The semiclassical limit of eigenfunctions of the Schrödinger equation and the Bohr-Sommerfeld quantization condition, revisited
aa r X i v : . [ m a t h . SP ] S e p THE SEMICLASSICAL LIMIT OF EIGENFUNCTIONS OF THESCHR ¨ODINGER EQUATION AND THE BOHR-SOMMERFELDQUANTIZATION CONDITION, REVISITED
D. R. YAFAEV
To Vasilij Mikhailovich Babich on his 80-th birthday
Abstract.
Consider the semiclassical limit, as the Planck constant ~ →
0, ofbound states of a quantum particle in a one-dimensional potential well. Wejustify the semiclassical asymptotics of eigenfunctions and recover the Bohr-Sommerfeld quantization condition. Introduction
We study the limit as ~ → ψ ( x ) = ψ ( x ; λ, ~ ) of theSchr¨odinger equation − ~ ψ ′′ ( x ) + v ( x ) ψ ( x ) = λψ ( x ) , v ( x ) = v ( x ) , ψ ∈ L ( R ) , (1.1)for λ close to some non-critical energy λ (that is v ′ ( x ) = 0 for x such that v ( x ) = λ ). We assume that the equation v ( x ) = λ has exactly two solutions (the turningpoints) x ± = x ± ( λ ) and that v ( x ) < λ for x ∈ ( x − , x + ). Thus, ( x − , x + ) is apotential well and the energy λ is separated from its bottom. We suppose thateigenfunctions ψ ( x ) are real and normalized, that is Z ∞−∞ ψ ( x ) dx = 1 . It is a common wisdom that the limit of ψ ( x ) = ψ ( x ; λ, ~ ) as ~ → x ± . In neigh-borhoods of the turning points the asymptotics of ψ ( x ) is more complicated andis given in terms of an Airy function. Surprisingly, we have not found a preciseformulation and a proof of this result in the literature. Our goal is to fill in thisgap. We follow here the scheme suggested by R. E. Langer and thoroughly exposedby F. W. Olver in his book [7].The detailed asymptotics of ψ ( x ) described in Theorems 2.5 and 4.4 allows oneto recover the classical Bohr-Sommerfeld quantization condition on λ (see Theo-rem 4.1). Actually, we prove somewhat more establishing a one-to-one correspon-dence between eigenvalues of the Schr¨odinger operator H ~ = − ~ d /dx + v ( x )from a neighborhood of a non-critical energy and points ( n + 1 / ~ where n is aninteger. This implies the semiclassical Weyl formula for the distribution of eigen-values of the operator H ~ as ~ → Mathematics Subject Classification.
D. R. YAFAEV in Theorem 5.2 the quantization condition for discontinuous functions v ( x ). Thisformula generalizes that of Bohr and Sommerfeld and is probably new.We note that the Bohr-Sommerfeld quantization condition is also well known inmuch more difficult multidimensional problems. In this context we mention book[1] by V. M. Babich and V. S. Buldyrev (where the ray approximation is used), book[3] by M. V. Fedoryuk and V. P. Maslov (where the Maslov canonical operator isused) as well as papers [5] by B. Helffer et D. Robert and [4] by B. Helffer, A.Martinez and D. Robert (where the methods of microlocal analysis are used).However, in the one-dimensional problem it is more natural to rely on methods ofordinary differential equations. Such an approach was developed by M. V. Fedoryuk(see his book [2]) for analytic potentials. In this case one can avoid a study ofturning points so that the Airy function does not appear. The asymptotics of eigenfunctions yields (see Proposition 4.5) asymptoticsof observables Z ∞−∞ w ( x ) ψ ( x ; λ, ~ ) dx (1.2)for sufficiently arbitrary functions w ( x ). For example, we can take for w ( x ) char-acteristic functions of Borel subsets of R or choose w ( x ) = v ( x ). This gives theasymptotics of the kinetic energy K ( λ, ~ ) := ~ Z ∞−∞ ψ ′ ( x ; λ, ~ ) dx = K cl ( λ ) + O ( ~ / ) (1.3)as ~ → λ from a neighborhood of the point λ . The leading term K cl ( λ ) (the index “ cl ” stands of course for the corresponding classical object) isgiven by the expression K cl ( λ ) = Z x + ( λ ) x − ( λ ) ( λ − v ( x )) / dx (cid:16)Z x + ( λ ) x − ( λ ) ( λ − v ( x )) − / dx (cid:17) − . (1.4)Note that the integrals here are taken over the classically allowed region and that K cl ( λ ) coincides (see subsection 4.3) with the averaged value of the kinetic energyof a particle of energy λ in classical mechanics.We emphasize that our derivation of the quantization condition and of asymp-totic formulas for observables (1.2) requires Airy functions although they do notenter into the final answer. However, we do not know how to avoid Airy functionswithout additional assumptions on v ( x ).2. Semiclassical solutions of the Schr¨odinger equation
It is convenient to rewrite equation (1.1) as − u ′′ ~ ( x ) + ~ − q ( x ) u ~ ( x ) = 0 , (2.1)where q ( x ) = q ( x ; λ ) = v ( x ) − λ. We need some regularity of the function v ( x ) and a weak condition on its behaviorat infinity. Assumption 2.1.
The function v ∈ C ( R ) and, for some ρ >
1, the function (cid:0) | q ( x ) | − q ′ ( x ) + q ( x ) − | q ′′ ( x ) | (cid:1)(cid:12)(cid:12)(cid:12) Z x | q ( y ) | / dy (cid:12)(cid:12)(cid:12) ρ EMICLASSICAL LIMIT OF EIGENFUNCTIONS 3 is bounded for sufficiently large | x | .The last condition is satisfied in all reasonable cases. For example, if v ( x ) → v > λ , it is sufficient to require that v ′ ( x ) + | v ′′ ( x ) | = O ( | x | − ρ ) , ρ > , | x | → ∞ . It is also satisfied if v ( x ) behaves at infinity as | x | α or e α | x | where α >
0; in thesecases ρ = 2.We consider the case of one potential well. To be more precise, we make thefollowing Assumption 2.2.
