A generalized virial theorem and the balance of kinetic and potential energies in the semiclassical limit
aa r X i v : . [ m a t h . SP ] S e p A GENERALIZED VIRIAL THEOREM AND THE BALANCE OFKINETIC AND POTENTIAL ENERGIES IN THESEMICLASSICAL LIMIT
D. R. YAFAEV
To the memory of Pierre Duclos
Abstract.
We obtain two-sided bounds on kinetic and potential energies ofa bound state of a quantum particle in the semiclassical limit, as the Planckconstant ~ →
0. Proofs of these results rely on the generalized virial theoremobtained in the paper as well as on a decay of eigenfunctions in the classicallyforbidden region. Introduction
Let us consider an eigenfunction ψ ~ ( x ) of the Schr¨odinger operator H ~ = − ~ ∆ + v ( x ) , v ( x ) = v ( x ) , (1.1)in the space L ( R d ), that is − ~ ∆ ψ ~ ( x ) + v ( x ) ψ ~ ( x ) = λ ~ ψ ~ ( x ) , ψ ~ ∈ L ( R d ) . (1.2)We always suppose that the function v ( x ) is semibounded from below, and hencewe can set min x ∈ R d v ( x ) = 0 . We typically assume that the potential v ( x ) contains wells and study ψ ~ ( x ) for λ ~ close to some non-critical value λ > ∇ v ( x ) = 0 for all x such that v ( x ) = λ ). In particular, λ ~ is separated from bottoms of potential wells. Theeigenfunctions ψ ~ ( x ) are supposed to be real and normalized, that is Z R d ψ ~ ( x ) dx = 1 . Our goal is to study the behavior of the kinetic K ( ψ ~ ) = ~ Z R d |∇ ψ ~ ( x ) | dx and potential U ( ψ ~ ) = Z R d v ( x ) ψ ~ ( x ) dx energies as ~ → λ ~ = K ( ψ ~ ) + U ( ψ ~ ) (1.3) Mathematics Subject Classification.
Key words and phrases.
Virial theorem, semiclassical limit, classically forbidden region, esti-mates of kinetic and potential energies.Partially supported by the project NONAa, ANR-08-BLANC-0228.
D. R. YAFAEV is close to λ . To be more precise, we discuss the following
Problem 1.1.
Is it true that K ( ψ ~ ) ≥ c > λ ~ in a neighborhood of a non-critical point λ > ~ ?In view of equation (1.3) this problem can be equivalently reformulated in termsof the potential energy U ( ψ ~ ) as the inequality U ( ψ ~ ) ≤ λ ~ − c. (1.5)Note that, for small ~ , the eigenfunctions ψ ~ are essentially localized (see [5, 8, 2])in the classically allowed region where v ( x ) ≤ λ ~ + ε (for an arbitrary ε > ψ ~ arenot too strongly localized in a neighborhood of the set v ( x ) = λ ~ .Let us discuss Problem 1.1 in a heuristic way. The first level of the discussionis very superficial. Actually, as ~ →
0, one might expect that the term − ~ ∆disappears so that in the limit we obtain the operator of multiplication by thefunction v ( x ). This operator has a continuous spectrum and its “eigenfunctions”are Dirac functions of the variable v ( x ) − λ . Such functions “live” in a neighborhoodof the set v ( x ) = λ which might eventually interfere with the positive answer toProblem 1.1.The second level is, on the contrary, quite deep and stipulates that, for small ~ ,the behavior of a quantum particle with Hamiltonian (1.1) is close to the behaviorof the corresponding classical particle, and hence the classical equations of motioncan be used. In this context we mention book [1] relying on the method of Maslovcanonical operator and papers [4, 3] relying on methods of microlocal analysis.We avoid this deep level using only the virial theorem and the fact that a quantumparticle of energy λ ~ should be essentially localized as ~ → Problem 1.1 arose by the proof of the limiting absorption principle for theHamiltonian K of a quantum particle moving in a magnetic field of an infinitestraight current (see [9]). This problem reduces to a study of eigenvalues λ ~ closeto a point λ > v ( x ) = v ln | x | , v > , x ∈ R . The limiting absorption principle for the operator K requires the estimate dλ ~ /d ~ ≥ c ~ − , c > . It looks somewhat exotic but in view of the formula (see, e.g., [6]) dλ ~ /d ~ = 2 ~ Z R d |∇ ψ ~ ( x ) | dx, it is equivalent to estimate (1.4). We also discuss a problem dual to Problem 1.1.
