If u Y 1,2 (R n+1 + ) solves Lu = 0 in R n+1 +, then Green s formula: u = D L,+ (Tr 0 u) + S L ( ν L,+ u) in R n+1 +. ( The double layer potential: D L,+ t = L,+ ν, t S L ), t > 0 t t S L h h 2, for all h Cc (R n ) n+1. 37

43 Endgame From the sup-on-slices estimate and the square function estimate, as well as the well-posedness of (N 2 ) for L 0 = diva and its adjoint, we can deduce invertibility of S L and associated operators through the jump relations, allowing us to conclude solvability of the Neumann problem for L = div(a + b 1 ) + b 2 + V. We may similarly deduce solvability of the Dirichlet and Regularity problems (D 2 ) s, (R 2 ) s. 38

44 Thanks for listening! Thank you. :) 39

45 References I Pascal Auscher and Andreas Axelsson. Weighted maximal regularity estimates and solvability of non-smooth elliptic systems I. In: Invent. Math (2011), pp url: (cit. on p. 29). M. Angeles Alfonseca, Pascal Auscher, Andreas Axelsson, Steve Hofmann, and Seick Kim. Analyticity of layer potentials and L 2 solvability of boundary value problems for divergence form elliptic equations with complex L coefficients. In: Adv. Math (2011), pp url: (cit. on pp. 29, 33). 40

46 References II Pascal Auscher, Andreas Axelsson, and Steve Hofmann. Functional calculus of Dirac operators and complex perturbations of Neumann and Dirichlet problems. In: J. Funct. Anal (2008), pp url: (cit. on p. 29). Pascal Auscher, Andreas Axelsson, and Alan McIntosh. Solvability of elliptic systems with square integrable boundary data. In: Ark. Mat (2010), pp url: (cit. on p. 29). Douglas Arnold, Guy David, Marcel Filoche, David Jerison, and Svitlana Mayboroda. Localization of eigenfunctions via an effective potential. Preprint. January arxiv: (cit. on p. 21). 41

47 References III Shmuel Agmon. Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of N-body Schrödinger operators. Vol. 29. Mathematical Notes. Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1982, p. 118 (cit. on p. 8). Pascal Auscher, Steve Hofmann, Michael Lacey, Alan McIntosh, and Ph. Tchamitchian. The solution of the Kato square root problem for second order elliptic operators on R n. In: Ann. of Math. (2) (2002), pp url: (cit. on p. 28). Ariel Barton. Elliptic partial differential equations with almost-real coefficients. In: Mem. Amer. Math. Soc (2013), pp. vi+108. url: https: //doi.org/ /s (cit. on p. 28). 42

48 References IV Ariel Barton. Layer potentials for general linear elliptic systems. In: Electron. J. Differential Equations (2017), Paper No. 309, 23 (cit. on pp. 37, 38). Besma Ben Ali. Maximal inequalities and Riesz transform estimates on L p spaces for magnetic Schrödinger operators I. In: J. Funct. Anal (2010), pp url: (cit. on p. 20). Luis A. Caffarelli, Eugene B. Fabes, and Carlos E. Kenig. Completely singular elliptic-harmonic measures. In: Indiana Univ. Math. J (1981), pp url: (cit. on p. 27). 43

49 References V Björn E. J. Dahlberg. Estimates of harmonic measure. In: Arch. Rational Mech. Anal (1977), pp url: (cit. on p. 27). Martin Dindos, Stefanie Petermichl, and Jill Pipher. The L p Dirichlet problem for second order elliptic operators and a p-adapted square function. In: J. Funct. Anal (2007), pp url: (cit. on p. 30). Eugene B. Fabes, David S. Jerison, and Carlos E. Kenig. Necessary and sufficient conditions for absolute continuity of elliptic-harmonic measure. In: Ann. of Math. (2) (1984), pp url: (cit. on p. 29). 44

50 References VI R. A. Fefferman, C. E. Kenig, and J. Pipher. The theory of weights and the Dirichlet problem for elliptic equations. In: Ann. of Math. (2) (1991), pp url: (cit. on p. 29). Ana Grau de la Herrán and Steve Hofmann. A local T b theorem with vector-valued testing functions. In: Some topics in harmonic analysis and applications. Vol. 34. Adv. Lect. Math. (ALM). Int. Press, Somerville, MA, 2016, pp (cit. on p. 41). Fritz Gesztesy, Steve Hofmann, and Roger Nichols. On stability of square root domains for non-self-adjoint operators under additive perturbations. In: Mathematika 62.1 (2016), pp url: (cit. on p. 32). 45

