ELEMENTARY SPECTRAL THEORY OF SOME SCHRÖDINGER OPERATORS

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1 ELEMENTARY SPECTRAL THEORY OF SOME SCHRÖDINGER OPERATORS Abstract. We prove some elementary properties regarding Schrödinger operators of the form ` V. In particular we prove for a certain class of potentials V that the spectrum of ` V has only eigenvalues below 0. We also prove a basic Lieb-Thirring inequality for large sums of eigenvalues for Schrödinger operators and show that this inequality is dual to a type of Sobolev inequality for fermionic particles. 1. Solutions of the Schrödinger equation The time independent Schrödinger equation (TISE) for a particle interacting with a potential V pxq in R n is given by pxq ` V pxqpxq Epxq (1) where is the (distributional) Laplacian. Here, V : R n Ñ R is some real valued measureable function. is a wave function and the physical interpretation is that pxq 2 dx is the probability density associated with finding a particle at the point x. We therefore require the normalization condition, }} 2 1, (2) with } } p denoting the L p norm on R n as usual. We are interested in the eigenfunctions and eigenvalues for which (1) holds. Formally associated with the equation (1) and the operator `V is the quadratic form given by Epq T ` V (3) where T pxq 2 dx, V V pxq pxq 2 dx. (4) R n R n Physically, T is the kinetic energy of the particle, and V is the potential energy. One of our tasks is to find a class of wave functions and assumptions on V so that the above definitions make sense. It is natural to assume that has finite kinetic energy, i.e., that P H 1 pr n q, where H 1 pr n q is the space of square integrable wave functions with square integrable weak first derivatives. We will later see that in the case n 3, a suitable assumption on the potential is that V P L 3{2 pr 3 q ` L 8 pr 3 q. Supposing that the definition of Epq makes sense, we define the ground state energy E 0 inf Epq : }} 2 1, P H 1 pr n q (. (5) We would also like to find assumptions under which E 0 ą 8. Minimizers of Epq, i.e. functions 0 so that Ep 0 q E 0 will turn out to be solutions to (1) with eigenvalue E 0. We will also define and investigate higher eigenvalues and obtain a complete description of the part of the spectrum of `V (under suitable assumptions on V ) that lies below 0. Note that physically the negative part of the spectrum corresponds to bound states. They are stationary with respect to the dynamics induced by the semigroup t Ñ e itp `V q. However, this is unimportant for us and we will not mention this in the remainder of the paper. From now on we will consider the physical case n 3. We comment that the situations for n ą 3 are relatively similar to the situation described below, as the nature of the Sobolev inequalities in dimensions n ľ 3 are similar. Similar statements for n 1 and n 2 can be made, and in general, 1

2 2 ELEMENTARY SPECTRAL THEORY OF SOME SCHRÖDINGER OPERATORS less regularity can be assumed on the potentials. In n 1, the potential can be any bounded Borel measure. Lemma 1. Let Epq be as in (3) and assume that the potential satisfies V pxq P L 3{2 pr 3 q ` L 8 pr 3 q. Then there exists a constant C so P H 1 pr 3 q, Proof. For P H 1 pr 3 q the Sobolev inequality Epq ľ 1 2 T C }} 2 (6) R 3 pxq 2 dx ľ S 3 ˆ R 3 pxq 6 dx 1{3 (7) holds, with S 3 3p2π 3 q 2{3 {4. By assumption, we can write V pxq wpxq ` hpxq with w P L 3{2 pr 3 q and h P L 8 pr 3 q. Define for λ ą 0, # wpxq, wpxq ĺ λ w λ pxq (8) 0, else. By dominated convergence, lim λñ8 }w w λ } 3{2 0. Since V pxq pwpxq w λ pxqq ` pw λ pxq ` hpxqq, for any δ ą 0 we can write V pxq v δ pxq ` vpxq, where }v δ } 3{2 ă δ and v P L 8 pr 3 q. Take δ S 3 {2. Then, V ĺ v δ pxq pxq 2 dx ` vpxq psipxq 2 dx R 3 R ˆ 3 2{3 ˆ 1{3 ĺ v δ pxq 3{2 dx pxq 6 dx ` }v} 8 }} 2 2 ĺ 1 2 T ` }v} 8 }} 2 2. (9) In the second line we have applied Hölder s inequality. In the last line we have applied the Sobolev inequality (7) and our bound on the L 3{2 norm of v δ. Note that the above argument also shows that under the assumption that V pxq P L 3{2 pr 3 q ` L 8 pr 3 q, the function Ñ V is well defined whenever P H 1 pr 3 q. Therefore, Epq T ` V ľ T V ľ 1 2 T }v} 8 }} 2 2. (10) In particular, Lemma 1 implies that E 0 ą 8. We will now turn to establishing the existence of a minimizer 0. The main technical element is the following, in which we establish weak semicontinuity of the potential. Lemma 2. Assume the potential V P L 3{2 pr 3 q ` L 8 pr 3 q. Additionally, assume that V vanishes at infinity, that is, for any a ą 0, tx : V pxq ą au ă 8, (11) where denotes the Lebesgue measure of a set. Then the function Ñ V is weakly (sequentially) continuous on H 1 pr 3 q. That is, if á in H 1 pr 3 q, then V Ñ V. Proof. Let á in H 1 pr 3 q. By the uniform boundedness principle and the fact that H 1 pr 3 q is a Hilbert space the H 1 pr 3 q norms of the are uniformly bounded. For δ ą 0, define V δ pxq # V pxq, V pxq ĺ δ 1 0, else. (12)