The equation v ( x ) = λ has two solutions x + = x + ( λ ) and x − = x − ( λ ). We suppose that x − < x + , v ( x ) < λ for x ∈ ( x − , x + ), v ( x ) > λ for x [ x − , x + ] and lim inf | x |→∞ v ( x ) > λ. Moreover, the function v belongs to the class C in some neighborhoods of thepoints x ± and ± v ′ ( x ± ) > λ , then it is also satisfied forall λ from some neighborhood of λ .Our goal in this section is to describe asymptotics as ~ → u + ( x ) = u + ( x ; λ, ~ ) and u − ( x ) = u − ( x ; λ, ~ ) of equation (2.1) exponentially decaying as x → + ∞ and x → −∞ , respectively. These asymptotics will be given in termsof an Airy function and are uniform with respect to x ∈ [ x , ∞ ) or x ∈ ( −∞ , x ]where x is an arbitrary point from the interval ( x − , x + ). Let us recall the definition of Airy functions and their necessary properties(see, e.g., [7]), for details). Consider the equation − w ′′ ( t ) + tw ( t ) = 0 (2.2)and denote by Ai ( t ) its solution with asymptoticsAi ( t ) = 2 − π − / t − / exp( − t / / O ( t − / )) , t → + ∞ . (2.3)Then Ai ( t ) = π − / | t | − / sin(2 | t | / / π/
4) + O ( | t | − / ) , t → −∞ . (2.4)Note that Ai ( t ) > t ≥ t ) of equation (2.2) is defined by its asymptotics as t → −∞ which differs from (2.4) only by the phase shift:Bi ( t ) = − π − / | t | − / sin(2 | t | / / − π/
4) + O ( | t | − / ) , t → −∞ . (2.5)For t ≥
0, this function is positive and satisfies the estimateBi ( t ) ≤ C (1 + t ) − / exp(2 t / / . (2.6)Here and below we denote by C and c different positive constants whose precisevalues are of no importance.We also use that all asymptotics (2.3), (2.4) and (2.5) can be differentiated in t .In particular, the Wronskian { Ai ( t ) , Bi − ( t ) } := Ai ′ ( t ) Bi ( t ) − Ai ( t ) Bi ′ ( t ) = − π − . D. R. YAFAEV
It follows thatBi ( s ) Ai − ( s ) − Bi ( t ) Ai − ( t ) = π − Z st Ai − ( τ ) dτ, s ≥ t ≥ . (2.7) To formulate results, we need the following auxiliary functions ξ ± ( x ) = ξ ± ( x ; λ ): ξ + ( x ) = (cid:16) Z xx + q ( y ) / dy (cid:17) / , x ≥ x + ,ξ + ( x ) = − (cid:16) Z x + x | q ( y ) | / dy (cid:17) / , x − < x ≤ x + , (2.8)and ξ − ( x ) = (cid:16) Z x − x q ( y ) / dy (cid:17) / , x ≤ x − ,ξ − ( x ) = − (cid:16) Z xx − | q ( y ) | / dy (cid:17) / , x − ≤ x < x + . Here is a list of properties of these functions. The following result is practically thesame as Lemma 3.1 from Chapter 11 of [7].
Lemma 2.3.
Let x ∈ ( x − , x + ) . Then ξ + ∈ C ( x , ∞ ) , ξ − ∈ C ( −∞ , x ) and ξ ± ( x ) → + ∞ as x → ±∞ . The derivatives ± ξ ′± ( x ) > , ξ ′± ( x ± ) = ±| v ′ ( x ± ) | / (2.9) and the functions ξ ± ( x ) satisfy the equation ξ ′± ( x ) ξ ± ( x ) = q ( x ) . (2.10)It follows from this lemma that the function p ± ( x ) = ( | ξ ′± ( x ) | − / ) ′′ | ξ ′± ( x ) | − / (2.11)is continuous. Moreover, using identity (2.10), we see that − p ± ( x ) = 5 ξ ± ( x ) − + ξ ± ( x ) (cid:0) q ( x ) − q ′′ ( x ) − q ( x ) − q ′ ( x ) (cid:1) , x = x ± , (2.12)and hence according to Assumption 2.1 | p ± ( x ) | ≤ C | ξ ± ( x ) | − / − ρ , ρ = 3 min { ρ − , } / > . (2.13) Let us construct solutions u ± ( x ) = u ± ( x ; λ, ~ ) of equation (2.1) withsemiclassical asymptotics as ~ → x → ±∞ . We define these solutionsby their asymptotics as x → ±∞ . Below all asymptotic relations are supposed tobe differentiable with respect to x . In this subsection, we only formulate results. Proposition 2.4.
Under Assumption for every fixed ~ > , equation (2.1) hasa ( unique ) solution u ± ( x ) such that u ± ( x ) =2 − π / ~ / q ( x ) − / exp (cid:16) ∓ ~ − Z xx ± q ( y ) / dy (cid:17) × (cid:16) O (cid:0)(cid:12)(cid:12) Z xx ± q ( y ) / dy (cid:12)(cid:12) − ρ (cid:1)(cid:17) where ρ = min { ρ − , } > as x → ±∞ . Uniform asymptotic formulas for u ± ( x ) are given in the following assertion. EMICLASSICAL LIMIT OF EIGENFUNCTIONS 5
Theorem 2.5.
Let Assumptions and hold. If ± x ≥ ± x ± , then the solutions u ± ( x ) = u ± ( x ; λ, ~ ) admit the representations u ± ( x ) = | ξ ′± ( x ) | − / Ai ( ~ − / ξ ± ( x )) (cid:0) ε ± ( x ; ~ ) (cid:1) (2.14) where the remainder satisfies the estimate | ε ± ( x ; λ, ~ ) | ≤ C ~ (1 + | ξ ± ( x ) | ) − ρ , ρ = 3 min { ρ − , } / > . (2.15) Let x ∈ ( x − , x + ) . On the interval [ x , x + ] ( on the interval [ x − , x ]) the function u + ( the function u − ) admits the representation u ± ( x ) = | ξ ′± ( x ) | − / Ai ( ~ − / ξ ± ( x )) + O ( ~ / ( ~ / + | x − x ± | ) − / ) . (2.16)Away from the points x ± , we can replace the Airy function Ai ( t ) by its asymp-totics (2.3) or (2.4). Indeed, in view of (2.9), we see that | ξ ± ( x ) | ≥ c | x − x ± | , c > , (2.17)and hence ~ − / ξ + ( x ) → ±∞ if ~ − / ( x − x + ) → ±∞ and ~ − / ξ − ( x ) → ±∞ if ~ − / ( x − x − ) → ∓∞ . This leads to the following result. Corollary 2.6.
