mechanics, starting from the idea that a free particle, i.e. subject to no outer force eld, will behave like a plane wave and can therefore be describ

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1 SELF-ADJOINTNESS OF SCHR ODINGER OPERATORS Andreas M. Hinz Center for Mathematics Technical University Munich, Germany Abstract Starting from the example of solving the Schrodinger equation, the concept of (essential) self-adjointness of a linear operator in a Hilbert space is developed. Some general criteria to prove this quality are presented and applied to Schrodinger operators of the form 4 + V in L 2 R d. Keywords: Schrodinger equation, Hilbert space, self-adjointness, Schrodinger operator 0 Introduction: solving the Schrodinger equation The Schrodinger equation (cf. (t; x) (4 + V (x)) (t; x) (0) may certainly be counted among the most important pieces of physics and mathematics in the twentieth century. It is rooted in the wave model of quantum ca.m.hinz

2 mechanics, starting from the idea that a free particle, i.e. subject to no outer force eld, will behave like a plane wave and can therefore be described by a wave function of the form 8 t 2 R 8 x 2 R d : (t; x) A e 2i(kxt) ; which propagates with constant speed jkj 2 ]0; 1[ into direction k 2 R d n f0g; d 2 N. (A 2 C n f0g is a normalization constant.) Using only the most fundamental physical laws of quantum theory, namely Einstein's and de Broglie's equations, one arrives at (0) if one wants to determine the time development of starting from some initial state 0(x) : (0; x): has to full a dierential equation of rst order in t, which is linear because of the superposition principle for waves. Therefore it is also evident that wave functions have to be complex valued. The Laplacian 4, acting on the space variable x only, represents kinetic energy, while the so-called potential (function) V in (0) embodies the potential energy induced by an external force eld and has therefore to be real-valued. As an example, one might think of an electron subject to the coulombic force of a charged nucleus. The classical approach to nd a non-trivial solution of equation (0) is separation of variables, i.e. the ansatz (t; x) f(t) u(x). Then, for a (t 0 ; x 0 ) 2 R 1+d with (t 0 ; x 0 ) 6 0, we have f(t) (t; x 0) u(x 0 ) ; u(x) (t 0; x) f(t 0 ) ; whence f 2 C 1 (R), u 2 C 2 R d and t f 0 u; 4 etc.), such that f 4u (we write i f 0 u f (4 + V )u: (1) Putting x x 0, we see that f fulls the ordinary dierential equation f 0 4u(x0 ) (t) i V (x 0 ) f(t) : if(t); u(x 0 ) whose general solution is f(t) c exp(it) with some c 2 C n f0g. If we insert this into (1), we nd that u has to full 8 x 2 R d : 4u(x) + V (x) u(x) u(x): (2) 2

3 Since jf(t)j jcj exp (im()t), the time evolution as given by f is bounded if and only if the eigenvalue is real. There are only a few cases, like e.g. the harmonic oscillator, where there are (suciently many) classical eigensolutions u of (2). In particular in view of possible singularities of the potential function V as in the Coulomb case, we are forced to extend the notion of solution, based on the following observation: let C 1 0 R d : ' 2 C 1 R d ; supp(') is bounded ; where the support of ' is dened by supp(') fx 2 R d ; '(x) 6 0g. Then for every u 2 C d 2 R for which V u is locally integrable, i.e. integrable after multiplication with any test function ' 2 C d 1 0 R, we have by partial integration: 8 ' 2 C d 1 0 R : f4u(x) + (V (x) ) u(x)g '(x) dx such that for these u, equation (2) is equivalent to 8 ' 2 C 1 0 R d : u(x) f4'(x) + (V (x) ) '(x)g dx; u(x) f4'(x) + (V (x) ) '(x)g dx 0: (3) As there is no regularity requirement on u in (3), we call every non-trivial locally integrable u for which V u is locally integrable as well and which fullls (3) a weak eigensolution for eigenvalue. We now have to nd a suitable function space H, in which 4 + V acts as an operator. For the sake of linearity, H has to have the canonical algebraic structure of a vector space over C. We dene D(S) u 2 H; 9 v 2 H 8 ' 2 C 1 0 R d : u(x) f4'(x) + V (x) '(x)g dx 8 u 2 D(S) : S(u) v: v(x) '(x) dx ; This denes a linear operator in H, i.e. D(S) is a linear subspace of H and 8 u; v 2 D(S); 2 C : S(u + v) Su + Sv: 3