3 ELEMENTARY SPECTRAL THEORY OF SOME SCHRÖDINGER OPERATORS 3 Note that lim δñ0 }V V δ } 3{2 0. Since, V V ĺ V δ pxq V pxq pxq 2 dx ` V δ pxq V pxq 2 dx ` V δ pxq ˇˇ 2 2ˇˇ dx R 3 ĺ }V δ V } 3{2 2S 3 ˆ sup } } H 1 ` V δ pxq ˇˇ pxq 2 pxq 2ˇˇ dx, (13) R 3 we have that lim sup Ñ8 V V ĺ C δ ` lim sup V δ pxq ˇˇ pxq 2 pxq 2ˇˇ dx, (14) Ñ8 R 3 where C δ is a constant independent of that goes to 0 as δ Ñ 0. Note we have used that }} H 1 ĺ sup } } H 1. We are therefore left with proving that for any δ ą 0, lim sup Ñ8 V şr 3 δ pxq ˇˇ pxq 2 pxq 2ˇˇdx 0. Let ε ą 0, and define A ε tx : V δ ą εu. By our assumptions on V we have A ε ă 8. By Theorem 8.6 of [LL], we have Ñ strongly in L r pa ε q, for 2 ĺ r ă 6 (this follows from the fact that á in H 1 pr 3 q and A ε is of finite measure). By the elementary inequality, ˇ ˇ 2 2ˇˇ ˆ ` ĺ ˆ `, (15) we have that 2 Ñ 2 strongly in L r{2 pa ε q. Let 1 ĺ s ĺ 8 be a number so that 1{s ` 2{r 1. Since V δ P L 8 pr 3 q, we have that V δ P L s pa ε q. It follows that, V δ pxq ˇˇ pxq 2 pxq 2ˇˇ dx ĺ V δ pxq ˇˇ pxq 2 pxq 2ˇˇ dx R 3 A ε ` V δ pxq ˇˇ pxq 2 pxq 2ˇˇ dx R 3 za ε ĺ }V δ } s r{2 ` 2ε ˆ sup } } 2 2 (16) In the last line we have applied Hölder s inequality. Taking lim sup on both sides and then ε Ñ 0 yields the claim. We record here the following Corollary of the above proof, which will be useful later. Corollary 1. Let U be a nonnegative potential satisfying the conditions of Lemma 2. Then, multiplication by? Upxq is a compact operator from H 1 pr 3 q to L 2 pr 3 q. That is, if á weakly in H 1 pr 3 q, then? U Ñ? U strongly in L 2 pr 3 q. Proof. Applying the Sobolev and Hölder inequalities as before, we see that multiplication by? U is indeed a bounded operator from H 1 pr 3 q to L 2 pr 3 q. To see that it is compact, we need to prove that if á weakly then lim Upxq pxq pxq 2 dx 0. (17) Ñ8 R 3 However, this precisely the content of the proof of Lemma 2. We are now in position to prove the existence of a minimizer for Epq. Theorem 1. Let V pxq be a real valued function satisfying V P L 3{2 pr 3 q ` L 8 pr 3 q. Assume furthermore that V vanishes at infinity. Let Epq T ` V as before and assume that E 0 inf Epq : P H 1 pr 3 q, }} 2 1 ( ă 0. (18) Then there exists a 0 P H 1 pr 3 q satisfying } 0 } 2 1 and Ep 0 q E 0. Furthermore, 0 satisfies the TISE, in the sense of distributions. 0 pxq ` V pxq 0 pxq E 0 pxq, (19)

4 4 ELEMENTARY SPECTRAL THEORY OF SOME SCHRÖDINGER OPERATORS Proof. By Lemma 1, E 0 ą 8. Let be a minimizing sequence of H 1 pr 3 q functions with } } 2 1 so that Ep q Ñ E 0. By (6), the H 1 pr 3 q norms of the are uniformly bounded, since the Ep q are a converging sequence of real numbers. Since H 1 pr 3 q is weakly compact, there exists a subsequence, which we continue to denote by, that converges weakly to some H 1 pr 3 q function 0. By the fact that norms are weakly lower semicontinuous, lim inf pxq 2 dx ľ 0 pxq 2 dx (20) Ñ8 R 3 R 3 lim inf pxq 2 dx ľ 0 pxq 2 dx. (21) Ñ8 By (20), } 0 } 2 ĺ 1. By (21) and Lemma 2 the function Ñ Epq is weakly lower semicontinuous, i.e., lim inf Ñ8 Ep q ľ lim inf T ` lim inf V ľ T 0 ` V 0 Ep 0 q. (22) fñ8 Ñ8 Therefore, E 0 ľ Ep 0 q. Since E 0 ă 0 by assumption, 0 cannot be the zero function, i.e., } 0 } 2 ą 0. We have 0 ą E 0 ľ Ep 0 q E p 0 { } 0 } 2 q } 0 } 2 2 ľ E 0 } 0 } 2 2, (23) and so } 0 } 2 ľ 1. But since we already had that } 0 } 2 ĺ 1, we must have } 0 } 2 1 and so Ep 0 q E 0. We now prove that 0 satisfies (19). For functions H 1 pr 3 q functions φ and ϕ define Epφ, ϕq φpxq Ď ϕpxqdx ` R 3 V pxqφpxqϕpxqdx. s R 3 (24) Obviously, Epφ, ϕq is well defined and Ep, q Epq. Let ε ą 0 and f P C 8 c pr 3 q any infinitely differentiable function of compact support. Let ε pxq 0 pxq ` εfpxq. Define for ε small enough the function, Rpεq Epε q p ε, ε q Ep 0q ` εep 0, fq ` εepf, 0 q ` ε 3 Epf, fq p 0, 0 q ` εp 0, fq ` εpf, 0 q ` ε 2 pf, fq, (25) where p, q is the inner product on L 2 pr 3 q. Clearly Rpεq is differentiable at ε 0 and since 0 is a minimizer, the derivative must be 0 there. We compute, Integrating by parts we have 0 d dε Rpεqˇˇˇˇε 0 Ep 0, fq ` Epf, 0 q Ep 0 q rp 0, fq ` pf, 0 qs (26) p f ` V f, 0 q ` p 0, f ` V fq E 0 rpf, 0 q ` p 0, fqs. (27) Because we take f to be purely imaginary or purely real, we see that the real and imaginary parts of 0 satisfy the TISE separately, and therefore all of 0 satisfies the TISE. We have shown that minimizers of the functional Epφq satisfy the TISE. The converse also holds: Lemma 3. Let the potential V be as in Theorem 1, and let P H 1 pr 3 q satisfy in the sense of distributions for E P R. Then, pxq ` V pxqpxq Epxq (28) Epq E }} 2 2 (29) In particular, if satisfies (28) with eigenvalue E 0, then minimizes the functional Epφq.