Suppose that δ ~ ~ − / ≥ c > in particular, δ ~ may be fixed ) .Then the functions u ± ( x ) have asymptotics u ± ( x ) = 2 − π / ~ / q ( x ) − / exp (cid:16) ∓ ~ − Z xx ± q ( y ) / dy (cid:17)(cid:0) O ( ~ | ξ ± ( x ) | − / ) (cid:1) (2.18) as ~ → uniformly in x ≥ x + + δ ~ for u + ( x ) and in x ≤ x − − δ ~ for u − ( x ) . Let x ∈ ( x − , x + ) . Then the functions u ± ( x ) have asymptotics u ± ( x ) = π / ~ / | q ( x ) | − / sin (cid:16) ± ~ − Z x ± x | q ( y ) | / dy + π/ (cid:17) + O ( ~ / | x − x ± | − / ) (2.19) as ~ → uniformly in x ∈ [ x , x + − δ ~ ] for u + ( x ) and uniformly in x ∈ [ x − + δ ~ , x ] for u − ( x ) . On the other hand, using estimates (2.17) and | Ai ( t ) | ≤ C (1+ | t | ) − / , we obtainuniform in ~ estimates of the functions u ± ( x ) in neighborhoods of the turningpoints. Corollary 2.7.
For sufficiently small | x − x ± | , the estimate | u ± ( x ) | ≤ C (1 + ~ − / | x − x ± | ) − / (2.20) holds with a constant C which does not depend on ~ . We note that all asymptotic relations (2.14), (2.16), (2.18) and (2.19) can bedifferentiated with respect to x . In particular, we have asymptotics u ′± ( x ) = ∓ π / ~ − / | q ( x ) | / cos (cid:16) ± ~ − Z x ± x | q ( y ) | / dy + π/ (cid:17) + O ( ~ / | x − x ± | − / ) (2.21) D. R. YAFAEV as ~ → x ∈ [ x , x + − δ ~ ] for u + ( x ) and uniformly in x ∈ [ x − + δ ~ , x ]for u − ( x ). All these relations can also be differentiated with respect to λ . Forexample, we have ∂u ± ( x ; λ, ~ ) /∂λ = ± − π / ~ − / | q ( x ; λ ) | − / Z x ± x | q ( y ; λ ) | − / dy × cos (cid:16) ± ~ − Z x ± x | q ( y ; λ ) | / dy + π/ (cid:17) + O ( ~ / | x − x ± | − / )as ~ → x ∈ [ x , x + − δ ~ ] for u + ( x ) and uniformly in x ∈ [ x − + δ ~ , x ]for u − ( x ). Let us now calculate the norm of the function u ± ( x ) in the space L ( x , ±∞ )where x ∈ ( x − , x + ). Actually, we will obtain a more general result. Proposition 2.8.
Let a function w ( x ) be differentiable on the interval ( x − , x + ) except a finite number of points. Suppose that w ( x ) and w ′ ( x ) are locally boundedfunctions and that, for some N , | w ( x ) | ≤ Cq ( x ) (cid:12)(cid:12)(cid:12)Z x | q ( y ) | / dy (cid:12)(cid:12)(cid:12) N (2.22) if | x | is large. Then under Assumptions and we have the asymptotic relation Z ±∞ x w ( x ) u ± ( x ) dx = 2 − π ~ / Z x ± x w ( x )( λ − v ( x )) − / dx + O ( ~ / ) . (2.23) Proof.
We will prove (2.23) for the sign “ + ”, omit this index and add ~ . Usingasymptotics (2.14), (2.15) and the estimate ξ ′ ( x ) ≥ c >
0, we see that Z x + +1 x + w ( x ) u ~ ( x ) dx ≤ C Z ∞ x + ξ ′ ( x ) Ai ( ~ − / ξ ( x )) dx = C ~ / . Similarly, using identity (2.10) and condition (2.22) we find that Z ∞ x + +1 w ( x ) u ~ ( x ) dx ≤ C Z ∞ x + +1 ξ ′ ( x ) ξ N/ ( x ) Ai ( ~ − / ξ ( x )) dx = O ( ~ ∞ ) . Suppose that δ ~ → ~ → δ ~ ~ − / ≥ c >
0. The integral of u ~ ( x )over ( x + − δ ~ , x + ) is estimated by Cδ ~ because according to (2.20) the functions u ~ ( x ) are uniformly bounded in a neighborhood of the point x + . On the interval( x , x + − δ ~ ), we have a relation Z x + − δ ~ x w ( x ) u ~ ( x ) dx = π ~ / Z x + − δ ~ x w ( x ) | q ( x ) | − / × sin (cid:16) ~ − Z x + x | q ( y ) | / dy + π/ (cid:17) dx + O ( ~ / δ − ~ ) . (2.24)Indeed, in view of asymptotics (2.19) we have to show that the integrals ~ / Z x + − δ ~ x | x − x + | − / dx and ~ / Z x + − δ ~ x | q ( x ) | − / | x − x + | − / dx are O ( ~ / δ − ~ ). The first of them equals C ~ / δ − / ~ which is O ( ~ / δ − ~ ) because ~ = O ( δ / ~ ). To estimate the second integral, we have to additionally take intoaccount that | q ( x ) | ≥ c ( x + − x ) , c > . (2.25) EMICLASSICAL LIMIT OF EIGENFUNCTIONS 7
Next, we replace sin ( · ) in the right-hand side of (2.24) by 1 /
2. Let us esti-mate the error. Integrating by parts separately on every interval where w ( x ) isdifferentiable, we see that Z x + − δ ~ x w ( x ) | q ( x ) | − / exp (cid:16) i ~ − Z x + x | q ( y ) | / dy (cid:17) dx = − − i ~ w ( x + − δ ~ ) q ( x + − δ ~ ) − exp (cid:16) i ~ − Z x + x + − δ ~ | q ( y ) | / dy (cid:17) +2 − i ~ Z x + − δ ~ x (cid:0) w ′ ( x ) q ( x ) − − v ′ ( x ) q ( x ) − w ( x ) (cid:1) × exp (cid:16) i ~ − Z x + x | q ( y ) | / dy (cid:17) dx + O ( ~ ) . The right-hand side here is bounded by C ~ (cid:16) | q ( x + − δ ~ ) | − + Z x + − δ ~ x q ( x ) − dx (cid:17) which in view of estimate (2.25) does not exceed C ~ δ − ~ . Thus, it follows from(2.24) that Z x + − δ ~ x w ( x ) u ~ ( x ) dx = 2 − π ~ / Z x + − δ ~ x w ( x )( λ − v ( x )) − / dx + O ( ~ / δ − ~ ) . Finally, making an error of order O ( ~ / δ / ~ ), we can extend the integral in theright-hand side to the whole interval ( x , x + ). Setting δ ~ = ~ / and putting theresults obtained together, we arrive at asymptotic relation (2.23). (cid:3) Of course (2.22) is a very mild restriction. It is satisfied for v ( x ) = w ( x ). It isalso true for all functions v ( x ) if w ( x ) if bounded by some power of | x | at infinityand is even less restrictive if v ( x ) → ∞ as | x | → ∞ . In particular, setting w ( x ) = 1,we obtain Corollary 2.9.