4 (For linear operators, the brackets for the argument are usually omitted, if no ambiguity is possible.) Let us suppose that we have suciently many weak eigensolutions e n 2 H n f0g for the eigenvalues n 2 R (they are then eigenfunctions of S, i.e. Se n n e n ), such that any u 2 H can be written as u with 2 C N 0 and accordingly (t; ) n e n exp (i n t) n e n exists in H for all t 2 R. This presupposes a metric structure on H, compatible with the algebraic one, which can only be achieved by a norm k k on H. Then we have (formally) for h 6 0 and setting f n (t) exp (i n t): 1 X 1 h f (t + h; x) (t; x)g fi n exp (i n t)g n e n (x) fn (t + h) f n (t) f 0 h n(t) n e n (x); and it would be desirable to have the right hand side tend to 0 as h! 0. Alas, this can not be expected in general! The way out is to assume further that the e n are mutually orthogonal. This necessitates the introduction of a compatible geometric structure in the form of an inner product s in H, which thus will become a unitary space, where two vectors x and y are called orthogonal, i s(x; y) 0. Then there is something like the theorem of Pythagoras, namely u n e n ) kuk 2 j n j 2 ; where we have assumed that ke n k 1 for all n. By the properties of an inner product we then have 8 m 2 N 0 : s(u; e m ) s n e n ; e m! n s(e n ; e m ) m : Writing h; i for s(; ), as is common practice, we obtain 1 X 1 2 h f (t + h; ) (t; )g fi n exp (i n t)g hu; e n i e n f n(t + h) f n (t) f 0 h n(t) 2 jhu; e n ij 2 : (4) 4

5 Now the convergence of the right hand side as h! 0 can be investigated with the aid of the following lemma. Lemma 0.1 Let (a nm ) m2n0 n2n 0 be a sequence of null sequences in C, with Then 8 n 2 N 0 8 m 2 N 0 : ja nm j b n and a nm! 0 as m! 1. Proof. For " > 0 choose N 2 N 0 8 m M 8 n 2 f0; : : : ; Ng : ja nm j < such that nn +1 b n < 1: b n < " 2 and then M 2 N 0 with " 2(N + 1). As f n(t + h) f n (t) f 0 h n(t) 2j nj, the right hand side of (4) tends to 0 for every t 2 R, if j n j 2 jhu; e n ij 2 < 1. We will see that this is the case if and only if the sum in the left hand side of (4) converges for every t 2 R to some t(t; ) 2 H. ( (t; ) will then be called dierentiable in H with respect to t 2 R. Rules from classical calculus can be carried over, like e.g. the product rule h; i 0 h 0 ; i + h; 0 i.) This will require the unitary space (H; s) to be complete, i.e. H to be a Hilbert space. Therefore, by Fischer's theorem (cf. [3]), H L 2 R d is the appropriate home for the Schrodinger operator S. This also leads to the probabilistic interpretation of the wave function. The operator T in H, which assigns the image T u to every u 2 D(T ) because So ( v 2 H; 8 u; v 2 D(T ) : hu; T vi * u; 2 n jhv; e n ij 2 < 1 n hv; e n ie n + n hv; hu; e n ie n i * is the unique solution of the initial value problem ) n hu; e n ie n 2 H, is (formally) symmetric, n hv; e n i hu; e n i n hu; e n ie n ; v (0; ) u; 8 t 2 R : i t(t; ) T (t; ); 5 + ht u; vi:

6 where uniqueness follows from k k 2 0 h ; i 0 h 0 ; i + h ; 0 i hit ; i + h ; it i i fht ; i h ; T ig 0; because then 0, if u 0. Finally, T S, because for any u 2 D(T ) and ' 2 C 1 0 T u(x)'(x) dx For u 2 D(T ), the above hu; e n i hu; e n i n hu; e n ie n (x)! n e n (x)'(x) dx '(x) dx R d, we have e n (x) f4'(x) + V (x)'(x)g dx u(x) f4'(x) + V (x)'(x)g dx: is therefore also a solution of i t (t; ) S (t; ). If D(S) 6 D(T ), there could be other solutions for the corresponding initial value problem. Uniqueness, however, can be guaranteed, if S T, which means in particular that S has to be (formally) symmetric too. Unfortunately, the domains of both S and T are given implicitly only and depend on properties of V. It is not even clear, if they contain a substantial set, namely a dense subspace of H. However, if we assume V ' 2 H for all ' 2 C 1 0 to local square integrability of V ), it is obvious that C 1 0 R d (this is equivalent R d D(S) and that S 0 ' 4' + V ' for the minimal Schrodinger operator S 0 : S C 1 0 R d. Conversely, by the denition of D(S), the maximal Schrodinger operator S is the adjoint operator of S 0 : S S 0. The symmetry of S 0 means that S 0 has exactly one self-adjoint extension, namely S S. (S 0 is then called essentially self-adjoint.) We will now make these notions more precise in Section 1 and show that essential self-adjointness of the minimal operator is also sucient for the unique solvability of the initial value problem for the Schrodinger equation, i.e. we will be able to prove the following. 6

7 Theorem 0.1 Let fe n ; n 2 N 0 g be an orthonormal basis of H consisting of eigenfunctions e n for eigenvalues n of S and assume that S 0 is essentially self-adjoint. Then for every u 2 D(S) the unique solution of the initial value problem for the Schrodinger equation (0; ) u; 8 t 2 R : i t(t; ) S (t; ); is given by 8 t 2 R 8 x 2 R d : (t; x) exp(i n t)hu; e n ie n (x): Self-adjointness will also be the key to solve the problem even if there is no orthonormal basis of H consisting of eigenfunctions of S. 1 Linear operators in Hilbert space Properties of operators reect the algebraic, metric and geometric structure of a Hilbert space H. Linearity is associated with the vector space, boundedness, closability and closedness with the norm, symmetry with the inner product and nally self-adjointness with completeness. Although these properties can be characterized in the corresponding more general settings, we will, for simplicity, concentrate on operators T H 2, i.e. dened on some subset D(T ) of H and with values in H. In view of our application to the solution of the Schrodinger equation, we will also limit our eorts to a non-trivial H over the eld C. (Many of the results in this and the next section are valid for real Hilbert spaces, but some of the proofs present subtleties, which we do not want to address in this note.) We assume familiarity with the basic properties of Hilbert spaces (see, e.g., [12]). Denition 1.1 T is a linear operator in H, i T is a linear subspace of H 2 and T \ (f0g H) f(0; 0)g. The domain of T is D(T ) fu 2 H; 9 v 2 H : (u; v) 2 T g, and we write T u for v. 7