5 ELEMENTARY SPECTRAL THEORY OF SOME SCHRÖDINGER OPERATORS 5 Proof. Let φ be a sequence of Cc 8 pr 3 q functions converging to in H 1 pr 3 q norm. By the Sobolev inequalities they also converge in L 6 pr 3 q norm. For every we have by taking the conugate of (28), pxq φ s pxqdx ` V pxqpxqφ s pxqdx E pxqφ s pxqdx. (30) R 3 Let us take Ñ 8 on both sides. Integrating by parts (which we may do because is an H 1 function) we obtain lim pxq φ s pxqdx lim pxq s φ pxqdx pxq 2 dx. (31) Ñ8 R 3 Ñ8 The last equality follows from Cauchy-Schwarz. Also by Cauchy-Schwarz, lim Ñ8 p, φ q p, q. Let V pxq wpxq ` hpxq, with w P L 3{2 pr 3 q and h P L 8 pr 3 q. By Hölder s inequality, ˇ V pxqpxqφ s pxqdx V pxq pxq 2 dx ˇ ĺ }w} 3{2 }} 6 } φ } 6 ` }h} 8 }} 2 } φ } 2 (32) Therefore, E }} 2 2 lim Ep, φ q lim Ñ8 Ñ8 R 3 pxq φ s pxqdx ` V pxqpxqφ s pxqdx Epq. R 3 (33) Remark. We did not require that V vanishes at infinity. We have shown that under suitable assumptions on V that the ground state energy E 0 is attained if it is negative. We would now like to turn to the definition of higher eigenvalues. Before, doing so we should make a comment on our choice of terminology. In the remainder of the paper, we will use the term orthogonal to refer to two functions that are orthogonal in L 2 pr 3 q, even though the Hilbert space we are considering is H 1 pr 3 q. We will also use orthonormal to denote a set of functions which are pairwise orthogonal in L 2 pr 3 q and have L 2 pr 3 q norms equal to 1. Let us now proceed with our discussion of higher eigenvalues. We define the first higher eigenvalue as E 1 inf Epq : P H 1 pr 3 q, }} 2 1, p 0, q 0 (. (34) Again, p, q is the inner product on L 2 pr 3 q. We are minimizing the functional Epq over the part of L 2 pr 3 q that is orthogonal to the minimizer 0. This leads us to a natural inductive definition. If the eigenvalue E 1 is attained by some function 1 (that is normalized to have L 2 norm 1 and is orthogonal to 0 ), then we can define the second higher eigenvalue E 2 by the same formula as in (34) but also require that the functions are orthogonal to 1. If this eigenvalue is attained, then we can define E 3, and so on and so forth. To make this precise, suppose that the first k eigenvalues have been defined and are attained. What we mean is that for each 0 ĺ ĺ k there is an H 1 pr 3 q function with } } 2 1 so that p, l q 0 for any 0 ĺ l ă, and Ep q E where E inf Epq : P H 1 pr 3 q, }} 2 1, p l, q 0, 0 ĺ l ă (. (35) Then we define the pk ` 1qth eigenvalue E k`1 by E k`1 inf Epq : P H 1 pr 3 q, }} 2 1, p l, q 0, 0 ĺ l ă k ` 1 (. (36) We continue to define eigenvalues until one is not attained. If an eigenvalue E k is not attained by some wave function k with the required orthogonality properties, then we stop this definition. Obviously we have E 0 ĺ E 1 ĺ... The following theorem says this process does not stop until we hit E k 0.

6 6 ELEMENTARY SPECTRAL THEORY OF SOME SCHRÖDINGER OPERATORS Theorem 2. Let V be as in Theorem 1 and assume that the pk ` 1qth eigenvalue is negative. This includes the assumption that the first k eigenfunctions exist and attain the first k eigenvalues. Then the pk ` 1qth eigenfunction also exists and satisfies k pxq ` V pxq k pxq E k k pxq (37) in the sense of distributions (note that the pk ` 1qth eigenvalue is E k since the first eigenvalue is E 0 ). Proof. The proof is nearly identical to that of Theorem 1. We take a minimizing sequence of normalized H 1 functions φ so that Epφ q Ñ E k. By Lemma 1 the H 1 norms of the φ l s are bounded and so we extract a subsequence (which we continue to denote by φ ) converging weakly to some k. The same argument of Theorem 1 shows us that E k Ep k q and } k } 2 1 if we can show that k is orthogonal to each, 0 ĺ i ă k. But Ñ p i, q defines a linear functional on H 1 pr 3 q and so 0 lim Ñ8 p i, φ q p i, k q. Let us now prove that (37) holds. Let f P C 8 c pr 3 q be a function so that pf, i q 0, for 0 ĺ i ă k. As in the proof of Theorem 1, define for ε small enough the function ε Ñ Rpεq Ep k ` εfq{ } k ` εf} 2 2. Arguing as in Theorem 1 by evaluating the derivative of Rpεq at ε 0, we see that the distribution D : p ` V E k q k satisfies Dpfq 0 for every C 8 c pr 3 q that is orthogonal to every i, for i ă k. It follows from Theorem 6.14 in [LL] that D k 1 i 0 c i i, (38) where c 0,..., c k 1 are constants. We would like to show that every constant c l is 0. Let φ be a sequence of Cc 8 pr 3 q functions converging to l in H 1 norm, for some l ă k. Taking the conugate of D, we have k 1 p i, φ q Ď k pxq φ pxqdx ` V pxq Ď k pxqφ pxqdx E k p k, φ q. (39) Above, we have integrated by parts which is ustifed because k P H 1 pr 3 q. Let us take Ñ 8 on both sides. The same arguments that appear in Lemma 3 show that we can pass the limit inside the integral on both sides of the above equality. We obtain c l Ď k pxq l pxqdx ` V pxq Ď k pxq l pxqdx (40) by the orthogonality of i s. Let now ϕ be a sequence of Cc 8 pr 3 q functions converging to k in H 1 norm. Since l satisfies the Schrödinger equation (37) with the eigenvalue E l, we have, after an integration by parts, Ďϕ pxq l pxqdx ` R 3 V pxqďϕ pxq l pxqdx E l pϕ, l q. R 3 (41) Again, we may take the limit Ñ 8 on both sides and pass the limit inside the integral. We obtain, Ď k pxq l pxqdx ` R 3 V pxq Ď k pxq l pxqdx E l p k, l q 0. R 3 (42) Comparing with (40) we see that each c l 0. The claim follows. Let us now prove some elementary properties of the eigenfunctions: Lemma 4. Consider our sequence of eigenvalues E 0 ĺ E 1... and our sequence of orthonormal eigenfunctions 0, 1... Then each eigenvalue has finite multiplicity. That is any number E k ă 0 occurs only finitely many times in our list of eigenvalues. Furthermore, let be any H 1 function satisfying (37) with eigenvalue E k. Then is a linear combination of the eigenfunctions that have an eigenvalue equal to E k.