The asymptotic relation holds: Z ±∞ x u ± ( x ) dx = 2 − π ~ / Z x ± x ( λ − v ( x )) − / dx + O ( ~ / ) . (2.26)3. Proof of Theorem
We will prove Theorem 2.5 for the sign “ + ” and omit this index. On thecontrary, we add the index ~ to emphasize the dependence on it of various objects.Let x ∈ ( x − , x + ), x ∈ ( x , ∞ ) and let the function ξ ( x ) be defined by formulas(2.8). According to Lemma 2.3 ξ ( x ) ∈ ( ξ , ∞ ) where ξ = ξ ( x ) and x can beconsidered as a function of ξ if ξ ∈ ( ξ , ∞ ).Let us make the change of variables x ξ in equation (2.1) and set u ~ ( x ) = ξ ′ ( x ) − / f ~ ( ~ − / ξ ( x )) . (3.1)Then using identity (2.10), we obtain that − f ′′ ~ ( ~ − / ξ ) + ~ − / ξf ~ ( ~ − / ξ ) = ~ / r ( ξ ) f ~ ( ~ − / ξ ) , (3.2)where r ( ξ ) = p ( x ( ξ )) (3.3) D. R. YAFAEV and p ( x ) is defined by formula (2.11). In view of (2.13) we have the estimate | r ( ξ ) | ≤ C (1 + | ξ | ) − / − ρ , ρ = 3 min { ρ − , } / > . (3.4)Setting in (3.2) t = ~ − / ξ , we get the following intermediary result. Lemma 3.1.
Let t = ~ − / ξ ( x ) , and let the functions u ~ ( x ) and f ~ ( t ) be relatedby formula (3.1) . Then equation (2.1) for x ≥ x is equivalent to the equation − f ′′ ~ ( t ) + tf ~ ( t ) = R ~ ( t ) f ~ ( t ) for t ≥ ξ ~ − / (3.5) where R ~ ( t ) = ~ / r ( ~ / t ) . (3.6) Let us reduce differential equation (3.5) to a Volterra integral equation. Set K ~ ( t, s ) = − π (cid:0) Ai ( t ) Bi ( s ) − Ai ( s ) Bi ( t ) (cid:1) R ~ ( s ) , s ≥ t, (3.7)and consider the equation f ~ ( t ) = Ai ( t ) + Z ∞ t K ~ ( t, s ) f ~ ( s ) ds. (3.8)Differentiating it twice, we see that its solution satisfies also differential equation(3.5). We will study equations (3.5) or (3.8) separately for t ≥ t ≤ Lemma 3.2.
For t ≥ , equation (3.5) has a solution f ~ ( t ) such that f ~ ( t ) = Ai ( t ) (cid:0) η ~ ( t ) (cid:1) (3.9) where | η ~ ( t ) | ≤ C ~ (1 + ~ / t ) − ρ , ρ = 3 min { ρ − , } / > . (3.10) Proof.
Making the multiplicative change of variables f ~ ( t ) = Ai ( t ) g ~ ( t ) (3.11)and using (2.7), we rewrite equation (3.8) as g ~ ( t ) = 1 − Z ∞ t L ~ ( t, s ) g ~ ( s ) ds, (3.12)where L ~ ( t, s ) = Ai ( t ) − K ~ ( t, s ) Ai ( s ) = Z st Ai − ( τ ) dτ Ai ( s ) R ~ ( s ) , s ≥ t. It follows from (2.3) that Z st Ai − ( τ ) dτ ≤ C exp(4 s / / | L ~ ( t, s ) | ≤ C ~ / s − / (1 + ~ / s ) − / − ρ , ≤ t ≤ s. (3.13)This estimate allows us to solve equation (3.12) by iterations. In particular, thesolution of (3.12) satisfies the estimate | g ~ ( t ) − | ≤ C Z ∞ t | L ~ ( t, s ) | ds. Now estimate (3.10) on the remainder η ~ ( t ) = g ~ ( t ) − (cid:3) EMICLASSICAL LIMIT OF EIGENFUNCTIONS 9
Putting together formulas (3.1) and (3.9), we obtain representation (2.14) with ε ~ ( x ) = η ~ ( ~ − / ξ ( x )). Estimate (3.10) implies estimate (2.15). This leads tothe assertion of Theorem 2.5 for x ≥ x + . In particular, for a fixed ~ , we getProposition 2.4.Next, we consider the case t ≤ Lemma 3.3.
For t ∈ [ ~ − / ξ , , the solution f ~ ( t ) of equation (3.5) satisfies theestimate | f ~ ( t ) − Ai ( t ) | ≤ C ~ (1 + | t | ) − / . (3.14) Proof.