51 References VII François Germinet and Abel Klein. Operator kernel estimates for functions of generalized Schrödinger operators. In: Proc. Amer. Math. Soc (2003), pp url: (cit. on p. 19). Steve Hofmann, Carlos Kenig, Svitlana Mayboroda, and Jill Pipher. Square function/non-tangential maximal function estimates and the Dirichlet problem for non-symmetric elliptic operators. In: J. Amer. Math. Soc (2015), pp url: https: //doi.org/ /s (cit. on p. 28). 46

52 References VIII Steve Hofmann, Carlos Kenig, Svitlana Mayboroda, and Jill Pipher. The regularity problem for second order elliptic operators with complex-valued bounded measurable coefficients. In: Math. Ann (2015), pp url: (cit. on p. 28). Steve Hofmann and John L. Lewis. The Dirichlet problem for parabolic operators with singular drift terms. In: Mem. Amer. Math. Soc (2001), pp. viii+113. url: (cit. on p. 30). 47

53 References IX Steve Hofmann, Svitlana Mayboroda, and Mihalis Mourgoglou. Layer potentials and boundary value problems for elliptic equations with complex L coefficients satisfying the small Carleson measure norm condition. In: Adv. Math. 270 (2015), pp url: (cit. on p. 29). David S. Jerison and Carlos E. Kenig. The Dirichlet problem in nonsmooth domains. In: Ann. of Math. (2) (1981), pp url: (cit. on p. 28). David S. Jerison and Carlos E. Kenig. The Neumann problem on Lipschitz domains. In: Bull. Amer. Math. Soc. (N.S.) 4.2 (1981), pp url: https: //doi.org/ /s (cit. on p. 27). 48

54 References X C. Kenig, H. Koch, J. Pipher, and T. Toro. A new approach to absolute continuity of elliptic measure, with applications to non-symmetric equations. In: Adv. Math (2000), pp url: (cit. on p. 28). Carlos E. Kenig and Jill Pipher. The Dirichlet problem for elliptic equations with drift terms. In: Publ. Mat (2001), pp url: (cit. on p. 30). Carlos E. Kenig and Jill Pipher. The Neumann problem for elliptic equations with nonsmooth coefficients. In: Invent. Math (1993), pp url: (cit. on pp. 28, 29). 49

55 References XI Carlos E. Kenig and Jill Pipher. The Neumann problem for elliptic equations with nonsmooth coefficients. II. In: Duke Math. J (1995). A celebration of John F. Nash, Jr., (1996). url: (cit. on p. 29). Carlos E. Kenig and David J. Rule. The regularity and Neumann problem for non-symmetric elliptic operators. In: Trans. Amer. Math. Soc (2009), pp url: (cit. on p. 28). Seick Kim and Georgios Sakellaris. Green s function for second order elliptic equations with singular lower order coefficients. Preprint. December arxiv: (cit. on pp. 34, 35). 50

56 References XII Kazuhiro Kurata and Satoko Sugano. A remark on estimates for uniformly elliptic operators on weighted L p spaces and Morrey spaces. In: Math. Nachr. 209 (2000), pp (cit. on p. 20). Kazuhiro Kurata. An estimate on the heat kernel of magnetic Schrödinger operators and uniformly elliptic operators with non-negative potentials. In: J. London Math. Soc. (2) 62.3 (2000), pp url: (cit. on p. 8). Svitlana Mayboroda and. Exponential decay estimates for fundamental solutions of Schrödinger-type operators. Preprint. January arxiv: (cit. on p. 5). 51

57 References XIII Georgios Sakellaris. Boundary Value Problems in Lipschitz Domains for Equations with Drifts. Thesis (Ph.D.) The University of Chicago. ProQuest LLC, Ann Arbor, MI, 2017, p url: openurl?url_ver=z &rft_val_fmt=info: ofi/fmt:kev:mtx:dissertation&res_dat=xri: pqm&rft_dat=xri:pqdiss: (cit. on p. 30). Zhong Wei Shen. On the Neumann problem for Schrödinger operators in Lipschitz domains. In: Indiana Univ. Math. J (1994), pp url: (cit. on p. 30). 52

58 References XIV Zhong Wei Shen. L p estimates for Schrödinger operators with certain potentials. In: Ann. Inst. Fourier (Grenoble) 45.2 (1995), pp url: http: // 45_2_513_0 (cit. on p. 8). Zhongwei Shen. Estimates in L p for magnetic Schrödinger operators. In: Indiana Univ. Math. J (1996), pp url: (cit. on p. 8). Zhongwei Shen. On fundamental solutions of generalized Schrödinger operators. In: J. Funct. Anal (1999), pp url: (cit. on p. 8). 53

59 References XV Gregory Verchota. Layer potentials and regularity for the Dirichlet problem for Laplace s equation in Lipschitz domains. In: J. Funct. Anal (1984), pp url: (cit. on p. 27). 54