NEW ZEALAND JOURNAL OF MATHEMATICS Volume 46 (016), 65-7 THE ESSENTIAL SELF-ADJOINTNESS OF SCHRÖDINGER OPERATORS WITH OSCILLATING POTENTIALS Adam D. Ward (Received 4 May, 016) In Memory of Boris Pavlov (1936-016) Abstract. Let Ω be a domain in R m with non-empty boundary and let H = V be a Schrödinger operator defined on C0 (Ω) where V = V 1 V L loc (Ω) is a real valued potential, V L (Ω) and V 1 (x) 1 µ (Ω) d(x). (1) Here d(x) is the Euclidean distance to the boundary of the domain and µ (Ω) is the non-negative variational constant associated to the L -Hardy inequality. In [1] it was shown that H is essentially self-adjoint and that the condition described by equation (1) is optimal on certain geometrically simple domains. In this paper investigate the essential self-adjointness of Schrödinger operators with potentials that oscillate around the critical limit described by equation (1). 1. Introduction The problem of the essential self-adjointness of Schrödinger operators has a long and distinguished history (see [1], [] and [3] for an overview). Heuristically, if we consider a domain Ω in R m which has non-empty boundary, then the Schrödinger operator H = V defined on C0 (Ω) is essentially self-adjoint provided that a particle under the influence of the potential V is unable to come into contact with the boundary of the domain. In [1] and [4] it was shown that if V = V 1 V L loc is a real valued potential so that V L (Ω) and V 1 (x) 1 µ (Ω) d(x), () then H is essentially self-adjoint. We refer to the term ( 1 µ (Ω) ) d(x) as the critical limit for essential self-adjointness. Here d(x) is the Euclidean distance to the boundary and { } µ (Ω) = inf ω W 1,0 (Ω) Ω ω(x) Ω ω(x) d(x) is the non-negative constant associated to the L -Hardy inequality. It is well nown that a domain Ω admits an L -Hardy inequality if and only if µ (Ω) is positive. 010 Mathematics Subject Classification Primary: 47B5, Secondary: 6D10. Key words and phrases: Essential Self-adjointness of Schrodinger Operators, L p-hardy Inequalities.
66 ADAM D. WARD Equation () casts the problem of the essential self-adjointness of Schrödinger operators on domains with non-empty boundary in terms of a balancing act between the quantum tunneling effect and the uncertainty principle. Suppose that a domain Ω does not admit an L -Hardy inequality so that µ (Ω) = 0. In this case equation () implies that the potential V must inflate lie d(x) in order to ensure that the probability of finding a particle under its influence at the boundary is zero. In other words, if the potential inflates at this rate, then this is sufficient to ensure the particle does not tunnel through any classically forbidden region and reach the boundary. On the other hand, suppose that a domain does not admit an L -Hardy inequality so that µ (Ω) > 0. This obviously relaxes the criteria for essential self-adjointness. The physical reason for this is that the value of µ (Ω) places limits on the certainty with which we can say that a particle is located at the boundary. As in [4], if we suppose that the quantum state of the particle is described by the unit wavefunction ω W,0(Ω), 1 then we can easily derive the inequality ( ) ( ) µ (Ω) d(x) ω(x) ω(x). (3) Ω Ω The second integral on the right hand side of equation (3) describes the total momentum of the particle, whilst the first integral gives a measure of the particle s proximity to the boundary. Hence, if µ (Ω) > 0, we see that one cannot confine a particle to a smaller neighborhood of the boundary of a domain without producing a corresponding increase in the particle s total momentum. We will simply state here that the value of µ (Ω) has an intimate dependence on the (possibly fractal) dimension of the boundary (see Chapter 5 of [1] & [5] for further details). Furthermore, the critical limit described by equation () can be shown to be optimal on certain geometrically simple domains (see Chapter 6 of [1]). In particular, on the half line Ω = (0, ), where it is well nown that µ (Ω) = 1/4, equation () implies that the Schrödinger operator H = d V defined on C0 (0, ) is essentially self-adjoint provided that V 1 (x) 3/4 x. However, if V 1 (x) is a real continuous potential so that V 1 (x) C x for some C < 3/4, then by Theorem X.10 of [] and Lemma.1 it must be the case that H is not essentially self-adjoint. The situation is maredly different if we allow the domain Ω to equal the whole of the Euclidean space. Following on from earlier wor by Titchmarsh [6] and Sears [7], Berezin & Shubin [8] were able to show that the Schrödinger operator H = V defined on C0 (R m ), where V is a real locally bounded potential, is essentially self-adjoint if V (x) x b but is not essentially self-adjoint if V (x) x ɛ b. The constants b and ɛ are both assumed to be positive. Herein lies a correspondence between the essential self-adjointness of Schrödinger operators and the concept of classical completeness. If one considers a particle moving along the half line Ω = (0, ), then starting from the elementary energy equation E = ẋ V (x) it is trivial to derive the following t( ) t(0) = x= x=0 ( ) 1 E V (x).