Let us rewrite equation (3.8) as f ~ ( t ) = f (0) ~ ( t ) + Z t K ~ ( t, s ) f ~ ( s ) ds, (3.15)where the new “free” term f (0) ~ ( t ) = Ai ( t ) + f (1) ~ ( t ) , f (1) ~ ( t ) = Z ∞ K ~ ( t, s ) f ~ ( s ) ds. (3.16)It follows from (2.3) and (2.6) thatAi ( t ) + Ai ( t ) Bi ( t ) ≤ C (1 + t ) − / , t ≥ , and from (2.4) and (2.5) that | Ai ( t ) | + | Bi ( t ) | ≤ C (1 + | t | ) − / , t ≤ . (3.17)Therefore using (3.4), (3.6) and (3.7), we find that | f (1) ~ ( t ) | ≤ C (cid:16) | Ai ( t ) | Z ∞ Ai ( s ) Bi ( s ) | R ~ ( s ) | ds + | Bi ( t ) | Z ∞ Ai ( s ) | R ~ ( s ) | ds (cid:17) ≤ C (1 + | t | ) − / ~ / Z ∞ s − / (1 + ~ / s ) − / − ρ ds ≤ C (1 + | t | ) − / ~ . (3.18)Let us now consider equation (3.15). By virtue of estimates (3.4) and (3.17) itskernel satisfies the bound | K ~ ( t, s ) | ≤ C ~ / (1 + | t | ) − / (1 + | s | ) − / r ( ~ / s ) , t ≤ s ≤ , (3.19)where the function r ( ~ / s ) can be estimated by a constant. Thus, solving (3.15)again by iterations, we obtain the estimate | f ~ ( t ) − f (0) ~ ( t ) | ≤ C Z t | K ~ ( t, s ) | (1 + | s | ) − / ds ≤ C ~ / Z t (1 + | s | ) − / ds ≤ C ~ / (1 + | t | ) / . (3.20)If t ∈ [ ~ − / ξ , (cid:3) In view of formula (3.1), this lemma yields the result of Theorem 2.5 for x ∈ [ x , x + ].Differentiating integral equation (3.8) with respect to t , we obtain asymptoticrelations for f ′ ~ ( t ) and then for u ′ ~ ( x ). This concludes the proof of Theorem 2.5. Semiclassical asymptotics of eigenfunctions
Let λ = λ ( ~ ) be an eigenvalue of the Schr¨odinger operator H ~ = − ~ d /dx + v ( x ) from a neighborhood of a non-critical point λ . Then the solutions u ± ( x ) areproportional: u − ( x ; λ, ~ ) = a ( λ, ~ ) u + ( x ; λ, ~ ) . (4.1)Choose an arbitrary interior point x of the interval ( x − ( λ ) , x + ( λ )). To calculatethe Wronskian of u + ( x ) and u − ( x ), we use asymptotic relations (2.19) and (2.21).Setting ϕ ± ( x ; λ ) = ± Z x ± ( λ ) x ( λ − v ( y )) / dy, x ∈ ( x − ( λ ) , x + ( λ )) , (4.2)we find that w ( λ, ~ ) = u + ( x ; λ, ~ ) u ′− ( x ; λ, ~ ) − u − ( x ; λ, ~ ) u ′ + ( x ; λ, ~ )= π ~ − / (cid:16) sin( ~ − ϕ + ( x ; λ ) + π/
4) cos( ~ − ϕ − ( x ; λ ) + π/ ~ − ϕ + ( x ; λ ) + π/
4) sin( ~ − ϕ − ( x ; λ + π/ (cid:17) + O ( ~ / )= π ~ − / sin( ~ − Φ( λ ) + π/
2) + O ( ~ / ) (4.3)where Φ( λ ) = ϕ + ( x ; λ ) + ϕ − ( x ; λ ) so thatΦ( λ ) = Z x + ( λ ) x − ( λ ) ( λ − v ( y )) / dy. (4.4)Since w ( λ, ~ ) = 0, we see thatsin( ~ − Φ( λ ) + π/
2) = O ( ~ )and hence Z x + ( λ ) x − ( λ ) ( λ − v ( x )) / dx = π ( n + 1 / ~ + O ( ~ ) (4.5)for some integer number n = n ( λ, ~ ). This gives us the famous Bohr-Sommerfeldquantization condition.Suppose now that a number π ( n + 1 / ~ belongs to a neighborhood of λ . Letus check that there exists an eigenvalue λ n ( ~ ) of the operator H ~ satisfying theestimate | Φ( λ n ( ~ )) − π ( n + 1 / ~ | ≤ C ~ . (4.6)Since u ± ∈ L ( R ± ), it suffices to show that w ( λ, ~ ) = 0 for some λ = λ n ( ~ )satisfying estimate (4.6). Using the equality λ − v ( x ± ( λ )) = 0, we find thatΦ ′ ( λ ) = 2 − Z x + ( λ ) x − ( λ ) ( λ − v ( y )) − / dy > . (4.7)Hence Φ is a one-to-one mapping of a neighborhood of λ on a neighborhood of µ = Φ( λ ). Set µ = Φ( λ ) and ǫ ( µ, ~ ) = π − ~ / w (Φ − ( µ ) , ~ ) − sin( ~ − µ + π/ . (4.8)In view of (4.3) this function satisfies the estimate | ǫ ( µ, ~ ) | ≤ C ~ with a constant C which does not depend on ~ and µ from a neighborhood of µ . We have to showthat the equation sin( ~ − µ + π/
2) + ǫ ( µ, ~ ) = 0 EMICLASSICAL LIMIT OF EIGENFUNCTIONS 11 has a solution µ n ( ~ ) obeying the estimate | µ n ( ~ ) − π ( n + 1 / ~ | ≤ C ~ . Setting s = ~ − µ + π/
2, we see that this assertion is equivalent to the existence ofa solution s = s n ( ~ ) of the equationsin s + ǫ ( ~ ( s − π/ , ~ ) = 0 (4.9)obeying the estimate | s n ( ~ ) − π ( n + 1) | ≤ C ~ . (4.10)The last fact is obvious because ǫ ( ~ ( s − π/ , ~ ) = O ( ~ ).Next, we will show that for every n there is only one eigenvalue of the operator H ~ satisfying (4.6). To that end, we have to check that equation (4.9) cannot havetwo solutions satisfying (4.10). Supposing the contrary, we find a point ˜ s = ˜ s n ( ~ )such that cos ˜ s = − ~ ∂ǫ∂µ ( ~ (˜ s − π/ , ~ ) (4.11)and ˜ s n ( ~ ) = π ( n + 1) + O ( ~ ). Observe that relation (4.3) can be differentiated in λ which yields ∂w ( λ, ~ ) /dλ = π Φ ′ ( λ ) ~ − / cos( ~ − Φ( λ ) + π/
2) + O ( ~ − / ) . It follows that function (4.8) obeys the estimate ∂ε ( µ, ~ ) /dµ = O (1). Thus, theright-hand side of equation (4.11) is O ( ~ ) while its left-hand side tends to ( − n +1 as ~ → ~ − ϕ − ( x ; λ ) + π/
4) + O ( ~ )= a ( λ, ~ ) (cid:0) sin( ~ − ϕ + ( x ; λ ) + π/
4) + O ( ~ ) (cid:1) (4.12)and cos( ~ − ϕ − ( x ; λ ) + π/
4) + O ( ~ )= − a ( λ, ~ ) (cid:0) cos( ~ − ϕ + ( x ; λ ) + π/
4) + O ( ~ ) (cid:1) . (4.13)Together, these two relations imply that | a ( λ, ~ ) | = 1 + O ( ~ ). Moreover, since ϕ + ( x ; λ ) + ϕ − ( x ; λ ) = π ( n + 1 / ~ + O ( ~ ) , it follows from (4.12) and (4.13) that a ( λ, ~ ) = ( − n + O ( ~ ) . (4.14)Thus, we have obtained the following result. Theorem 4.1.