8 For 2 C, the subspace (T ) 1 (f0g) is called the eigenspace of (and T ); if this eigenspace is non-trivial, is called an eigenvalue of T, and every non-trivial element of the eigenspace is called an eigenvector (or eigenfunction, if H is a function space) for (and T ). The most handsome operators are those which are bounded. Lemma 1.1 If T is a linear operator in H, then T is continuous, i T is bounded, i.e. kt k : sup fkt uk; u 2 D(T ); kuk 1g < 1. Proof. If T is continuous, then there is a > 0 such that kt vk < 1 for every v 2 D(T ) with kvk < 2 and consequently kt uk 1 kt (u)k < 1 for every u 2 D(T ) with kuk 1. Conversely, if T is bounded and u; v 2 D(T ) with ku vk <, then kt u T vk kt (u v)k kt k ku vk kt k. Unfortunately, dierential operators in L 2 R d are not bounded in general and therefore we have to resign ourselves to closedness or even closability. Denition 1.2 A linear operator T in H is closable, i its closure T is a linear operator. T is called closed, i T T. Corollary 1.1 Let T be a bounded linear operator in H. Then T is closable, and T is the only bounded extension of T with domain D(T ). In particular, T is closed if and only if D(T ) is closed. Proof. Let D(T ) (u n ) n2n! u 2 D(T ). Then u 7! lim n!1 T u n denes the operator T on D(T ) D(T ): since kt u n T u N k kt k ku n u N k, (T u n ) n2n is a Cauchy sequence; moreover, the limit is independent of the choice of the sequence (u n ) n2n approximating u, as can be seen by observing that the images of the mixed sequence built from two such sequences converge too. Linearity of T is obvious. 8

9 Further, kt uk lim n!1 kt u n k lim n!1 kt k ku n k kt k kuk; such that kt k kt k; kt k kt k is trivial since T T. u 2 D If T e is a bounded extension of T with D et D(T ), then for every et, there is a sequence D(T ) (u n ) n2n! u, such that T u n T e un! T e u by continuity of e T, granted by Lemma 1.1. The operator S of section 0 is closed, as can be seen directly or by recourse to the fact that S S 0. Lemma 1.2 Let T be a linear operator in H. Then T : (u; v) 2 H 2 ; 8 ' 2 D(T ) : hu; T 'i hv; 'i denes a (closed) linear operator in H, called the adjoint of T, if and only if T is densely dened, i.e. D(T ) H. Proof. Obviously, T is a linear subspace of H 2, and its closedness follows from the continuity of the inner product. T is a linear operator if and only if D(T ) f0g. Corollary 1.2 Let T be a densely dened linear operator in H. Then T T ; in particular, T is closable if and only if T is densely dened, in which case T T. Proof. We have T T f(t '; ') 2 H 2 ; ' 2 D (T )g f(u; v) 2 H 2 ; 8 ' 2 D (T ) : hu; T 'i hv; 'ig T, and the equivalence follows from Lemma 1.2. T T T T. If we apply this to T, we get Another type of closable operators are symmetric operators. 9

10 Denition 1.3 A linear operator T in H is called symmetric, i T is densely dened and T T. Corollary 1.3 Let T be a symmetric operator in H. Then T is closable, and T is symmetric. Proof. By Corollary 1.2, T is closable and T T. As T T, and T is closed by Lemma 1.2, we get T T. Symmetric operators have other nice features. Lemma 1.3 Let T be a symmetric operator in H. Then a) 8 u 2 D(T ) : ht u; ui 2 R; in particular all eigenvalues of T are real. b) Eigenspaces for dierent eigenvalues of T are orthogonal. c) 8 2 C 8 u 2 D(T ) : k(t )uk jim()j kuk. Proof. a) ht u; ui hu; T ui ht u; ui. If u is an eigenvector for the eigenvalue, then kuk 2 hu; ui ht u; ui 2 R and consequently 2 R. b) Let e and f be eigenvalues with eigenvectors e and f. Then ( e f)he; fi ht e; fi he; T fi 0; whence from e 6 f f, we obtain he; fi 0. c) We may assume kuk 1. Then k(t )uk jh(t )u; uij jht re()u; ui i im()j jim()j; the latter since ht u; ui 2 R from (a). We are now ready for the decisive step to prove Theorem 0.1. Theorem 1.1 Let M be an orthonormal basis of H and 2 R M. Then ( X X D(T ) u 2 H; 2 e jhu; eij 2 < 1 ; T u e hu; eie; e2m e2m ) denes a self-adjoint operator, i.e. T T. 10