7 ELEMENTARY SPECTRAL THEORY OF SOME SCHRÖDINGER OPERATORS 7 Proof. Assume that there is some E k that occurs in our list of eigenvalues infinitely many times with E k ă 0 (i.e., E k E k`1...). By the theorem ust proved, there are then infinitely many orthonormal eigenfunctions, ľ k, each with eigenvalue E E k. Since any sequence of orthonormal functions converges weakly to 0 in L 2, we have that á 0 in L 2. By Lemma 1 the H 1 norms of the s are uniformly bounded, and so we may pass to a subsequence converging weakly in H 1. But this subsequence must have the same weak limit in L 2 as the entire sequence does, so this subsequence, which we continue denote by, must converge to 0 weakly in H 1. By Lemma 2 we must have that lim Ñ8 V 0. But, 0 ą E k lim Ñ8 Ep q lim Ñ8 `T ` V ľ 0. (43) we therefore have a contradiction and the first claim is proven. Let now be an H 1 pr 3 q function satisfying the TISE with eigenvalue E k ă 0. Say that E k has multiplicty l, so that k,..., k`l 1 are the l orthonormal eigenfunctions with eigenvalue E k. WLOG, assume that E k 1 ă E k. Let φ be a sequence of Cc 8 pr 3 q functions converging to i in H 1 norm, with i ă k. Since satisfies the TISE with eigenvalue E k we have that E k p, i q lim E k p, φ q lim pxq s φ pxq dx ` V pxqpxqφ s pxqdx Ñ8 Ñ8 R 3 R 3 pxq s i pxq dx ` V pxqpxq s i pxqdx. (44) The integration by parts and passing the limits through the integrals is ustifed by the same argument appearing in Lemma 3. However, since i satisfies the TISE with eigenvalue E i, the same argument shows that E i p, i q pxq s i pxq dx ` V pxqpxq s i pxqdx. (45) Since E i E k, p, i q 0. Therefore is orthogonal to each i, with i ă k. Assume that pxq řk`l 1 i k p i, q i pxq is not the 0 function (if it is, then we are done - note that this the sum is ust the proection of onto the eigenspace of E k ). Let now, řk`l 1 pxq i k p i, q i pxq ppxq pxq řk`l 1 (46) p i, q i pxq 2 i k Then p is a normalized wave function that is orthogonal to each i, for i ĺ k ` l 1. Therefore, Ep p q ľ E k`l ą E k. However, p clearly satisfies the TISE with eigenvalue Ek and so by Lemma 3, Ep p q E k. This is a contradiction, and so pxq k`l 1 i k p i, q i pxq. (47) With a bit more work regarding defining the operator ` V as an unbounded operator on L 2, what we have proven can be turned into a statement about its spectrum. What we have shown is that the spectrum of ` V that lies below 0 consists purely of eigenvalues. The general picture is that above 0 one expects continuous spectrum which is related to scattering. 2. Lieb-Thirring Inequalities We turn now to a slightly different, however related, topic regarding the operator ` V, specifically that of bounds on large sums of its negative eigenvalues. We begin with a heuristic discussion which will hopefully motivate the topic. The semiclassical approach to quantum mechanics goes back to some of the earliest days of its development. The main idea was to literally quantize the classical

8 8 ELEMENTARY SPECTRAL THEORY OF SOME SCHRÖDINGER OPERATORS phase space (i.e. the 2d-dimensional space consisting of pairs pp, xq, with p the momentum of a particle in n dimensions and x its position) by saying that every allowed unit of p2πq n of phase space volume can support a single quantum state. This prescription allows one to calculate the number of negative eigenvalues by integration as follows: E 0 «R p2πq n Θp p 2 V pxqqdpdx. (48) nˆrn Here, Θptq is the step function which is 0 for t ă 0 and 1 otherwise. The heuristic ustification of (48) is as follows. The RHS is the volume of phase space where the classical energy of the particle, p 2 ` V pxq, is negative. Dividing this volume by the normalization p2πq n we obtain the number of quantum states that this part of phase space can support. This is equal to the LHS, the number of nonpositive eigenvalues (here 0 0 1). It is easy to do the p integral first; for every fixed x, the p integration ust gives the volume of the d dimensional ball of radius a V pxq, where V pxq maxt0, V pxqu is the negative part of the potential. Hence, E 0 1 V pxq n{2 dx. (49) p4πq n{2 Γpn{2 ` 1q R n Above, Γptq is the gamma function. One can go further and compute moments of the negative eigenvalues, ř E γ for γ ľ 0. Now every volume p2πq n of phase space where p 2 ` V pxq ă 0 contributes p 2 ` V pxq γ to this sum. By this reasoning, E γ «p 2 ` V pxq γ dpdx. (50) p 2`V pxqă0 Again, one can do the p integration first and arrive at E γ «L cl γ,n V pxq γ`n{2 dx, (51) R n where L cl γ,n is the classical constant L cl γ,n p2πq n R n : p ĺ1 p1 p 2 q γ dp Γpγ ` 1q p4πq n{2 Γpγ ` 1 ` n{2q. (52) The interesting fact is that under suitable assumptions on V, the formula (51) is actually asymptotically correct in the semiclassical limit, where we scale V Ñ λv and send λ Ñ 8 (see Theorem in [LL] for the case γ 1). In physics, this corresponds to taking the value of Planck s constant Ñ 0. We will not be interested in this semiclassical limit, but we will be interested in whether or not the equality in (51) can be turned into an inequality for the sums of moments of negative eigenvalues which holds for any potential V. Our next theorem says that this can, in fact, be achieved. We record here suitable assumptions on the potential V pxq, which generalize the assumptions we made in the case n 3. Suitable assumptions on V : R n Ñ R will from now on mean: $ & L n{2 pr n q ` L 8 pr n q, n ľ 3, V P L 1`ε pr 2 q ` L 8 pr 2 q, n 2, (53) % L 1 pr 1 q ` L 8 pr 1 q, n 1