Let Assumptions and hold for a point λ . Suppose thatan eigenvalue λ = λ ( ~ ) of the operator H ~ belongs to a neighborhood of λ . Thennecessarily condition (4.5) is satisfied with some integer number n = n ( λ, ~ ) . Con-versely, for every n such that π ( n + 1 / ~ belongs to a neighborhood of Φ( λ ) , thereexists an eigenvalue λ n ( ~ ) of the operator H ~ satisfying estimate (4.6) with a con-stant C not depending on n and ~ . Such an eigenvalue λ n ( ~ ) is unique. Moreover,the coefficient a ( λ, ~ ) in (4.1) has asymptotics (4.14) where n is the same numberas in (4.5) . Corollary 4.2.
Let an interval ( a , a ) belong to a neighborhood of a point λ satisfying Assumptions and . Then the total number N ~ of eigenvalues ofthe operator H ~ in this interval equals N ~ = π − (Φ( a ) − Φ( a )) ~ − + ǫ ( ~ ) (4.15) where | ǫ ( ~ ) | ≤ for sufficiently small ~ .Proof. According to Theorem 4.1 there is exactly one eigenvalue of the operator H ~ in a neighborhood of size C ~ of every point Φ − ( π ( n +1 / ~ ). These neighborhoodshave empty intersections for sufficiently small ~ . Thus, N ~ equals the number ofpoints π ( n + 1 / ~ lying in the interval (Φ( a ) , Φ( a )). Clearly, this number equalsthe right-hand side of (4.15). (cid:3) Remark 4.3.
Suppose that Assumptions 2.1 and 2.2 hold true for all λ ∈ [ a , a ].Then remainders in different asymptotic formulas of this paper can be estimateduniformly in λ ∈ [ a , a ]. Formula (4.15) also remains true for such ( a , a ).Note that definition (4.4) can be rewritten asΦ( λ ) = 2 − Z Z p + v ( x ) ≤ λ dpdx. (4.16)Indeed, integrating in the right-hand side first over p we obtain the right-hand sideof (4.4). It follows that the asymptotic coefficient in (4.15) is the volume of a partof the phase space:Φ( a ) − Φ( a ) = 2 − mes { ( x, p ) ∈ R : a ≤ p + v ( x ) ≤ a } . Thus, relation (4.15) is the semiclassical Weyl formula with a strong estimate ofthe remainder.
Let us denote by ψ ( x ) = ψ ( x ; λ, ~ ) the eigenfunction of the operator H ~ corresponding to its eigenvalue λ . We suppose that ψ = ψ ∈ L ( R + ) and k ψ k = 1which fixes ψ up to a sign. Clearly, ψ ( x ) = c ± u ± ( x ) , c ± = c ± ( λ, ~ ) , (4.17)where according to (4.14) | c + ( λ, ~ ) | = | c − ( λ, ~ ) | (1 + O ( ~ )) . Therefore it follows from Corollary 2.9 that | c ± ( λ, ~ ) | = 2 / π − / ~ − / (cid:16) Z x + ( λ ) x − ( λ ) ( λ − v ( x )) − / dx (cid:17) − / + O ( ~ / ) , (4.18)which in view of Theorem 2.5 yields the following result. Theorem 4.4.
Under the assumptions of Theorem , let us denote by ψ ( λ, ~ ) the real normalized eigenfunction ( defined up to a sign ) of the operator H ~ corre-sponding to its eigenvalue λ = λ ( ~ ) . Let x be an arbitrary point from the interval ( x − ( λ ) , x + ( λ )) . Then, for x ∈ ( x , ∞ ) , asymptotics of ψ ( x ; λ, ~ ) as ~ → is givenby formulas (4.17) , (4.18) for the sign “+” and asymptotic relations of Theorem for the function u + ( x ; λ, ~ ) . Similarly, for x ∈ ( −∞ , x ) , asymptotics of ψ ( x ; λ, ~ ) as ~ → is given by formulas (4.17) , (4.18) for the sign “ − ” and asymptotic rela-tions of Theorem for the function u − ( x ; λ, ~ ) . In neighborhoods of the turningpoints, the estimate | ψ ( x ; λ, ~ ) | ≤ C ( ~ / + | x − x ± | ) − / (4.19) EMICLASSICAL LIMIT OF EIGENFUNCTIONS 13 holds with a constant C which does not depend on ~ . In view of formula (2.23), this result can be supplemented by the following
Proposition 4.5.
Let a function w satisfy the assumptions of Proposition .Then under the assumptions of Theorem we have Z ∞−∞ w ( x ) ψ ( x ; λ, ~ ) dx = Z x + ( λ ) x − ( λ ) w ( x )( λ − v ( x )) − / dx × (cid:16)Z x + ( λ ) x − ( λ ) ( λ − v ( x )) − / dx (cid:17) − + O ( ~ / ) . (4.20)In particular, this relation applies to the potential energy V ( λ, ~ ) = Z ∞−∞ v ( x ) ψ ( x ; λ, ~ ) dx and by virtue of the energy conservation K ( λ, ~ ) + V ( λ, ~ ) = λ , we also obtain theasymptotics of the kinetic energy. Corollary 4.6.