ESSENTIAL SELF-ADJOINTNESS OF SCHRÖDINGER OPERATORS 67 The expression above effectively describes the time it taes for a particle under the influence of the potential V to move from the origin to infinity. On the one hand, if V (x) x b, then this quantity will be infinite and we say that the system is classically complete. On the other hand, if V (x) x ɛ b, then this quantity will be finite and we say that the system is classically incomplete since the particle can escape to infinity in a finite amount of time. Although the concepts of classical completeness and the essential self-adjointness of Schrödinger operators are in some sense complimentary, the correspondence is by no means perfect. In [9] Rauch & Reed exploit quantum phenomena not present in classical mechanics to produce examples of systems which are classically complete but not essentially self-adjoint and vice versa. Nevertheless, the inability of a particle to escape to infinity in a finite amount of time is a fruitful way to thin of the essential self-adjointness of Schrödinger operators on R m. Indeed, Eastham, Evans & Mcleod [3] show that if there exists a sequence of sufficiently thic concentric layers on which V (x) is much larger than x b, then the operator H = V defined on C0 (R m ) is essentially self-adjoint irrespective of the behavior of the potential outside of these layers. The physical interpretation here is that these layers provide sufficient barriers to the outward progress of the particle so that it cannot escape to infinity in a finite amount of time. In contrast, on the half line Ω = (0, ) Rauch & Reed [9] show that if V (x) is much smaller than x b, except on a sequence of layers where the potential spies and is unbounded above, then as long as these layers are sufficiently thin the operator H = d V defined on C0 (0, ) is not essentially self-adjoint. The physical interpretation here is that the quantum particle can tunnel through each of the spies and is still able to escape to infinity in a finite amount of time. Returning to the case where Ω is a domain in R m with non-empty boundary, the following natural questions arise. Suppose that V 1 (x) > 1 µ(ω) d(x) everywhere on Ω except for on a sequence of layers where the potential dips below this critical limit. Would H still be essentially self-adjoint in this case? Conversely, suppose that V 1 (x) < 1 µ(ω) d(x) everywhere on Ω except for on a sequence of layers upon which the potential exceeds this critical limit. Would H then cease to be non-essentially selfadjoint? In this paper we provide a partial answer to these questions by considering the simplest possible case where Ω = (0, ). The main theorem of the paper, whose validity we establish by modifying the analysis given in [9], is stated below. Theorem 1.1. Let Ω = (0, ). Define the Schrödinger operators H a = d V a and H b = d V b with domains of definition D(H a ) = D(H b ) = C0 (Ω). We assume that V a and V b are real valued potentials of the form V a = V a,1 V a, and V b = V b,1 V b, with V a,, V b, L (Ω) and where for some C 1 > 3 4 > C V a,1 = C 1 x σ j (x), V b,1 = C x σ j (x). Here the σ (x) are narrow disjoint spies with radius of support r centered at 1. We ( assume that 0 < r < 1/[4( 1)] so that the spies are disjoint, σ (x) C0 1 r, 1 r ) and that 0 σ (x) σ ( 1 ) = C 3 where C 3 > max { C 1 3 4, 3 4 C } so that Va,1 (1/) < 3 4 < V b,1 (1/). If the r are chosen to be