11 Proof. Obviously, D(T ) is a subspace of H, and D(T ) H since span(m) D(T ). The existence of T u is guaranteed by Fourier expansion in H, which also yields linearity of T. Moreover, by Parseval's identity, X X 8 u; v 2 D(T ) : hu; T vi hu; eih e v; ei e u; eihv; ei e2m e2mh ht u; vi; such that T is symmetric. If u 2 D (T ), then e hu; ei hu; T ei ht u; ei for every e 2 M D(T ) and consequently u 2 D(T ), whence T T. This theorem now completes the proof of Theorem 0.1, because the selfadjoint operator T cannot have a strict extension S which is symmetric, by virtue of T T S S T. We now try to liberate ourselves from the assumption of the existence of an orthonormal basis consisting of eigenfunctions. We observe that the operator T in Theorem 1.1 can be rewritten as ( X X D(T ) u 2 H; 2 kp uk 2 < 1 ; T u P u; 2R 2R ) where P u is the projection of u to the eigenspace of. (A projector X P is a symmetric operator with D(P ) H and P 2 P.) Here, the sum 2 kp uk 2 2R is built up in a monoton increasing way in countably many positive steps, if we let grow from 1 to 1. This suggests a generalization by replacing the sum with an integral. To this end, we observe further that the family (P ) 2R, being orthogonal, X i.e. P P 0 for 6, by Lemma 1.3b, generates, by putting E P, a spectral family (E ) 2R. Denition 1.4 A family (E ) 2R of projectors in H is called a spectral family, i it is non-decreasing, i.e. 8 8 u 2 H : he u; ui he u; ui, right-continuous, i.e. 8 2 R : E lim n!1 E + 1 n : E +, 11

12 and E 1 : lim n!1 E n 0; E 1 : lim n!1 E n 1. D(T ) Then, with E : lim E n!1 1, n ( u 2 H; X 2R 2 ke uk 2 ke uk 2 < 1 ) ; T u X 2R (E E ) u; or, using the Cauchy-Stieltjes integral, D(T ) u 2 H; 2 dke uk 2 < 1 ; 8 v 2 H : ht u; vi dhe u; vi: The famous spectral theorem of von Neumann (cf. [9]) states that the connection between spectral families and self-adjoint operators in H is not restricted to those with a complete set of eigenvectors. Theorem 1.2 Let (E ) 2R be a spectral family in H and f 2 C(R). Then D u 2 H; jf()j 2 dke uk 2 < 1 is dense in H and 8 v 2 H : hf(t )u; vi f() dhe u; vi denes a linear operator f(t ) in H with D (f(t )) D D (f(t ) ), 8 v 2 H : hf(t ) u; vi f() dhe u; vi; R and kf(t )uk 2 jf()j 2 dke uk 2 kf(t ) uk 2. For f(), the corresponding map from the set of spectral families to the set of self-adjoint operators in H is bijective. We will not attempt to give a proof of the spectral theorem here; for a convenient approach, see [8] (cf. also [12, Theorems 7.14,7.17]). In view of equation (4), we put f() exp(it), which yields the unitary operators U(t) : exp(it t). (A unitary operator U is a Hilbert space homomorphism, i.e. U : H! H is bijective and linear and 8 u; v 2 H : huu; Uvi hu; vi; by 12

13 the polarization identity, the latter property is equivalent to U being isometric, i.e. 8 u 2 H : kuuk kuk). As we will see, the U(t); t 2 R; are forming an evolution group. Denition 1.5 A family of operators (U(t)) t2r is called an evolution group, i 8 t 2 R : U(t) is a unitary operator on H, U(0) 1, 8 s; t 2 R : U(s + t) U(s)U(t), 8 u 2 H : s! t ) U(s)u! U(t)u. This notion is the basis for Stone's theorem (cf. [11]), which we will not prove either (cf. [12, Theorem 7.38]). Theorem 1.3 Let (U(t)) t2r be an evolution group in H and dene the operator A by D(A) fu 2 H; : U()u is dierentiable at 0g ; Au 0 (0): Then the operator ia is self-adjoint. We are now able to generalize Theorem 0.1 to every self-adjoint operator. Theorem 1.4 Let T be a self-adjoint operator in H. Then for every u 2 D(T ) the unique solution of the initial value problem (0) u; 8 t 2 R : i 0 (t) T (t); is given by (t) U(t)u exp(it t)u. 13