9 ELEMENTARY SPECTRAL THEORY OF SOME SCHRÖDINGER OPERATORS 9 Theorem 3. Fix γ ľ 0. Assume that the potential V V` V satisfies (53) and that V P L γ`d{2 pr n q. Let E 0 ă E 1 ĺ E 2... be the negative eigenvalues, if there are any. Then, for suitable n, there is a constant L γ,n so that E γ ĺ L γ,n V pxq R γ`n{2 dx. (54) n ľ0 By suitable n, we mean that the inequality holds for the following pairs of γ and n: γ ľ 1 2 for n 1, γ ą 0 for n 2, γ ľ 0 for n 3. (55) For any other pair, there is a choice of potential V that violates (54). We can take the constants to be $ & pn ` γqγpγ{2q 2 Γpγ ` 1 ` n{2q, if n ą 1, γ ą 0 L γ,n p4πq n{2 2 γ γ ˆ or n 1, γ ľ 1, (56) %? π{pγ2 1{4q, if n 1, γ ą 1{2 Note that our assumptions on the potential here are weaker than the assumptions of Theorem 2, and so we are not guaranteed discrete spectra below 0. However, our definition of the eigenvalues is still valid: we define inductively each higher eigenvalue only if the previous eigenvalues are achieved by requiring that the functions we are minimizing over are orthogonal to the previous eigenfunctions. Note, however, that through an application of the min-max principle (i.e., Theorem 12.1 in [LL]), it is easy to see that the operator ` V can only have eigenvalues below 0, by the assumptions on V. The proof of (54) in all cases except n ľ 3, γ 0 and n 1, γ 1{2 is due to E.H. Lieb and W. Thirring [LT]. The inequalities in the case of γ 0, n ľ 3 were proven independently in [C], [L] and [R], each by completely different methods and are known as the CLR bounds (for an amusing anecdote of B. Simon regarding the three almost simultaneuous proofs see [Si]. For a correction to this anecdote see [SRY]). The proof in the case γ 1{2, n 1 came much later in [1]. It is also one of the few cases where the sharp constant is currently known [HLT]. It is an open (if not as active as it once was) area of research to compute the sharp constants in (54). It is known in some cases that L γ,n L cl γ,n, while in other cases that L γ,n ą L cl γ,n. Unfortunately the sharp constant is not known in the physically most interesting case, γ 1, n 3. It is conectured that L cl 1,3 L 1,3 [LT]. It is also known that for γ ă 1, L γ,n ą L cl γ,n The Birman-Schwinger Principle. Birman [B] and Schwinger [Sch] independently discovered that the problem of computing the number of eigenvalues of ` V that lie below some number can be recast as a problem of computing the number of eigenvalues of an integral kernal operator. For U ľ 0, U P L γ`n{2 pr n q consider the eigenvalue equation p Uq E with P H 1 pr n q and E ą 0. Define φpxq a Upxqpxq. Our eigenvalue equation then says that p ` Eqpxq a Upxqφpxq, or equivalently, p ` Eq 1a Upxqφpxq. Therefore if pxq is an eigenvalue of ` E, then φ satisfies φ K E φ (57) where K E (called the Birman-Schwinger kernal) is the integral kernal operator given by K E px, yq a 1 Upxq ` E px, yqa Upyq (58)

10 10 ELEMENTARY SPECTRAL THEORY OF SOME SCHRÖDINGER OPERATORS where p `Eq 1 px, yq is the usual Green s function for `E (i.e., the Yukawa potential). Explicity, 1 E px, yq G Epx yq, 8 G E pxq p4πtq n{2 2 x exp 4t 0 Et dt. (59) Elementary properties of the Yukawa potential are proven in Theorem 6.23 of [LL]. We record here a few facts that we will use in our study of the Birman-Schwinger kernel. Proposition 1. The Yukawa potential G E pxq is in L q pr n q if 1 ĺ q ĺ 8 if n 1, 1 ĺ q ă 8, if n 2, or 1 ĺ q ă n{pn 2q if n ľ 3. If f P L p pr n q for some 1 ĺ p ĺ 8, then upxq G E fpxq P L s pr n q where, p ĺ s ĺ 8 if n 1 ; p ĺ s ĺ 8 when p ą 1 and n 2; 1 ĺ s ĺ 8 when p 1 and n 2 ; p ĺ s ĺ np{pn 2pq when 1 ă p ă n{2 and n ľ 3; p ĺ s ĺ 8 when p ľ n{2 and n ľ 3; 1 ĺ s ĺ n{pn 2q when p 1 and n ľ 3. Lastly, the Fourier transform of G E is given by 1 yg E pkq p2πkq 2 (60) ` E We collect some elementary properties of the Birman Schwinger operator. Because we will not provide proofs of the n 1, γ 1{2 and n ľ 3, γ 0 cases of the LT inequalities (54), we will always assume that γ ą 0 and if n 1 that γ ą 1{2. Lemma 5. The Birman-Schwinger operator is a bounded operator from L 2 to L 2. It is positive; that is, it satisfies pf, K E fq ľ 0 for every f P L 2. It is compact; that is, if f á f weakly in L 2 pr n q, then K E f Ñ K E f strongly in L 2 pr n q. It is monotonically decreasing in E; that is, if E ă E 1, then pf, K E fq ľ pf, K E 1fq for every f. Remark. That K E is compact actually follows from the fact that it is Schatten class; that is, for m large enough, trpk E q m ă 8. The fact that this implies that K E is compact requires quite a bit of functional analysis machinery, and we will therefore look for a different proof of the compactness of K E. Proof. Let us first prove boundedness. For any L 2 pr n q function f a straightforward calculation using Hölder s inequality shows that? Uf is in L r pr n q with r 2pγ ` n{2q.. (61) 1 ` γ ` n{2 Furthermore, the L r norm of? Uf is bounded by a constant times the L 2 norm of f. Note that 1 ă r ă 2. For L 2 pr n q functions f and g,? pk E g, fq fpxq s UpxqGE px yqgpyq? Upyqdxdy, (62) R nˆrn if the functions on the RHS are integrable. However, Young s inequality (Theorem 4.2 in [LL]) states that, ˇ ˇpfpxq s? UpxqqpG E px yqqpgpyq? Upyqqˇ dxdy ĺ C?? r Uf Ug }G E } q (63) r R nˆr n where 2{r `1{q 2 and C a constant. It is straightforward but tedious to check that with r as in (61) and q as in Proposition 1 that this equality for r and q can be satisfied. Since the L r norm of? Uf is bounded by a constant times the L 2 norm of f, we have f Ñ pk E g, fq is a bounded linear functional on L 2 with norm less than a constant times the L 2 norm of g, and so the mapping g Ñ K E g P L 2 is bounded.