Problem 1.2.
Is it true that K ( ψ ~ ) ≤ λ ~ − c, c > , for all λ in a neighborhood of a non-critical point λ > ~ ? BALANCE OF ENERGIES 3
This fact is equivalent to the estimate U ( ψ ~ ) ≥ c > ψ ~ are not too strongly localized at thebottom of the potential well.2. A generalized virial theorem
Below the operator H ~ is always defined by formula (1.1). The followingresult generalizes the classical virial theorem. Theorem 2.1.
Suppose that v ∈ C ( R d ) . Let a = ¯ a ∈ C ( R d ) , and let its fourderivatives be bounded. Then eigenfunctions ψ ~ of the operator H ~ satisfy an iden-tity Z R d (cid:16) ~ d X j,k =1 a jk ∂ k ψ ~ ∂ j ψ ~ − ~ (∆ a ) ψ ~ − h∇ a, ∇ v i ψ ~ (cid:17) dx = 0 , (2.1) where a jk = ∂ a/∂x j ∂x k .Proof. Let A = d X j =1 (cid:0) a j ( x ) ∂ j + ∂ j a j ( x ) (cid:1) , a j = ∂a/∂x j , be a general self-adjoint first order differential operator. Then the commutators[ − ∆ , A ] = − d X j,k =1 ∂ j a jk ∂ k − ∆ a and [ v, A ] = − h∇ a, ∇ v i . It remains to use that, for a function ψ ~ satisfying equation (1.2), the identity([ H ~ , A ] ψ ~ , ψ ~ ) = 0holds. (cid:3) Note that since eigenfunctions corresponding to isolated eigenvalues decay expo-nentially, identity (2.1) requires practically no assumptions on the behavior of thefunction a ( x ) as | x | → ∞ . However we consider only bounded functions a ( x ).If a ( x ) = x , then identity (2.1) reduces to the usual (see, e.g., [7]) form2 ~ Z R d |∇ ψ ~ ( x ) | dx = Z R d rv r ( x ) ψ ~ ( x ) dx, r = | x | , (2.2)of the virial theorem. Combining equations (1.3) and (2.2) we see that Z R d (cid:0) − rv r ( x ) + v ( x ) (cid:1) ψ ~ ( x ) dx = λ ~ , (2.3)where the eigenfunctions ψ ~ are real and normalized. As is well known, for homogeneous potentials, the kinetic K ( ψ ~ ) and po-tential U ( ψ ~ ) energies are related to the total energy by exact equalities. Proposition 2.2.
Suppose that v ( tx ) = t α v ( x ) , t > , α > . (2.4) Then K ( ψ ~ ) = α ( α + 2) − λ ~ and U ( ψ ~ ) = 2( α + 2) − λ ~ . (2.5) D. R. YAFAEV
Indeed, it suffices to use that rv r ( x ) = αv ( x ) in (2.3).This result remains true for some α <
0; however to define H ~ as a self-adjointoperator, we have to require that | α | be not too large. The case of homogeneous potentials is of course exceptional. In general,one cannot expect (if d >
1) to have even a semiclassical asymptotics of the kinetic(or potential) energy. Let us consider a simple
Example 2.3.