14 Proof. By virtue of Theorem 1.2 it is clear that U(t) exp(it t) is dened on all of H and isometric. Obviously, U(0) 1. Furthermore, for u; v 2 H and t 2 R, we have he U(t)u; vi hu(t)u; E vi exp(it)dhe u; E vi whence, for s 2 R, hu(s)u(t)u; vi 1 exp(is)d he U(t)u; vi exp(is)d 1 exp(it)dhe u; vi; exp(it)dhe u; vi exp(i(s + t))dhe u; vi hu(s + t)u; vi : Surjectivity of U(t) now follows from U(t)U(t) 1. Next we show that it u is the derivative of U()u at 0 for any u 2 D(T ). For h > 0 we have 1 2 h fu(h)u ug + it u exp(ih) 1 h + i For " > 0, R > 0 can be chosen independently of h, such that R 1 exp(ih) 1 + i h 2 dke uk R exp(ih) 1 + i h R 4 jj 2 dke uk With this R xed, we proceed further with R R exp(ih) 1 2 R + i h dke uk 2 h2 4 R4 R 1 R 2 dke uk 2 : 2 dke uk 2 jj 2 dke uk 2 < " 2 : dke uk 2 h2 4 R4 kuk 2 ; which is smaller than " 2 for suciently small h. To prove continuity of U()u for any u 2 H, we choose an approximating sequence D(T ) 3 u n! u and observe that ku(s)u U(t)uk ku(s)u U(s)u n k + ku(s)u n U(t)u n k + ku(t)u n U(t)uk 14

15 2ku u n k + ku(s)u n U(t)u n k; which can be made small by rst choosing n large and then using the fact already established that U()u n is dierentiable and consequently continuous. So we have proved that (U(t)) t2r is an evolution group and T ia for the operator A of Theorem 1.3. By that theorem, ia is self-adjoint, such that T ia. we have Now let : U()u for u 2 D(T ). Then (0) u is obvious. For t; h 2 R 1 (h) u h f (t + h) (t)g U(t) ; h and therefore, by continuity of the operator U(t), i 0 (t) iu(t)au iau(t)u T (t): Uniqueness of the solution can be shown based on symmetry of T as above for Theorem 0.1. Solving the Schrodinger equation (0) has now been entirely reduced to estabishing self-adjointness of the operator S of Section 0 or any property of the handier operator S 0 which in turn will guarantee self-adjointness of S. We will address the issue of suitable criteria in the following section. 2 Criteria for (essential) self-adjointness The equivalence of self-adjointness of the operator T and the existence of an evolution group (exp(it t)) t2r, as expressed by Theorems 1.3 and 1.4, suggests the following idea: representing exp(it t) formally by the compound interest formula lim 1 + it t n, we should have n!1 n H D T nt i 1 T n t i D(T ): This, in fact, leads to the fundamental criterion for self-adjointness. 15

16 Theorem 2.1 Let T be a symmetric operator in H. equivalent. o) T is self-adjoint, Then the following are i) 8 2 C n R : (T )D(T ) H, ii) 9 2 C : (T )D(T ) H (T )D(T ). Proof. Let T be self-adjoint and 2 C n R. Then is not an eigenvalue of T by Lemma 1.3a, whence T is injective. The operator (T ) 1 is bounded by virtue of Lemma 1.3c, and closed, since T is closed. By Corollary 1.1, the domain (T )D(T ) of (T ) 1 is closed as well. Furthermore, ((T )D(T )) ((T ) ) 1 (f0g) (T ) 1 (f0g) (T ) 1 (f0g) f0g; such that (T )D(T ) (T )D(T ) H. The inclusion from (i) to (ii) is trivial. Let 2 C with (T )D(T ) H (T )D(T ). Since T is symmetric, we only have to show that D (T ) D(T ) to obtain self-adjointness of T. For u 2 D (T ), there is a v 2 D(T ) D (T ) with (T )v (T ) u, since T maps D(T ) onto H. From T v T v it follows that (T ) (u v) (T ) u (T )v 0 and consequently u v 2 (T ) 1 (f0g) ((T )D(T )) f0g; that is u v 2 D(T ). In order to use Theorem 2.1, it is necessary to establish symmetry of the operator rst. This is, in general, not easy for an operator with maximal domain like S in Section 0. On the other hand, for a small, symmetric operator, like S 0, it is not evident that it has a self-adjoint extension at all. We will now introduce an important class of operators which do have self-adjoint extensions. 16

17 Denition 2.1 Let T be a symmetric operator in H. Then : inf fht u; ui; u 2 D(T ); kuk 1g is called the lower bound of T, and T is called semi-bounded (from below), i > 1. For semi-bounded operators, the existence of a distinguished self-adjoint extension, the Friedrichs extension can be proved. Theorem 2.2 In H let T be a semi-bounded operator with lower bound. Dene H T : u 2 H; 9 (u n ) n2n D(T ) : u n! u and 8 " > 0 9 N 2 N 8 n N : jht (u n u N ); u n u N ij < "g : Then T F : T (H T \ D (T )) denes a self-adjoint extension of T with lower bound. Sketch of proof. We may assume 1. Then (u; v) ht u; vi denes an inner product in D(T ) and every Cauchy sequence in (D(T ); ) is also a Cauchy sequence in (H; s), such that the completion of (D(T ); ) can be identied with (H T ; ). Since is continuous, we have 8 u 2 H T \ D (T ) 8 v 2 H T : (u; v) ht u; vi: As T T F, it follows that T F is symmetric and again by continuity of that it has the same lower bound as T. To establish self-adjointness of T F by Theorem 2.1, it suces to show that T F D (T F ) H. For u 2 H, ' 7! h'; ui denes a bounded linear functional on (H T ; ), and the representation theorem of Frechet and Riesz guarantees the existence of v 2 H T with h'; ui ('; v) for every ' 2 H T. In particular, for ' 2 D(T ) we have hv; T 'i hv; T 'i (v; ') hu; 'i, whence v 2 H T \ D (T ) D (T F ) and T F v T v u. 17