11 ELEMENTARY SPECTRAL THEORY OF SOME SCHRÖDINGER OPERATORS 11 Let us now prove positivity. First consider f P Cc 8 pr n q. Then, by our assumptions on U,? Uf P L 2 pr n q, and by Proposition 1, G E p? Ufq P L 2 pr n q, and denotes convolution as usual. By Plancherel s theorem, pf, K E fq p? Uf, G E?Ufq p z? { Uf, G E?Ufq, (64) with p h denoting the Fourier transform of h. Because? Uf P L 2 pr n q and because of the regularity on G E stated in Proposition 1 (i.e., which L q spaces G E is in), we may apply Theorem 5.8 of [LL] to turn the Fourier transform of a convolution into the product of the Fourier transforms: { G E?Uf y G E z? Uf. (65) Here, it is crucial which L p spaces our functions are in. Since the Fourier transform of G E is positive by Proposition 1, we have immediately by (64) and (65) that pf, K E fq ľ 0 for f P C 8 c pr n q. By density of C 8 c pr n q in L 2 pr n q, positivity of K E for general f P L 2 pr n q follows immediately. Additionally, monotonicity also follows immediately from the Fourier characterization of G E. For if E ă E 1, then y G E pkq ľ y G E 1pkq. By our above argument, pf, K E fq ľ pf, K E 1fq for every f P C 8 c pr n q and by density this extends to all of L 2 pr n q. Lastly, we prove compactness. We first comment that Corollary 1 holds for our U pxq because of our (stronger) assumption that U P L n{2`γ pr n q, even if n 3 (Corollary 1 is stated only in the case n 3). The proof in the case n 3 is identical and is the content of Theorem 11.4 in [LL]. We take the result without proof. Our claim will therefore follow if we can prove that f Ñ G E?Uf is a bounded mapping from L 2 pr n q to H 1 pr n q. We already have argued that f Ñ? Uf is a bounded mapping of L 2 pr n q to L r pr n q with r as in (61). We therefore need only show that g Ñ G E g is a bounded mapping of L r pr n q into H 1 pr n q. However, that G E g is in L 2 pr n q follows from Proposition 1 (again, this is straightforward but tedious to check). By the Fourier characterization of H 1 pr n q (see, e.g., Theorem 7.9 of [LL]) we need only prove that k { G E gpkq P L 2 pr n q (i.e., that its first derivative is square integrable). Again, the constant r and the regularity of G E given in Proposition 1 are such that we may apply Theorem 5.8 of [LL] to write {G E gpkq y G E pkqpgpkq. (66) Here, pgpkq P L r1 pr n q, with r 1 the dual index to r (here, we are applying the L r Fourier transform which exists because r ĺ 2). We have then that k G { E g 2 2 ĺ p2πq 2 R n 2 pgpkq p2πkq 2 dk (67) ` E where we have used that k 2 {pr2πks 2 ` Eq ĺ p2πq 2. By Hölder s inequality, the RHS is bounded above by a power of the L r1 norm of pg times a constant. By Hausdorff-Young inequality (Theorem 5.7 in [LL]), we have that }pg} r 1 ĺ C }g} r for a constant C. We therefore conclude that g Ñ G E g is indeed a bounded linear map of L r pr n q into H 1 pr n q. This proves our claim. We have now characterized K E as a compact and positive integral kernal operator on L 2 pr n q. By the spectral theorem, it therefore has a list of eigenvalues, where denote the th eigenvalue of K E by λ E, which are nonnegative, decreasing in, and converge to 0. With this in hand, we prove the Birman-Schwinger principle: Lemma 6. Let N E puq denote the number of eigenvalues of U that are less than E. Then N E puq equals the number of eigenvalues of K E that are greater than 1.