Set v ( x ) = | x | α + | x | α , x = ( x , x ) ∈ R , α j > , j = 1 , . (2.6)For every λ > u ∈ (2( α + 2) − λ, α + 2) − λ ) , there exist sequences of eigenvalues λ ~ and eigenfunctions ψ ~ of the operator H ~ such that λ ~ → λ and U ( ψ ~ ) → u as ~ → a ( j ) n , n = 1 , . . . , be eigenvalues of the one-dimensional operators − D j + | x j | α j , j = 1 ,
2. Eigenvalues λ ~ of operator (1.1) with potential (2.6) aregiven by the formula λ ~ = λ (1) ~ + λ (2) ~ (2.7)where λ ( j ) ~ = ~ γ j a ( j ) n j , j = 1 , γ j = 2 α j ( α j + 2) − and n j = 1 , . . . are arbitrary.Normalized eigenfunctions ψ ~ are scaled products of normalized eigenfunctions ϕ ( j ) n j of these two one-dimensional operators, that is ψ ~ ( x ) = ~ − ( β + β ) / ϕ (1) n ( ~ − β x ) ϕ (2) n ( ~ − β x ) , β j = 2( α j + 2) − . For this eigenfunction, the potential energy equals U ( ψ ~ ) = ~ γ Z R | x | α ϕ (1) n ( x ) dx + ~ γ Z R | x | α ϕ (2) n ( x ) dx so that in view of the second formula (2.5) U ( ψ ~ ) = β ~ γ a (1) n + β ~ γ a (2) n . (2.8)Let us now take into account that a ( j ) n = c j n γ j (1 + o (1)) , c j > , n → ∞ . Pick some numbers µ j > n j = [( µ j c − j ) /γ j ~ − ] where [ b ] is the integerpart of a number b . Then λ ( j ) ~ = ~ γ j a ( j ) n j → µ j (2.9)as ~ →
0. Moreover, it follows from (2.8) thatlim ~ → U ( ψ ~ ) = 2( α + 2) − µ + 2( α + 2) − µ . (2.10)If µ + µ = λ , then according to (2.7) and (2.9) λ ~ → λ . However the limit (2.10)may take arbitrary values between 2( α + 2) − λ and 2( α + 2) − λ . BALANCE OF ENERGIES 5 Estimates of the kinetic energy
In addition to the virial theorem, we need results on a decay of eigenfunc-tions in the classically forbidden region. We suppose thatlim inf | x |→∞ v ( x ) =: v ∞ > λ ~ which belong to a neighborhood of some non-criticalenergy λ ∈ (0 , v ∞ ). Our construction works for v ∈ C ( R d ), but in order to usethe results on a decay of eigenfunctions we assume that v ∈ C ∞ ( R d ). As always,eigenfunctions ψ ~ of the operator H ~ are real and normalized. Let the sets F ( λ )and G ( λ ) be defined by the formulas F ( λ ) = { x ∈ R d : v ( x ) < λ } , G ( λ ) = R d \ F ( λ ) . Then (see [5], [8] as well as [2] and references therein) for all fixed ε > Z G ( λ ~ + ε ) (cid:0) ~ |∇ ψ ~ | + ψ ~ ( x ) (cid:1) dx → ~ →
0. Actually, eigenfunctions ψ ~ decay exponentially as ~ → G ( λ ~ + ε ), but we do not need this result.Relation (3.1) can be supplemented by an estimate of the potential energy. Ofcourse the next lemma is useful only in the case when v ( x ) is not bounded atinfinity. Lemma 3.1.
For all ε > , we have Z G ( λ ~ + ε ) v ( x ) ψ ~ ( x ) dx → as ~ → .Proof. In view of (3.1) it suffices to check that, for some R ,lim ~ → Z | x |≥ R v ( x ) ψ ~ ( x ) dx = 0 . (3.2)Choose R such that v ( x ) ≥ λ ~ + ε for | x | ≥ R/
2. Let η ∈ C ∞ ( R d ) be such that η ( x ) ≥ η ( x ) = 0 for | x | ≤ R/ η ( x ) = 1 for | x | ≥ R . Multiplying equation(1.2) by η and integrating by parts, we see that Z R d η (cid:0) ~ |∇ ψ ~ | + vψ ~ (cid:1) dx = λ ~ Z R d ηψ ~ dx − ~ Z R d h∇ ψ ~ , ∇ η i ψdx. (3.3)Let us consider the right-hand side. The first integral tends to zero as ~ → η ( x ) = 0 in the classically allowed region. Using the Schwarz inequalityand relation (1.3), we estimate the second term by ~ λ / ~ max |∇ η ( x ) | . Thereforeexpression (3.3) tends to zero as ~ → (cid:3) Relations (2.1) and (3.1) can be combined. The simplest example is givenin the next statement.