18 If a symmetric operator has a self-adjoint extension, it is not clear whether the latter, and consequently the corresonding spectral family, is unique. Denition 2.2 A densely dened operator T in H is called essentially selfadjoint, i it has exactly one self-adjoint extension. An obvious sucient condition for essential self-adjointness of T is the selfadjointness of its closure T, because then for every self-adjoint extension T e of T we have T T e T et T T, that is e T T. For symmetric T we can therefore employ the criteria of Theorem 2.1 as applied to T. In practice, this is not too easy, but it is unavoidable, because self-adjointness of T turns out to be also necessary for essential self-adjointness of T. The proof of the latter fact can be based on the following lemma. Lemma 2.1 Let T be a symmteric operator in H. Then U 7! e T with D et (U 1)H; e T i(u + 1)(U 1) 1 ; denes a bijection between the set of all unitary extensions U of (T i)(t + i) 1 and the set of all self-adjoint extensions e T of T. Proof. The domain of the mapping under consideration makes sense, because (T + i) is injective by Lemma 1.3a. Let U be a unitary extension of (T i)(t + i) 1. Then such that D(T ) D(e T ). 8 ' 2 D(T ) : ' (U 1) 1 2 i(t + i)' ; (5) U 1 is injective: let u 2 H with Uu u; then from (5) we get 8 ' 2 D(T ) : hu; 'i 1 i hu; (U 1)(T + i)'i 2 1 i fhuu; U(T + i)'i hu; (T + i)'ig 0; 2 18

19 the latter because U is unitary. Since D(T ) is dense in H, we get u 0. Again from (5) we deduce such that e T T. 8 ' 2 D(T ) : T e 1 ' (U + 1)(T + i)' T '; 2 et is symmetric, because for all u; v 2 H we have D(U 1)u; e T (U 1)v E i h(u 1)u; (U + 1)vi i (huu; vi hu; Uvi) Finally, such that whence also et + i D i (huu; Uvi + hu; Uvi huu; vi hu; vi) i h(u + 1)u; (U 1)vi 8 u 2 H : u et 8 u 2 H : U et i D et H and D et (U 1)u; (U 1)v E : et 1 + i (U 1) 2 iu ; (6) et + i (U 1)u et i (U 1)u; (7) H. By Theorem 2.1, e T is self-adjoint. Formula (7) implies that the inverse mapping must be given by U et i et + i 1 for any self-adjoint extension e T of T. In fact, the mappings e T i : D et! H are surjective by Theorem 2.1 and injective by Lemma 1.3a. U is isometric, since 8 ' 2 D et : T e i ' 2 e T ' 2 + k'k 2 and therefore 8 u 2 H : kuk 2 T e 1 et 2 + i u + et 2 + i 1 u kuuk 2 : 19

20 Finally, since e T is an extension of T, we have 8 ' 2 D(T ) : U(T + i)' U et + i ' et i ' (T i)': The mapping in question is surjective: since u 2 H : (U 1)u et i et + i u u 2i et + i u; et we have (U 1)H D 8 u 2 H : and whence e T i(u + 1)(U 1) 1. et + i (U 1)u 2iu i(u + 1)u + i(u 1)u; The mapping in question is injective, because by (6) and (7) we have 8 u 2 H : et i et + i 1 u et i (U 1) 1 2 iu Uu: We will now collect criteria for essential self-adjointness of symmetric operators. Theorem 2.3 Let T be a symmetric operator in H. equivalent. o) T is essentially self-adjoint, Then the following are i) T is self-adjoint, ii) T is symmetric, iii) 8 2 C n R : (T )D(T ) H, iv) 9 2 C n R : (T )D(T ) H (T )D(T ), and if T is semi-bounded with lower bound, v) 8 < : (T )D(T ) H, vi) 9 < : (T )D(T ) H. Proof. Let T be self-adjoint. T is densely dened, because T T, and from Corollary 1.2 we get T T T ; in particular, T is symmetric. Conversely, 20

21 let us assume that T is symmetric. Then, by Corollaries 1.2 and 1.3, we have T T and T is symmetric. Therefore, T T T T T, whence T T. So we have established the equivalence of (i) and (ii). To prove the mutual equivalence of (i), (iii) and (iv), we only have to show, in view of Theorem 2.1, that T D T (T )D(T ) (8) for any symmetric operator T and 2 C n R. This follows from u 2 T D T, 9 (' n ) n2n D(T ) : ' n! ' 2 D T ; (T )' n! T ' u, u 2 (T )D(T ); where we have made use of the continuity of (T ) 1 from Lemma 1.3c. The same argument also applies in the case of a semi-bounded operator for <, because for any normalized ' 2 D(T ) we have k(t )'k jh(t )'; 'ij > 0: So the implications from (v) through (vi) to (i) are proved. For the implication from (i) to (v), we remark that for any self-adjoint operator e T and which is not an eigenvalue of T e we have et D 1 1 et et (f0g) et (f0g) f0g et D et and therefore H. In particular, this is true for any less than the lower bound of e T, which clearly cannot be an eigenvalue. In our situation, T is self-adjoint and semi-bounded with lower bound and so T 1 is densely dened, closed and, as above, bounded for any <. But then, by Corollary 1.1, its domain T D T (T )D(T ) is H. We have already seen earlier that (i) implies (o). Now let T have the unique self-adjoint extension e T and assume T is not self-adjoint. Then by Theorem

