The Erwin Schrödinger International Boltzmanngasse 9 Institute for Mathematical Physics A-1090 Wien, Austria

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1 ESI The Erwin Schrödinger International Boltzmanngasse 9 Institute for athematical Physics A-1090 Wien, Austria Regularity of the nodal sets of solutions to Schrödinger equations. Hoffmann-Ostenhof T. Hoffmann-Ostenhof N. Nadirashvili Vienna, Preprint ESI 36 (1993) July 9, 1993 Supported by Federal inistry of Science and Research, Austria

2 REGULARITY OF THE NODAL SETS OF SOLUTIONS TO SCHRÖDINGER EQUATIONS. Hoffmann-Ostenhof 1 T. Hoffmann-Ostenhof 2,3 N. Nadirashvili 3 Institut für athematik, Universität Wien 1 Institut für Theoretische Chemie, Universität Wien 2 International Erwin Schrödinger Institute for athematical Physics 3 1. Introduction Let Ω be an open set in R n and let V : Ω R be real valued with V L 1 loc (Ω). We consider real valued solutions u 0 which satisfy (1.1) u = V u in Ω in the distributional sense. In a recent paper two of us [HO2] investigated the local behaviour of such solutions under rather mild assumptions on the potential V, namely we assumed that V K n,δ (Ω) for some δ > 0, see e.g. [AS,S], where the class K n,δ is defined by requiring that V (y) (1.2) lim sup χ Ω dy = 0 ε 0 x R n x y n 2+δ x y <ε Here χ Ω denotes the characteristic function of Ω. One of our main results was Theorem 1.1. Suppose u 0 is a real valued solution of (1.1). Let x 0 Ω then either there is a harmonic homogenous polynomial P 0 of degree such that (1.3) u(x) = P (x x 0 ) + Φ(x) with (1.4) Φ(x) = O( x x 0 +min(1,δ ) ) δ < δ for x x 0 or u vanishes at x 0 faster than polynomially, that is lim x x 0 α u(x) = 0 x x 0 Typeset by AS-TEX Typeset by AS-TEX

3 2. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, N.Nadirashvili for every α > 0. It is known [AS,S] that V K n,δ (Ω), δ < 1 implies u C 0,δ (Ω) where C 0,δ denotes Hölder continuity. Suppose u has a zero of first order at x 0 so that u = P 1 (x x 0 ) + Φ in a neighbourhood of x 0 according to Theorem 1.1. (1.3) and (1.4) implies that for δ < δ 1 u(x) P 1 (x x 0 ) lim = 0 x x 0 x x 0 1+δ so that u is at x 0 smoother than at points for which u 0. So the question arises whether the zero sets of solutions of Schrödinger equations are in fact smoother than the corresponding solutions. Let us illustrate this with an explicit example. According to the theorem of Cauchy and Kowalewski there is a small disk B ρ = {(x, y) R 2 : x 2 + y 2 < ρ 2 } such that with v = v in B ρ v = x y x3 1 2 x2 y + higher order terms and with v(0, y) = y, v x (0, y) = 1. Now let in B ρ, u be defined by { x y for x 0 u = v for x > 0 then u = V u with { 1 x > 0 V = V (x, y) = 0 x 0 and a simple calculation shows that the nodal line of u is given by y = x for x 0 y = x 1 3 x3 + O(x 5 ) for x > 0. Hence y(x) has a second derivative and one sided third derivatives. But u itself already has a jump in the second derivative for every (x, y) with y = 0 and x 0. Results on this additional regularity of nodal sets together with some proofs will be presented in this announcement, the full paper will appear elsewhere. 2. Regularity of nodal sets Without loss of generality we consider (1.1) in B R0 we assume V K n,δ (B R0 ). Let = {x R n : x < R 0 } and (2.1) N u = {x B R0 : u = 0}