Under the assumptions of Theorem asymptotic relation (1.3) holds with the leading term K cl ( λ ) given by (1.4) . Since K cl ( λ ) >
0, for small ~ the kinetic energy K ( λ, ~ ) ≥ c > V ( λ, ~ ) ≤ λ − c . This implies that the eigenfunctions ψ ( x ; λ, ~ )are not too strongly localized in neighborhoods of the turning points x ± ( λ ). In viewof estimate (4.19), this statement can be reinforced. Proposition 4.7.
Let the assumptions of Theorem hold, and let k ψ ( λ, ~ ) k = 1 .Then Z x ± ( λ )+ δx ± ( λ ) − δ ψ ( x ; λ, ~ ) dx ≤ Cδ / where the constant C does not depend on ~ . Remark 4.8.
It follows from asymptotics (2.14) and (4.18) that | ψ ( x ± ( λ ); λ, ~ ) | = α ± ( λ ) ~ − / (1 + O ( ~ / )) , where the coefficient α ± ( λ ) = 2 / π − / (cid:16) Z x + ( λ ) x − ( λ ) ( λ − v ( x )) − / dx (cid:17) − / | v ′ ( x ± ( λ )) | − / Ai (0) = 0 . This contradicts the assertion of Theorem 7.1 of [8] that normalized eigenfunctionsare uniformly bounded in neighborhoods of turning points.
Recall that a classical particle (of mass m and energy λ ) moves periodically(see, e.g., [6]) in a potential well bounded by the points x − = x − ( λ ) and x + = x + ( λ )such that v ( x ± ) = λ . Let us check that the asymptotic coefficient K cl in (1.3)coincides with the averaged over the period T = T ( λ ) value K av = T − Z T K ( t ) dt of the classical kinetic energy K ( t ) = mx ′ ( t ) / λ − v ( x ( t )) . Since dt = x ′ ( t ) − dx = ( m/ / ( λ − v ( x )) − / dx, the period is given by the formula T = 2 Z x + x − dtdx dx = (2 m ) / Z x + x − ( λ − v ( x )) − / dx and K av = T − Z T ( λ − v ( x ( t ))) dt = 2( m/ / T − Z x + x − ( λ − v ( x )) / dx. Putting these two relations together, we obtain for K av the same expression (1.4)as for K cl . This proves the equality K cl ( λ ) = K av ( λ ) . We also note that K cl ( λ ) = (cid:0) d ln Φ( λ ) /dλ (cid:1) − where the function Φ( λ ) is defined by formulas (4.4) or, equivalently, (4.16). Forthe proof of this equality, it suffices to plug representation (4.7) for the functionΦ ′ ( λ ) into formula (1.4). 5. Discontinuous potentials
Away from the turning points, assumptions on v ( x ) can be somewhatrelaxed. Consider, for example, an interval ( x − + δ, x + − δ ) where δ >
0. There, itsuffices to require that v ∈ C and that v ′ be absolutely continuous so that v ′′ ∈ L (instead of v ∈ C ). In this case the function r ( ξ ) defined by formulas (2.11) and(3.3) belongs to L only so that the factor r ( ~ / s ) in the right-hand side of estimate(3.19) cannot be neglected. Therefore (cf. (3.20)) we have the estimate Z t | K ~ ( t, s ) | (1 + | s | ) − / ds ≤ C ~ / (1 + | t | ) − / Z t | r ( ~ / s ) | ds ≤ C ~ / (1 + | t | ) − / Z ξ ( x − + δ ) | r ( s ) | ds. It follows that instead of (3.14) we have a slightly weaker estimate with ~ / inplace of ~ in the right-hand side. All other estimates remain unchanged. Thus,Theorem 2.5 is true with a little bit weaker estimates of remainders in asymptoticformulas for u ± ( x ; λ, ~ ) inside the interval ( x − + δ, x + − δ ). Repeating the argumentsof Section 4, we get the following result. Proposition 5.1.
Under the assumptions above, all results of Theorem and ofCorollary about eigenvalues of the operators H ~ remain true with the remain-ders O ( ~ / ) in (4.5) , O ( ~ / ) in (4.14) and C ~ / in the right-hand side of (4.6) .Theorem about corresponding eigenfunctions remains also true with the remain-ders O ( ~ / | x − x ± | − / ) in (2.19) , O ( ~ − / | x − x ± | − / ) in (2.21) and O ( ~ / ) in (4.18) . Our goal in this subsection is to extend the results of Section 4 to functions v ( x ) with a singular point x inside a potential well.We suppose that Assumption 2.1 holds everywhere except a point x and thatAssumption 2.2 holds for some λ such that x is an interior point of the interval EMICLASSICAL LIMIT OF EIGENFUNCTIONS 15 ( x − ( λ ) , x + ( λ )). We assume that v ( x ) has finite limits at x but the left andright limits might be different. Finally, we require that v ′ ∈ L ( x , x ± δ ) and v ′′ ∈ L ( x , x ± δ ) for some δ > u + ( x ) and u − ( x ) of equation (1.1) on the in-tervals ( x , ∞ ) and ( −∞ , x ), respectively. Define, as usual, the function r ( ξ ) byformulas (2.11) and (3.3). Since r ∈ L ( x , x ± δ ), the limits u ± ( x ±
0) and u ′± ( x ±
0) exist, and we can use formulas (2.19) and (2.21) for these limits (withslightly weaker estimates of the remainders – see subs. 5.1). It follows that theWronskian w ( λ, ~ ) of u + and u − calculated at the point x is given by the expres-sion (cf. (4.3)) π ~ − / (cid:16) p ( x , λ ) sin( ~ − ϕ + ( x ; λ ) + π/
4) cos( ~ − ϕ − ( x ; λ ) + π/ p ( x , λ ) − cos( ~ − ϕ + ( x ; λ ) + π/