12 12 ELEMENTARY SPECTRAL THEORY OF SOME SCHRÖDINGER OPERATORS Proof. We have already seen that every solution of the Schrödinger equation gives rise to an eigenfunction φ? U of the Birman Schwinger kernel. Note that φ cannot be 0. If it were, this would imply that E which is impossible for an H 1 pr n q function. Now, if φ is an L 2 pr n q function satisfying K E φ φ, then we define p ` Eq 1? Uφ. Our argument using the Fourier characterization of G E to prove compactness of the Birman-Schwinger kernel in the proof of Lemma 5 shows that is an H 1 pr n q function. We have then p ` Eq? Uφ? UK E φ U (68) and so is a solution of the Schrödinger equation with eigenvalue E. This one-to-one correspondence between and φ implies that the multiplicities of the eigenvalue E of U and the eigenvalue 1 of K E are the same. Since the Birman-Schwinger kernel is decreasing in E, we see that the function E Ñ λ E, the th eigenvalue of K E, is monotonically decreasing. In particular, λ 1 E (the first and largest eigenvalue of K E ) is a monotonically decreasing function of E. Now if E is very large, then every eigenvalue of K E will lie below 1 because K E Ñ 0 uniformly as E Ñ 8 (this follows from the Fourier characterization of G E used in the proof of Lemma 5) and we will also have that N E puq 0 (our assumptions on U imply that U is bounded below - Lemma 1). Now as we start to decrease E, eventually we will have λ 1 E 1 for some E, when λ1 E crosses the threshold 1. At this point, we have N E puq 1 (by our one-to-one correspondence between φ and described above) and there is precisely one eigenvalue of K E above 1 (here we assume that the ground state is unique - if it is not, then the first m functions E Ñ λ E all cross 1 simultaneously and so N E puq m too). Now we continue to decrease E. Each time that, for some, the function E Ñ λ E crosses the threshold 1, we get another eigenvalue of U at this value E and N E puq increases by 1. On the other hand, everytime N E puq increases by 1, we get another eigenvalue of K E, and by monotonicty, this eigenvalue is larger than 1 for all larger E. This proves the claim. Remark. We have implicitly assumed that the functions E Ñ λ E are continuous in E. However it is easy to see that this is the case. For 0 ă E ĺ E 1, we have from the Fourier representation of G E, the inequality 0 ĺ K E K E 1 ĺ rpe 1 Eq{E 1 sk E. By the min-max principle (see, e.g., Thm 12.1 of [LL]) the eigenvalues of K E differ from the corresponding eigenvalues of K E 1 by at most pe 1 Eq{E 1 times the norm of K E. This proves continuity Proof of the LT inequalities (54).. By the min-max principle, the eigenvalues of ` V are all larger than the eigenvalues of V so it is no loss of generality to assume that V V. We continue to denote U V. Lemma 6 implies that N E puq ĺ N pmq E : pλ E qm, (69) where the RHS is possibly 8. The RHS is ust the trace of pk E q m. Note that pk E q m is perfectly well defined as a (positive) operator through the functional calculus since K E ľ 0. We therefore have N E puq ĺ N pmq E trp? UG E? Uq m ĺ trpuq m{2 pg E q m puq m{2 Upxq m G E p0qdx R ˆ n 1 R pp2πkq 2 ` Eq m Upxq m dx. (70) n R n Above we have used the operator trace inequality trpb 1{2 AB 1{2 q m ĺ tr B m{2 A m B m{2 which holds for positive A and B. A proof of this inequality is in [LS]. Since puq m{2 pg E q m puq m{2 is an integral kernel

13 ELEMENTARY SPECTRAL THEORY OF SOME SCHRÖDINGER OPERATORS 13 operator, its trace is ust the integral over its diagonal entries, which we have applied in the third line. The fourth line follows by the Fourier characterization of G E in Proposition 1. The integral over k is finite iff 2m ą n, in which case it is equal to 1 Γpm n{2q R pp2πkq 2 p4πq n{2 E m`n{2. (71) ` Eqm Γpmq n We wish to employ the bound (70). We write E γ γ 8 0 N E puqe γ 1 de, (72) which follows by integration by parts and noting that the derivative of N E puq is ust a sum of delta functions at the numbers E. We cannot however use (70) directly in (72) or we would be led to a divergent integral. We instead note that N E puq ĺ N E{2 ppu E{2q`q (where pzq` : maxt0, zu). This follows from the fact that the number of eigenvalues for U below E must be the same as the number of eigenvalues below E{2 for U ` E{2, and so N E puq N E{2 pu E{2q. But by the min-max principle deleting the positive part of the potential only decreases the eigenvalues and so N E{2 pu E{2q ĺ N E{2 ppu E{2q`q. Using this bound in (72) and then using the bound (70) but with pu E{2q` in place of U, we obtain E γ ĺ p4πq n{2 Γpm n{2q γ Γpmq 8 We do the E-integration first. A computation shows R n ˆ Upxq E 2 m pa Eq s`e 1 t de A s`t`1 p1 yq s y t dy 0 ` m`n{2 ˆE dxe γ 1 de. (73) 2 A s`t`1 Γps ` 1qΓpt ` 1q{Γps ` t ` 2q (74) Therefore, E γ ĺ p4πq n{2 2 γ Γpm n{2qγp m ` γ ` n{2q γm Upxq γ`n{2 dx. (75) Γpγ ` 1 ` n{2q R n In order for the E-integration in (73) to be finite, we require m ` n{2 ` γ ą 0. Recall that we also require m ą n{2. We therefore require γ ` n{2 ą m ą n{2. By choosing m pγ ` nq{2 when n ą 1 or n 1, γ ľ 1 and m 1 for other cases, we obtain the claims (54) except in the critical cases n ľ 3, γ 0 and n 1, γ 1{2. 3. Kinetic Energy Inequalities We will now give a useful application of the LT inequalities (54). In the case γ 1 we have, E ĺ L 1,n V pxq R 1`n{2 dx. (76) n Our goal is to use (76) to obtain an upper bound on the kinetic energy of N quantum particles. More precisely, let P H 1 pr Nn q be a function with L 2 norm 1. is a wave function describing N quantum particles and the function n ρ pxq : px 1,..., x N q 2 dx 1... dx y i...dx N (77) R pn 1qn

14 14 ELEMENTARY SPECTRAL THEORY OF SOME SCHRÖDINGER OPERATORS (where the hat indicates that dx x i is omitted in the integration) is the probability density associated with finding a particle at the point x P R n. The kinetic energy of the N particles is ust the sum of the kinetic energies of each of the individual particles: N T xi px 1,..., x N q 2 dx 1...dx N. (78) R Nn The single particle density matrix associated with is given by N γ p1q px, x1 q px 1,..., x i 1, x,..., x N qpx 1,..., x i 1, x 1,..., x N qdx 1... dx y i...dx N. (79) It is easy to see that γ p1q largest eigenvalue is denoted γ p1q is a positive integral kernal operator on L2 pr n q satisfying tr γ p1q N. Its 8. It is bounded above by N. If is antisymmetric function (i.e. describes fermions) then γ p1q 8 is less than 1. With these definitions we can state the following fundamental kinetic energy inequality: Theorem 4. With the kinetic energy and density defined above, the inequality (76) implies K T ľ ρ γ p1q {p pxq p 1 dx (80) p1 8 R n where p 1 ` n{2 (the power appearing in (76)) p 1 p{pp 1q (the dual index to p) and K satisfying ppl 1,n q p1 pp 1 Kq p 1. (81) Proof. Consider now the operator H ` V acting on H 1 pr n q, i.e., single particle wave functions, where V is a potential to be determined. We then consider the N-body operator N K N H i (82) with H i acting as H on the ith particle. With the aid of the one particle density matrix γ p1q write p, K N q tr Hγ p1q we can ı. (83) Let us pause to comment on the above notation. Instead of worrying about defining as an unbounded operator on a dense domain, we are interpreting the expectation p, K N q by the formally associated quadratic form, i.e., N p, K N q T ` V px i q px 1,..., x N q 2 dx 1...dx N. (84) R Nn Note that N V px i q px 1,..., x N q 2 dx 1...dx N R Nn V pxqρ pxqdx R n (85) For positive trace class operators A on L 2 pr n q we interpret trphaq as follows. Since A has the decomposition A ř 8 λ iϕ i pϕ i, q for orthonormal i P L 2 pr n q and positive λ i, we can define 8 tr rhas : λ i Epϕ i q (86) with Epφq as before. We define the trace wherever the RHS makes sense (in particular, whever the sum is absolutely summable and every ϕ P H 1 pr n q.)