Proposition 3.2.
Let the function rv r ( x ) v ( x ) − be bounded as | x | → ∞ . Choosesome λ > . Suppose that, for some ε ∈ (0 , λ ) , the potential v ( x ) admitsrepresentation (2.4) in the region F ( λ + ε ) . Then for λ ~ ∈ ( λ − ε / , λ + ε / ,we have, as ~ → , the asymptotic relations K ( ψ ~ ) = α ( α + 2) − λ ~ + o (1) and U ( ψ ~ ) = 2( α + 2) − λ ~ + o (1) . (3.4) D. R. YAFAEV
Indeed, it follows from (2.3) and Lemma 3.1 that Z F ( λ + ε ) (cid:0) − rv r ( x ) + v ( x ) (cid:1) ψ ~ ( x ) dx = λ ~ + o (1) . By virtue of (2.4) the integral in the left-hand side equals2 − ( α + 2) Z F ( λ + ε ) v ( x ) ψ ~ ( x ) dx. Therefore using again Lemma 3.1, we obtain (3.4).Estimates (1.4) on the kinetic energy or, equivalently, (1.5) on the potential en-ergy can be obtained under much weaker assumptions on v ( x ). In view of (3.1) theprincipal difficulty is to exclude that eigenfunctions ψ ~ ( x ) are localized in neigh-borhoods of the surfaces v ( x ) = λ ~ . Let us formulate the main result of this paper.
Theorem 3.3.
Let λ > be a non-critical energy, and let for some ( sufficientlysmall ) ε > and all λ ∈ ( λ − ε , λ + ε ) F ( λ ) = N [ n =1 F n ( λ ) , F n ( λ ) ∩ F m ( λ ) = ∅ for n = m. (3.5) Suppose that for all n = 1 , . . . , N and some points x n ∈ F n ( λ − ε ) an inequality h x − x n , ∇ v ( x ) i ≥ c > , x ∈ F n ( λ + ε ) ∩ G ( λ − ε ) , (3.6) holds. Denote by ψ ~ normalized eigenfunctions ψ ~ of the operator H ~ correspondingto eigenvalues λ ~ in a neighborhood ( λ − ε / , λ + ε / of the point λ . Theninequality (1.4) is true for sufficiently small ~ . The assumptions of this theorem are, actually, very mild. Roughly speaking,we suppose that the classically allowed region F ( λ ) consists of a finite number ofpotential wells. Condition (3.6) means that v ( x ) increases as x passes through theboundary of F n ( λ ). This is consistent with the fact that F n ( λ ) is a potential wellof v ( x ) for the energy λ . If N = 1, then setting x = 0, we obtain that inequality(3.6) reduces to the condition v r ( x ) ≥ c > Lemma 3.4.
For all ε > and all δ ∈ (0 , λ ~ ) , we have Z F ( λ ~ + ε ) v ( x ) ψ ~ ( x ) dx ≤ λ ~ + ε − δ Z F ( λ ~ − δ ) ψ ~ ( x ) dx. Proof.