68 ADAM D. WARD sufficiently small, then H a is essentially self-adjoint whereas H b is not essentially self-adjoint. We have decided to structure this paper as follows. In Section we present various definitions, notations and well nown results. We will also prove a technical lemma that will be crucial to the proof of Theorem 1.1 which is given in Section 3.. Definitions, Notations & Technical Lemmas We begin this section by recalling the definition of symmetric, self-adjoint and essentially self-adjoint operators. Definition.1. Let A be a densely defined, linear operator on the Hilbert space H. A is said to be symmetric if A A so that D(A) D(A ) and Ax = A x for all x D(A). Definition.. Let A be a densely defined linear operator on the Hilbert space H. A is said to be self-adjoint if A = A, so that D(A) = D(A ) and Ax = A x for all x D(A). Definition.3. Let A be a densely defined linear symmetric operator on the Hilbert space H. A is said to be essentially self-adjoint if its closure is self-adjoint, i.e. if Ā = Ā A. We will prove Theorem 1.1 by appealing to the following result which is often referred to as the symmetric form of the Kato-Rellich Theorem. Lemma.1. [, Theorem X.13] Let A and C be densely defined linear symmetric operators on the Hilbert space H. Suppose that there exists a dense linear subspace D D(A), D(C), b > 0 and a [0, 1) so that for all φ D (A C) φ a ( Aφ Cφ ) b φ. Then A is essentially self-adjoint on D if and only if C is essentially self-adjoint on D. In order to proceed we mae the following definitions. Definition.4. For each N let λ ± regions { I 4 = x > 0 x 1/ 1 4( 1) { I = x > 0 x 1/ 1 ( 1) = 1/ ± 1 / [( 1)] and define the It follows immediately from the definitions above that I 4 I, I 1 I = and if σ (x) represents the function described in the statement of Theorem 1.1, then supp σ (x) (1/ r, 1/ r ) I 4. Definition.5. Define the real valued function N(x) C 0(, ) so that N(x) is monotone non-decreasing on the interval (, 1), is equal to 1 on [ 1, 1] and is monotone non-increasing on the interval (1, ). For each N define the dilated function N (x) = N ( 4( 1)x ). }, }.
ESSENTIAL SELF-ADJOINTNESS OF SCHRÖDINGER OPERATORS 69 It is easy to show that N (x 1/) C0 (I ), N (x 1/) = 1 for all x I 4 and that N (x 1/) C ( 1) and N (x 1/) C ( 1). To complete this section we prove the following lemma. The analysis here is a modification of that given in [9]. Lemma.. There exists a finite uniform constant C > 0 so that the following estimate holds for all N and φ C0 (0, ) ( ) 1 sup φ(x) C x I 4 6 φ (x) L (I ) φ(x) L (I ) Proof. Choose some N and φ(x) C0 (0, ). Using the Fundamental Theorem of Calculus and the properties of the function N (x) decsribed above, it is easy to establish that for all x I 4 x t ( ) φ(x) = N (s 1/) φ(s) ds dt. (4) Applying integration by parts twice to equation (4) and using the fact that N (x 1/) = 1 on I 4 we arrive at the following equality φ(x) = x x t N (t 1/) φ(t) dt N (s 1/) φ(s) N (s 1/) φ (s) ds dt so that by taing the modulus of both sides and extending the intervals of integration we obtain φ(x) φ(x) N (t 1/) φ(t) dt N (s 1/) φ(s) ds dt N (s 1/) φ (s) ds dt. Next, by applying the Cauchy-Schwarz inequality and noting that N (x) 1, we derive ( ( (λ λ ) ( N (t 1/) λ dt ( (λ λ ) N (s 1/) ds ( 1 λ ds φ(t) dt ( φ (s) ds φ(s) ds C 1 ( 1) 1 φ(x) L(I ) C 3 ( 1) 3 φ (x) L(I ) Since the estimate above holds for all x I 4, the result follows immediately after the application of some elementary inequalities.