22 and (8) there is a v 2 (T + i)d(t ) with kvk 1 (one may have to replace T by T ). Since the operator dened on H by u u 2hu; viv is unitary, so is U : e U, where e U is the unitary operator corresponding to e T according to Lemma 2.1. Furthermore, U (T + i)d(t ) e U (T + i)d(t ) (T i)(t + i) 1. As Uv e Uv 6 0, U 6 e U, and Lemma 2.1 guarantees the existence of a selfadjoint extension of T dierent from e T. So (o) implies (i). We are now prepared to apply these general results to the situation of the Schrodinger operator of Section 0. 3 Application to Schrodinger operators Throughout this section, we are concerned with the operators of Section 0, namely the minimal Schrodinger operator S V on C d 1 0 R with the potential V 2 L d 2;loc R, whence S 0 is a linear operator in H L 2 R d. Furthermore, V is assumed to be real-valued, such that S 0 is symmetric. As before, the adjoint operator S 0 of S 0 will be denoted by S. Note that for all u 2 D(S) both 4u and V u are locally integrable. As S 0 represents an energy operator with hs 0 '; 'i kr'k 2 + V j'j 2 (9) interpreted as the expected value of the total energy of the (normalized) state ', the operator S 0 will be semi-bounded in most physical applications. The Friedrichs extension S 0F is then distinguished among all self-adjoint extensions of S 0 by the fact that every element of its domain can be approximated in L 2 R d by a sequence of test functions which is convergent in the energy norm given by (9). (If the lower bound of S 0 is not positive, one has to add (1 + )k'k 2 to produce a norm.) We illustrate the situation with the kinetic energy operator, i.e. the case where V 0. Then obviously H S0 W 1 2 R d. On the other hand, u 2 D(S) means that both u and 4u are in L 2 R d. By an interpolation argument (cf. [4, Lemma 3a]), u 2 W 1 2 R d, such that S S 0F and S 0 is essentially self-adjoint. 22

23 The methods used in literature to prove essential self-adjointness of S 0 if V 6 0 were mainly perturbational (cf. [7]; for an historical outline, see [5, Chapter 3]). We want to present a more direct approach, based on Theorem 2.3, namely we will prove that (S 0 )D(S 0 ) f0g for (some) <, that is, we have to show that is not an eigenvalue of S. direction is the following. What one can achive in that Proposition 3.1 Let V 2 L 2;loc R d be real-valued and S V on C 1 0 R d be semi-bounded with lower bound. Then there are no locally bounded eigenfunctions for S S 0 and <. Proof. We may assume 0. Let u 2 L 2 show that u 0. By [4, Lemma 3a], u 2 W 1 2;loc that u is real-valued. d R \ L 1;loc R d ; Su 0. We will d R, and we may also assume For " > 0 and k 2 N consider : u " k 2, where u " is the regularized u (cf., j j e.g., [5, Denition and Lemma 2.2]) and k, with a mesa function k (smooth, with values in [0; 1], 1 in B 0; 1 2, 0 outside B (0; 1)). Then 0 hu; S 0 ( " )i hu; 4 " i + hu; V " i R ru r " + hv u; " i R ru " r + h(v u) " ; i R k k jru " jk u " k ru " r k + hv u " ; u " ki 2 + h(v u) " V u " ; u " ki 2 hu " k ; S 0 (u " k )i kjr k j u " k 2 + h(v u) " V u " ; u " k 2i ku " k k 2 max j0 j 2 k 2 ku " k 2 L 2 (B(0;k)) k(v u) " V u " k L1 (B(0;k)) kuk L1(B(0;k+")). As "! 0, we get ku k k 2 max j0 j 2 kuk 2 ; k 2 since both (V u) " and V u " tend to V u in L 1 (B (0; k)). For k! 1, we arrive at kuk 2 0, whence u 0. 23