Observe that Z F ( λ ~ − δ ) v ( x ) ψ ~ ( x ) dx ≤ ( λ ~ − δ ) Z F ( λ ~ − δ ) ψ ~ ( x ) dx and Z F ( λ ~ + ε ) ∩ G ( λ ~ − δ ) v ( x ) ψ ~ ( x ) dx ≤ ( λ ~ + ε ) Z G ( λ ~ − δ ) ψ ~ ( x ) dx = ( λ ~ + ε ) (cid:0) − Z F ( λ ~ − δ ) ψ ~ ( x ) dx (cid:1) . So it suffices to put these two estimates together. (cid:3)
BALANCE OF ENERGIES 7
Combining Lemmas 3.1 and 3.4, we see that for all ε > δ ∈ (0 , λ ~ ), U ( ψ ~ ) ≤ λ ~ + ε − δ Z F ( λ ~ − δ ) ψ ~ ( x ) dx + σ ( ε, ~ ) (3.7)where σ ( ε, ~ ) → ~ → ε is fixed . The notation σ ( ε, ~ ) will also be usedbelow.The second estimate relies on the virial theorem which we need in the followingform. Lemma 3.5.
Let x n ∈ F n ( λ + ε ) be arbitrary points. Then N X n =1 Z F n ( λ + ε ) (cid:16) ~ |∇ ψ ~ ( x ) | − h x − x n , ∇ v ( x ) i ψ ~ ( x ) (cid:17) dx = o (1) , ~ → . (3.8) Proof.
Let us use Theorem 2.1 for a suitable function a ( x ) which we constructnow. Choose functions ϕ n ∈ C ∞ ( R d ) such that ϕ n ( x ) = 1 for x ∈ F n ( λ + ε )and ϕ n ( x ) = 0 away from some neighborhoods of F n ( λ + ε ) so that supp ϕ n ∩ supp ϕ m = ∅ if n = m . We define the function a ( x ) by the equality a ( x ) = N X n =1 | x − x n | ϕ n ( x ) . (3.9)Neglecting in (2.1) the classically forbidden region, we see that Z F ( λ + ε ) (cid:16) ~ d X j,k =1 a jk ∂ k ψ ~ ∂ j ψ ~ − ~ (∆ a ) ψ ~ − h∇ a, ∇ v i ψ ~ (cid:17) dx = o (1) (3.10)as ~ →
0. If x ∈ F n ( λ + ε ), then according to (3.9) we have ( ∇ a )( x ) = 2( x − x n ), a jj ( x ) = 2 and a jk ( x ) = 0 if j = k . Thus, relation (3.8) follows from (3.10). (cid:3) Lemma 3.6.
Let assumption (3.6) hold for some points x n ∈ F n ( λ − ε ) , and set c = max n sup x ∈ F n ( λ − ε ) ( −h x − x n , ∇ v ( x ) i ) . (3.11) Then ~ Z R d |∇ ψ ~ ( x ) | dx ≥ c − c Z F ( λ − ε ) ψ ~ ( x ) dx + σ ( ε , ~ ) (3.12) where c = c + c and σ ( ε , ~ ) → as ~ → .Proof. According to (3.6) and (3.11), we have Z F n ( λ + ε ) h x − x n , ∇ v ( x ) i ψ ~ ( x ) dx ≥ c Z F n ( λ + ε ) ∩ G ( λ − ε ) ψ ~ ( x ) dx − c Z F n ( λ − ε ) ψ ~ ( x ) dx. Summing these estimates over n = 1 , · · · , N and using (3.8), we see that2 ~ Z F ( λ + ε ) |∇ ψ ~ ( x ) | dx ≥ c Z F ( λ + ε ) ∩ G ( λ − ε ) ψ ~ ( x ) dx − c Z F ( λ − ε ) ψ ~ ( x ) dx + o (1) D. R. YAFAEV as ~ →
0. The first integral in the right-hand side equals 1 minus the integrals of ψ ~ ( x ) over F ( λ − ε ) and G ( λ + ε ). The integral over G ( λ + ε ) tends to zerobecause G ( λ + ε ) lies in the classically forbidden region. (cid:3) Using the energy conservation (1.3) and the obvious inclusion F ( λ − ε ) ⊂ F ( λ ~ − ε / U ( ψ ~ ) ≤ λ ~ − − c + 2 − c Z F ( λ ~ − ε / ψ ~ ( x ) dx + σ ( ε , ~ ) . (3.13)If c ≤
0, then (3.13) directly implies (1.5). So below we assume c > proof of Theorem 3.3. Let us compareestimates (3.7) where we set δ = ε / X = Z F ( λ ~ − ε / ψ ~ ( x ) dx is small, then we use (3.13). If it is big, we use (3.7). To be more precise, estimates(3.7) and (3.13) imply that U ( ψ ~ ) ≤ λ ~ + 2 − max ≤ X ≤ min { ε − ε X, − c + c X } + σ ( ε, ε , ~ ) , where σ ( ε, ε , ~ ) → ~ → ε and ε . Observe thatmax ≤ X ≤ min { ε − ε X, − c + c X } ≤ ε − ε c (1 + c ) − (if ε ≤ ε is arbitrary small, this yields estimates (1.5) and hence (1.4).In these estimates c is any number smaller than 2 − ε c (1 + c + c ) − . Our lower bound on the potential energy (and hence an upper bound onthe kinetic energy) is almost trivial.