70 ADAM D. WARD 3. Proof of Main Theorem Proof. Consider the operators H a,0 = d V a,0 and H b,0 = d V b,0 with domains of definition D(H a,0 ) = D(H b,0 ) = C0 (Ω) where V a,0 = V a σ j(x) and V b,0 = V b σ j(x). By Theorem 6..4 of [1] it must be the case that H a,0 is essentially self-adjoint. In contrast, by Lemma.1 and Theorem X.10 of [], it must be the case that H b,0 is not essentially self-adjoint. We will show that the operator H a is self-adjoint, the non-essential self-adjointness of H b following from an analogous argument. In this regard, since H a,0 and H a are evidently densely defined linear symmetric operators, by Lemma.1 it suffices to show that there exists some a [0, 1) and b > 0 so that for all φ(x) C0 (Ω) we have σ j (x) φ(x) a ( H a,0 φ H a φ ) b φ. (5) Indeed, choosing arbitrary φ C0 (Ω) and N we can use the result of Lemma. to derive the following inequalities 1/ r σ (x) φ(x) = σ (x) φ(x) C r 4 sup φ(x) 1/ r x I 4 ( ) Cr 4 6 φ (x) L (I ) φ(x) L (I ) Cr 4 ( 6 H a,0 φ(x) L (I ) 6 V a, (x) φ(x) L (I ) 6 C 1 x φ(x) L (I ) φ(x) L (I ) ) C r 4 ( 6 H a,0 φ(x) L (I ) φ(x) L (I ) ) a H a,0 φ(x) L (I ) b φ(x) L (I ) where the last inequality follows from choosing r sufficiently small so that C r a and C r 6 b for some a [0, 1) and b > 0. Hence, for all φ C 0 (Ω) and each N we have that σ (x) φ(x) a H a,0 φ(x) L(I ) b φ(x) L(I ). (6) Since supp σ (x) (1/ r, 1/ r ) I 4 I and each of the I are disjoint, using equation (6) we ultimately obtain σ j (x) φ(x) σ j (x) φ(x) a H a,0 φ(x) L(I ) b φ(x) L(I ) a H a,0 φ(x) b φ(x) ( ) a H a,0 φ(x) H a φ(x) b φ(x). Since the validity of equation (5) has been established, the proof is now complete.
ESSENTIAL SELF-ADJOINTNESS OF SCHRÖDINGER OPERATORS 71 References [1] A.D. Ward, On Essential Self-Adjointness, Confining Potentials & the L p - Hardy Inequality, PhD Thesis, NZIAS Massey University NZ, 014. [] M.C. Reed and B. Simon, Methods of Modern Mathematical Physics - Vol. Fourier Analysis & Self-Adjointness, Academic Press, New Yor, 1975. [3] M.S.P. Eastham, W.D. Evans and J.B. Mcleod, The essential self-adjointness of Schrödinger type operators, Archive for Rational Mechanics and Analysis, 60 () (1976), pp. 185-04. [4] A.D. Ward, The essential self-adjointness of Schrödinger operators on domains with non-empty boundary, Manuscripta Mathematica, First Published Online January 016, ISSN: 143-1785, pp. 1-14, http://.doi.org/10.1007/s009-016-080-8. [5] A.D. Ward, On the variational constant associated to the L p -Hardy inequality, (To Appear in Journal of the Australian Mathematical Society). [6] E.C. Titchmarsh, Eigenfunction Expansions Associated with Second Order Differential Equations, Second Edition, Oxford University Press, 196. [7] D. B. Sears, Note on the uniqueness of Green s functions associated with certain differential equations, Canadian J. Math, (1950), pp. 314-35. [8] F. A. Berezin and M. A. Shubin, The Schrödinger Equation, Kluwer, Dordrecht, 1991. [9] J. Rauch and M. Reed, Two examples illustrating the differences between classical and quantum mechanics, Communications in Mathematical Physics, 9 () (1973), pp. 105-111. [10] M.C. Reed and B. Simon, Methods of Modern Mathematical Physics - Vol. 1 Functional Analysis, Academic Press, New Yor, 197. A.D. Ward NZ Institue for Advanced Study, Massey University, Private Bag 10 904, North Shore MSC, 0745 Aucland, New Zealand. adward.wor@gmail.com