15 ELEMENTARY SPECTRAL THEORY OF SOME SCHRÖDINGER OPERATORS 15 It is the content of Chapter 3 of [LS] that the equality (83) holds (see the footnote on page 45). We now continue with the proof. The minimum of of tr rhas over all positive trace-class operators A with A 8 ĺ γ p1q 8 is clearly given by the sum of the negative eigenvalues of H times γ p1q 8. That is, the optimal choice of A is γ p1q 8 times the proection onto the negative spectral subspace of H. Therefore, p, K N q ľ γ p1q 8 E. (87) We now apply (76): T ` V pxqρ pxqdx p, K N q ľ γ p1q 8L 1,n V pxq R R p dx. (88) n n We now make a choice for V. Let V pxq Cρ pxq 1{pp 1q where C ą 0 is a constant to be determined. We obtain the inequality T ľ C ρ pxq p 1dx γ p1q 8L 1,n C p ρ pxq p 1 dx. (89) R n R n We know optimize over C and choose C pp γ p1q 8L 1,n q p1 {p. This proves the claim. An interesting fact is that the kinetic energy inequality is equivalent to the LT inequality for γ 1. This is captured by the following theorem: Theorem 5. Suppose that (80) holds for every N particle wave function (for every N), with the constant K as in (81). Then (76) holds. Proof. Let be the N particle wave function formed by the Slater determinant of the eigenfunctions for the N lowest eigenvalues of ` V. More precisely, if E are the negative eigenvalues of ` V and p ` V q E, then It is straightforward to check that px 1,..., x N q pn!q 1{2 dett i px qu N i,. (90) N 1 0 E T ` V pxqρ pxqdx. R n (91) For of this form, we have that γ p1q 8 ĺ 1. Applying (80) we have T ` V pxqρ pxqdx ľ K ρ pxq p 1 dx rppl 1,n q 1{p V pxqsrppl q 1{p 1,n ρ pxqsdx R n R n R n ľ K ρ pxq p 1dx 1 R p 1 rppl q 1{p 1,n ρ pxqs p 1 dx n R n 1 rppl 1,n q 1{p V pxqs p dx L 1,n V pxq p dx. (92) p R n R n The third equality comes from Hölder s inequality followed by ab ĺ a p {p ` b p1 {p 1. The equality comes from the definition of the constants in 81. Taking N Ñ 8 yields the claim. What these two theorems show are that the inequalities (in the case of antisymmetric ) tr p ` V q ĺ V pxq p dx (93) and p, N xi q ľ K ρ pxq p 1 dx (94) R n

16 16 ELEMENTARY SPECTRAL THEORY OF SOME SCHRÖDINGER OPERATORS are dual to each other, in the sense that the integrands on the RHS are a certain Legendre transform of each other. The same holds for the LHS. For suitable convex functions, taking a double Legendre transform reproduces the original function. Thus, Theorems 4 and 5 represent an attractive method of computing sharp constants in the LT inequality in the physical case of interest, that is with γ 1. If one can compute the sharp constant in the kinetic energy inequality (80), then one immediately obtains the sharp constant in (76). References [B] M. Sh. Birman, The spectrum of singular boundary problems, Math. Sb. 55, (1971). [C] M. Cwikel, Weak type estimates for singular values and the number of bound states of Schrödinger operators, Ann. of Math. 106, (1977). [HLT] D. Hundertmark, E.H. Lieb, and L.E. Thomas, A sharp bound for an eigenvalue moment of the one-dimensional Schrödinger operator Adv. Theor. Math. Phys. 2, (1998). [L] E.H. Lieb, The Number of Bound States of One-Body Schrödinger Operators and the Weyl Problem, Proc. Amer. Math. Soc. Symposia in Pure Math. 36, (1980). [LL] E.H. Lieb, M. Loss, Analysis, American Mathematical Society, Providence, Rhode Island (2001). [LS] E.H. Lieb, R. Seiringer, The Stability of Matter in Quantum Mechanics, Cambridge University Press, New York (2010). [LT] E.H. Lieb, W. Thirring, Inequalities for the Moments of the Eigenvalues of the Schrödinger Hamiltonian and Their Relation to Sobolev Inequalities, in Studies in Mathematical Physics, E. Lieb, B. Simon, A. Wightman, eds., Princeton University Press, (1976). [R] G.V. Rosenblum, Distribution of the discrete spectrum of singular differential operators, Izv. Vyss. Ucebn. Zaved. Matematika 164, (1976). [Si] B. Simon, Mathematical physics at Princeton in the 1970s, Bulletin of the IAMP, July [Sch] J. Schwinger, On the bound states of a given potential, Proc. Nat. Acad. Sci. U. S.A. 47, (1961). [SRY] M. Solomyak, G. Rozenblum, D. Yafaev, Letter to the Editor: On the early history of the CLR estimate, Bulletin of the IAMP, October [1] [W] T. Weidl, On the Lieb-Thirring constants L γ,1 for γ ľ 1{2, Commun. Math. Phys. 178, (1996).