24 So we are left with the problem of nding sucient conditions for V which guarantee local boundedness of weak eigensolutions, i.e. solutions of 4u V u in L 1;loc R d (again we assume 0). We apply Green's representation formula 8 ' 2 C 1 0 R d : ' s( y)4'(y)dy; where s(z) 1 jzj 1 d3 d, with! d the area of the unit sphere in R d, to! d 1 j j ' r0 (x )u " ; here r with a mesa function, and u " is the r regularized u as above. Then, for "! 0, we get u(x) eu(r 0 ; x) s(x y) r0 (x y)4u(y)dy; where eu(r 0 ; ) 2 C d 1 R. So it suces to investigate v r : ueu(r; ) at r r 0 for local regularity. Since v r0 v r 2 C d 1 R and jv r (x)j s(x y)j4u(y)jdy, this suggests the following denitions. B(x;r) Denition 3.1 ( b R d means R d is open and bounded.) d R : p : R d! C measurable; K 0 loc K loc R d : Then 4u 2 K 0 loc 8 b R d 9 r 1 : sup s(x y)jp(y)jdy < 1 ; x2 B(x;r) p : R d! C measurable; 8 b R d : lim r!0 sup x2 B(x;r) s(x y)jp(y)jdy 0 : R d implies u 2 L 1;loc R d and 4u 2 K loc R d guarantees continuity of u. The problem with this is the question: under which assumption on V do we have 4u V u 2 K 0 loc R d Of course, V 2 K 0 loc R d would be sucient, if we knew u 2 L 1;loc R d but that is what we are eager to prove! The way out is an iteration process necessitating the local Kato condition V 2 K loc R d. Moreover, with the interpretation of ju(x)j 2 as the density of the probability to encounter the particle described by u at x in mind, it is clear that 24

25 a local singularity of u at x can only occur, if the potential is strongly attractive at x, i.e. if the negative part V : maxf0; V g of V is very singular at x. So it seems reasonable that only a local assumption on V is necessary. This is substantiated by Kato's inequality 4juj sign(u)4u in L 1;loc R d (cf. [2, p. 357{ 359]) with the consequence that 4juj V juj V juj. So we can make use of the following regularity statement, which can be found in [5, Theorem 2.1]. Proposition 3.2 Let p 2 K d loc R be real-valued, v 2 L 1;loc R d non-negative with pv 2 L d 1;loc R and (4 + p)v 0. Then v 2 L 1;loc R d. Together with Proposition 3.1 and Theorem 2.3 we arrive at Theorem 3.1 Let V 2 L 2;loc S V on C 1 0 d R be real-valued, V 2 K loc R d and such that d R is semi-bounded. Then S 0 is essentially self-adjoint in L 2 R d. For d 3, L 2;loc R d K loc R d, and therefore the assumption on V is always fullled. Although some assumption on V is needed for d 5, the local Kato condition on V is not necessary for essential self-adjointness of the operator S 0, as can be seen from [2, VII Proposition 4.1]. The case d 4 seems to be open. Another open problem for d 4 is Jorgens's conjecture (1972) (cf. [1]). Conjecture 3.1 Let V; W 2 L 2;loc R d be real-valued and V W. If 4+V C 1 0 R d is bounded from below and essentially self-adjoint in L d 2 R, R d is essentially self-adjoint in L d 2 R. then 4 + W C 1 0 Let us nally mention that boundedness from below of S 0 can be guaranteed by the assumption of a global Kato condition V 2 K R d (where R d Denition 3.1) due to relative form boundedness with respect to 4 (cf. [5, Corollary 3.3]). Moreover, by truncating the negative part of the potential, we may even use Theorem 3.1 to obtain essential self-adjointness of Schrodinger operators which are not bounded from below, namely allowing for a behavior of the potential like O (jxj 2 ) at innity (cf. [5, Theorem 3.4]). 25 in

26 Theorem 3.2 Let V 2 L 2;loc R d be real-valued and V 2 K d R + O (jxj 2 ). R d is essentially self-adjoint in L d 2 R. Then 4 + V C 1 0 Under the same assumptions on V, this approach allows to treat magnetic Schrodinger operators as well, i.e. (r i b) 2 + V C 1 0 R d, as long as b is continuously dierentiable as a function from R d to R d ; if one employs a method of Leinfelder and Simader, one can even cover the most general case, where b 2 L d d 4;loc R and r b 2 L2;loc R d (cf. [6, Theorem 2.5]). References [1] H. L. Cycon, On the stability of self-adjointness of Schrodinger operators under positive perturbations, Proc. Roy. Soc. Edinburgh Sect. A {173(1980). [2] D. E. Edmunds and W. D. Evans, Spectral Theory and Dierential Operators, Clarendon Press, Oxford (1987). [3] E. Fischer, Sur la convergence en moyenne, C. R. Acad. Sci. Paris { 1024(1907). [4] A. M. Hinz, Asymptotic Behavior of Solutions of 4v + qv v and the Distance of to the Essential Spectrum, Math {182(1987). [5] A. M. Hinz, Regularity of solutions for singular Schrodinger equations, Rev. Math. Phys. 4 95{161(1992). [6] A. M. Hinz and G. Stolz, Polynomial boundedness of eigensolutions and the spectrum of Schrodinger operators, Math. Ann {211(1992). [7] T. Kato, Perturbation Theory for Linear Operators, Springer, New York (1966). [8] H. Leinfelder, A Geometric Proof of the Spectral Theorem for Unbounded Self- Adjoint Operators, Math. Ann {96(1979). [9] J. v. Neumann, Mathematische Grundlagen der Quantenmechanik, Springer, Berlin (1932). [10] E. Schrodinger, Abhandlungen zur Wellenmechanik, Johann Ambrosius Barth, Leipzig (1927). [11] M. H. Stone, Linear Transformations in Hilbert Space and Their Applications to Analysis, American Mathematical Society, New York (1932). [12] J. Weidmann, Linear Operators in Hilbert Spaces, Springer, New York (1980). 26