4 Nodal sets 3 and let N u (1) = {x N u : u vanishes of first order at x} so that for each x 0 N u (1) there is a P (x0) 1 (x x 0 ) 0 with (2.2) u(x) = P (x0) 1 (x x 0 ) + Φ(x) for x x 0 according to Theorem 1.1. Theorem 2.1. Pick x 0 N u (1) and assume δ 1. Then for each δ < δ and for sufficiently small ε > 0, N u (1) B ε (x 0 ) is a C 1,δ -hypersurface. Remarks. (i) By a C 1,δ hypersurface we mean that N u (1) B ε (x 0 ) can be represented as the graph of a C 1,δ (R n 1 ) function. (ii) Theorem 2.1 is sharp in the sense that δ > δ is not possible. We do not know whether δ = δ might be allowed. (iii) We shall later discuss the case of smoother potentials, say V K n,δ (Ω) for δ (1, 2) or V C k,α (Ω). C k,α denotes the usual Hölder spaces. Sketch of the proof. We first state a Lemma which is a sharpening of Theorem 1.1. Lemma 2.1. Let x 0 N u B R1 (0) with R 1 = R 0 /2 and suppose sup x BR0 u = C 1 then for every δ < δ, there is a C 2 such that for x < R 0 (2.3) u(x) P (x0) 1 (x x 0 ) C 2 x x 0 1+δ, where C 2 = C 2 (V, C 1, δ δ, n, R 0 ) Remark. The V -dependence can be made explicit via a suitable norm of V. The important fact is that C 2 does not depend on x 0. If x 0 happens to be a higher order zero of u then P (x0) 1 (x x 0 ) 0. The proof of Lemma 2.1 relies heavily on the techniques which have been developed in [HO2] in order to prove Theorem 1.1. Some additional technical complications arise, causing however no entirely new problems. Naturally a complete proof is, as already the proof of Theorem 1.1 somewhat involved. Now define for x 0, x 1 N u (2.4) ( u)(x i ) = ( P (xi) )(x i ) = a i, i = 0, 1 then it can be shown via Lemma 2.1 Lemma 2.2. Let x 0, x 1 N u B R2, R 2 = R 0 /4 then for δ < δ there is a constant C 3 such that (2.5) a 1 a 2 C 3 x 0 x 1 δ. with C 3 = C 3 (C 2 ), C 2 the constant given according to (2.3). For later purposes we prove the following more general statement.

5 4. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, N.Nadirashvili Lemma 2.2. Suppose P (0) and P (1) are polynominals of degree and that P (0) (1) (x) P (x x 1) c x 1 +δ for x 2 x 1. Then for every -th partial derivative there exists a constant C(, n) such that (P (0) (1) (x) P (x x 1)) cc(, n) x 1 δ Proof. Let Q (x) = P (0) (1) (x) P (x x 1). In the one-dimensional case Q (x) C x 1 +δ implies via a classical inequality of Chebyshev that d dx Q (x) C x 1 δ To obtain the corresponding estimate for the n-dimensional case we consider directional derivatives of -th order. There the one-dimensional estimate obviously holds. The partial derivatives can then be estimated by linear combinations of the directional derivatives [Bo]. We shall now proceed in the following way: we assume that x 0 N u (1) and that (2.6) ( u)(x 0 ) = (λ, 0,..., 0) λe 1, λ 0. We shall first show that in a neighbourhood of x 0 the nodal set u(x) = 0 can be represented as the graph of a uniquely determined continuous function such that ϕ : B γ (πx 0 ) R n 1 R (2.7) u(ϕ(y), y) = 0 y B γ (πx 0 ). thereby π denotes the orthogonal projection from R n to S := {x R n x 1 = 0} and B γ (πx 0 ) = {y S πx 0 y < γ} with γ > 0 small enough. Proposition 2.1. For sufficiently small ρ > 0 is injective. π : N u B ρ (x 0 ) S Proof. We start with some rather obvious observations and definitions. Since u(x 0 ) = 0, x 0 G where G = {x B R2 : u(x) > 0}. Let x G. We say ε x is an n 1 dimensional affine hyperplane to G at x if for every sequence of points x i G with x i x dist(x i, ε x ) = o( x i x ) for i.

6 Nodal sets 5 Let y G and d y (x) := (x y, ( u)(y)) then ε x0 = {x R n : d x0 (x) = 0}. Now define for x G N u (1) ( u)(x) = ( u)(x) ( u)(x) then by Lemma 2.2 it is straight forward to show that for small ρ (2.8) (( u)(x 0 ), ( u)(x)) > 1 2 for x N u B ρ (x 0 ) with B ρ (x 0 ) = {x R n : x x 0 ρ }. Set ρ = ρ in Proposition 2.1. Now suppose that Proposition 2.1 is wrong. Then there exist x, x B ρ (x 0 ) N u, x x such that π( x) = π(x) := x = (0, x 2, x 3,..., x n ). Let E = π 1 ( x) B ρ (x 0 ) N u. E is a closed set with cardinality 2. If E has an accumulation point y then π 1 ( x) ε y, but ε y = {x R n : (x y, ( u)(y)) = 0} and this implies (( u)(x 0 ), ( u)(y)) = 0 contradicting (2.8). So there are z 1, z 2 E where z 1, z 2 only differ in the first coordinate such that g(z 1, z 2 ) = {x R n : x = tz 1 + (1 t)z 2, t (0, 1)} satisfies g(z 1, z 2 ) N u = implying u(x) 0 and x g(z 1, z 2 ). But (( u)(x 0 ), ( u)(z j )) = u x 1 (x 0 ) u x 1 (z j ) sgn u x 1 (z 1 ) sgn u x 1 (z 2 ), j = 1, 2, hence sgn (( u)(x 0 ), ( u)(z 1 )) sgn (( u)(x 0 ), ( u)(z 2 )) again contradicting (2.8). This proves the proposition. Now we have to show the y B γ (πx 0 ), γ small enough, there is a t R such that (t, y) B ρ (x 0 ) N u. But this is an immediate consequence of the continuity of u : Denote x 0 = (x 01,..., x 0n ), let y B γ and x ± := (x 0,1 ± ε, y) such that x ± B ρ (x 0 ). Since u(x 0 ) = λe 1 and u(x 0 ) = 0, sgn u(x 0,1 + ε, x 0,2... x 0,n ) sgn u(x 0,1 ε, x 0,2... x 0n ) for ε small enough. Since u is continuous we also have y B γ, for γ sufficiently small, sgn u(x 0,1 + ε, y) sgn u(x 0,1 ε, y). Hence by the intermediate value theorem there is a t with t x 0,1 < ε such that u(t, y) = 0. This implies (2.7). Further via an implicit function theorem [H, Satz 170.1] (which is not standard but ideal for our purposes) we conclude that ϕ C 1,δ (B γ ) so that N u (1) is indeed locally the graph of a C 1,δ function. We give now a brief discussion of the higher regularity of nodal sets if V is assumed to be more regular. Theorem 2.2. (i) Suppose (1.1) holds with V K n,δ (Ω) with δ (1,2]. Then N u (1) is locally a C 2,δ -hypersurface for δ < δ 1. (ii) Suppose (1.1) holds with V C k,α (Ω) with α (0, 1) then N u (1) is locally a C k+3,α -hypersurface. The idea of the proof is basically the same as for the proof of Theorem 2.1. However we have to replace Theorem 1.1 by a more detailed result.