Proposition 3.7.
Let, for some λ > and ε > , representation (3.5) hold for λ = λ + ε . Suppose that for all n = 1 , . . . , N there exist points x n ∈ F n ( λ ) suchthat the estimates h x − x n , ∇ v ( x ) i ≤ c v ( x ) , x ∈ F n ( λ ) , (3.14) are satisfied with some constant c > . Then U ( ψ ~ ) ≥ cλ ~ (3.15) for all λ ~ ∈ ( λ − ε / , λ + ε / , an arbitrary c < c + 2) − and sufficientlysmall ~ .Proof. Comparing relation (3.8) with assumption (3.14), we see that2 ~ Z F ( λ + ε ) |∇ ψ ~ ( x ) | dx ≤ c Z F ( λ + ε ) v ( x ) ψ ~ ( x ) dx + σ ( ε , ~ ) . Then using relation (3.1) and Lemma 3.1, we obtain the estimate2 K ( ψ ~ ) ≤ c U ( ψ ~ ) + o ( ~ ) . In view of the energy conservation (1.3), this yields (3.15). (cid:3)
BALANCE OF ENERGIES 9
Assumption (3.14) essentially means that, inside every well F n ( λ ), the function v ( x ) may equal zero only at the point x n . If, for example, v ( x ) = v n | x − x n | α n , v n > , α n > , x ∈ F n ( λ ) , n = 1 , . . . , N, then estimate (3.14) holds with c = max { α , . . . , α N } .I thank B. Helffer and D. Robert for useful discussions. References [1] M. V. Fedoryuk and V. P. Maslov, Semi-classical approximation in quantum mechanics,Amsterdam, Reidel, 1981.[2] B. Helffer,
Semi-classical analysis for the Schr¨odinger operator and applications , LectureNotes in Mathematics, v.1336, Springer Verlag, 1988.[3] B. Helffer, A. Martinez and D. Robert, Ergodicit´e et limite semi-classique, Comm. Math.Phys., , 313-326, 1987.[4] B. Helffer and D. Robert, Puits de potentiels g´en´eralis´es, Ann. Institut H. Poincar´e, phys.th´eor., , No 3, 291-331, 1984.[5] B. Helffer and J. Sj¨ostrand, Multiple wells in the semi-classical limit. I, Comm. Part. Diff.Eq., , 337-408, 1984.[6] T. Kato, Perturbation theory for linear operators , Springer Verlag, Berlin, Heidelberg, NewYork, 1984.[7] M. Reed and B. Simon,
Methods of Modern Mathematical Physics
IV, Academic Press, 1978.[8] B. Simon, Semiclassical analysis of low lying eigenvalues, II, Tunneling, Annals of Math., , 89-118, 1984.[9] D. Yafaev, A particle in a magnetic field of an infinite rectilinear current, Math. Phys., Analand Geometry, , 219-230, 2003. IRMAR, Universit´e de Rennes I, Campus de Beaulieu, 35042 Rennes Cedex, FRANCE
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