7 6. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, N.Nadirashvili Theorem (i) Suppose (1.1) holds and V K n,δ (Ω), δ (1, 2]. Let x 0 Ω then either there exist two harmonic homogenous polynomials P 0, P +1 of degree, + 1 respectively such that with u(x) = P (x x 0 ) + P +1 (x x 0 ) + Φ(x) Φ(x) = O( x x 0 +δ ) δ < δ for x x 0 or u vanishes at x 0 faster than polynomially. (ii) Suppose (1.1) holds and V C k,α (Ω), α (0, 1). Let x 0 Ω, then there is a polynomial of degree + k + 2 such that p(x) = P (x) + P +1 (x) + p 1 with P, P +1 again harmonic and homogenous of degree, + 1 respectively and P 0, and p 1 (x) vanishes at least of order + 2 at zero. We have then u(x) = p(x x 0 ) + Φ(x x 0 ) with Φ(x) = O( x ) +k+2+α Remarks. (a) Under the conditions of part (ii) of this theorem strong unique continuation is well known. Also (ii) is related to the classical Schauder estimates, see e.g. [GT]. (b) The proof of Theorem 2.3 again uses the techniques of [HO2] but some iterations are necessary. (c) For the coulombic case a more detailed version of Theorem 1.1 was recently shown in [HO2 S1] and [HO2 S2]. An investigation of the regularity of the nodal sets for this important case should be possible along the present lines. Starting from Theorem 2.3 the proof of Theorem 2.2 follows the same ideas as the proof of Theorem 2.1. So one first proves a suitable analog of Lemma 2.1, uses Lemma 2.2 and proceeds essentially as we did above in order to prove Theorem 2.1. References [AS]. Aizenman, B. Simon,, Brownian motion and Harnack inequality for Schrödinger operators, Commun Pure Applied athematics 35 (1982), [BO] J. Boman, Differentiability of a function and of its composition with functions of one variable, ath. Scand. 20 (1967), [GT] D.Gilbarg, N.S.Trudinger, Elliptic partial differential equations, Springer, Berlin, [HO2]. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, Local properties of solutions of Schrödinger equations, Comm. PDE 17 (1992), [HO2S1].Hoffmann-Ostenhof, T.Hoffmann-Ostenhof and H.Stremnitzer, Electronic wavefunctions near coalescence points, Phys. Rev. Letters 68 (1992), [HO2S2]. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof and H. Stremnitzer, Local properties of Coulombic wavefunctions, submitted for publication.

8 Nodal sets 7 [H] H. Heuser, Lehrbuch der Analysis Teil 2, B. G. Teubner, Stuttgart, [S] B. Simon, Scrödinger semigroups, Bull. Am. ath. Soc. 7 (1982), Hoffmann-Ostenhof: Institut f ür athematik, Universit ät Wien, Strudlhofgasse 4, A-1090 Wien, Austria T. Hoffmann-Ostenhof: Institut f ür Theoretische Chemie, Universit ät Wien, Währinger Strasse 17, A-1090 Wien, Austria and International Erwin Schr ödinger Institute for athematical Physics, Pasteurgasse 6/7, A-1090 Wien, Austria address: hoho@itc.univie.ac.at N. Nadirashvili: International Erwin Schr ödinger Institute for athematical Physics, Pasteurgasse 6/7, A-1090 Wien, Austria and Institute of Earth Physics, oscow, Russia