4) sin( ~ − ϕ − ( x ; λ ) + π/ (cid:17) + O (1)where p ( x , λ ) = (cid:0) λ − v ( x − (cid:1) / (cid:0) λ − v ( x + 0) (cid:1) − / . (5.1)Let an eigenvalue λ of the operator H ~ be close to λ . Since w ( λ, ~ ) = 0, we seethat p ( x , λ ) sin( ~ − ϕ + ( x ; λ ) + π/
4) cos( ~ − ϕ − ( x ; λ ) + π/ p ( x , λ ) − sin( ~ − ϕ + ( x ; λ ) + π/
4) cos( ~ − ϕ − ( x ; λ ) + π/
4) = O ( ~ / ) . (5.2)Formula (5.2) yields a generalization of the Bohr-Sommerfeld quantization condi-tion (4.5) and reduces to it if v ( x + 0) = v ( x − a ( λ, ~ ) be defined by equality (4.1). To calculate | a ( λ, ~ ) | , weuse again relations (4.12) and (4.13). However, additional factors | q ( x − | − / and | q ( x − | / appear now in their left-hand sides. Similarly, additional factors | q ( x + 0) | − / and | q ( x + 0) | / appear in their right-hand sides. This implies that a ( λ, ~ ) = p ( x , λ ) cos ( ~ − ϕ − ( x ; λ ) + π/ p − ( x , λ ) sin ( ~ − ϕ − ( x ; λ ) + π/
4) + O ( ~ / )= (cid:0) p ( x , λ ) sin ( ~ − ϕ + ( x ; λ ) + π/ p − ( x , λ ) cos ( ~ − ϕ + ( x ; λ ) + π/ (cid:1) − + O ( ~ / ) . (5.3)As before, using formula (2.26) (where x = x ) and the normalization condition k ψ k = 1, we obtain explicit expressions for the absolute values of constants c ± ( λ, ~ )in (4.17): | c + ( λ, ~ ) | = 2 / π − / ~ − / (cid:16) Z x + ( λ ) x ( λ − v ( x )) − / dx + a − ( λ, ~ ) Z x x − ( λ ) ( λ − v ( x )) − / dx (cid:17) − / + O ( ~ / ) (5.4)and | c − ( λ, ~ ) | = 2 / π − / ~ − / (cid:16) a ( λ, ~ ) Z x + ( λ ) x ( λ − v ( x )) − / dx + Z x x − ( λ ) ( λ − v ( x )) − / dx (cid:17) − / + O ( ~ / ) . (5.5) Thus, Theorems 4.1 and 4.4 can be supplemented by the following result.
Theorem 5.2.
Under the assumptions above, let an eigenvalue λ = λ ( ~ ) of theoperator H ~ belong to a neighborhood of λ . Then necessarily condition (5.2) issatisfied with the numbers ϕ ± ( x ; λ ) and p ( x , λ ) defined by (4.2) and (5.1) , re-spectively. All assertions ( for x = x ) of Theorem about the correspondingnormalized eigenfunction ψ ( x ; λ, ~ ) are true with the constants c ± ( λ, ~ ) whose ab-solute values are determined by formulas (5.3) , (5.4) and (5.5) . Remark 5.3.
If the functions v ′ ( x ) and v ′′ ( x ) are bounded in a neighborhood ofthe point x , then even in the case v ( x + 0) = v ( x −
0) estimates of all remaindersare the same as in Section 4. Thus, we have O ( ~ ) in (5.2) and O ( ~ / ) in (5.3) –(5.5). Remark 5.4.
Let under the assumptions above v ( x + 0) = v ( x − v ′ ( x ) is notrequired to be continuous at the point x . In particular, we see that jumps ofderivatives of the function v ( x ) at the point x are inessential. Let us consider an explicit example: v ( x ) = a + + v + x α + for x > v ( x ) = a − + v − | x | α − for x < , (5.6)where v ± > α ± >
0. Then all λ > max { a + , a − } are non-critical, the equation v ( x ) = λ has two solutions x + > x − < x − , x + ) is a potential well. Thepoint x = 0 might be singular and p (0 , λ ) = ( λ − a − ) / ( λ − a + ) − / .For potentials (5.6), the integrals in formulas (4.4) and (4.18) can be calculatedin terms of the beta function B. Observe that x + = ( λv − ) /α if v ( x ) = vx α for x >
0. For the integrals over (0 , x + ), we have Z x + ( λ − vx α ) / dx = λ / ( λ/v ) /α α − B(3 / , /α ) (5.7)and Z x + ( λ − vx α ) − / dx = λ − / ( λ/v ) /α α − B(1 / , /α ) . (5.8)The integrals over ( x − ,
0) can be calculated quite similarly.It follows that the quantization condition (5.2) holds with ϕ + (0 , λ ) = λ / /α + + v − /α + + α − B(3 / , /α + ) ,ϕ − (0 , λ ) = λ / /α − − v − /α − − α − − B(3 / , /α − ) , where λ ± = λ − a ± . In particular, in the case a + = a − =: a the Bohr-Sommerfeldquantization condition reads as( λ − a ) / /α + v − /α + + α − B(3 / , /α + )+ ( λ − a ) / /α − v − /α − − α − − B(3 / , /α − ) = π ~ ( n + 1 /
2) + O ( ~ ) . (5.9)Plugging expressions (5.7) and (5.8) into (1.4) we also find that K cl ( λ ) = λ / /α + + v − α + + α − B(3 / , /α + ) + λ / /α − − v − α − − α − − B(3 / , /α − ) λ − / /α + + v − α + + α − B(1 / , /α + ) + λ − / /α − − v − α − − α − − B(1 / , /α − ) . Observe that Theorems 4.1 and 4.4 can be applied to potential (5.6) if a + = a − and α ± ≥
2. If a + = a − but α ± ∈ [1 , EMICLASSICAL LIMIT OF EIGENFUNCTIONS 17 this case O ( ~ ) in (5.9) should be replaced by O ( ~ / ). If a ± are arbitrary and α ± ≥
1, then the conditions of Theorem 5.2 are satisfied. Moreover, according toRemark 5.3 in the case α ± ≥
2, the estimates of the remainders can be improved.Finally, we note that if α j <
1, then v ′′ L ( − δ, δ ) so that the semiclassicalapproximation does not directly work (even for a + = a − ) although all formulasabove remain meaningful. Let us briefly consider the problem on the half-axis. We now suppose thatequation (1.1) is satisfied for x ≥ ψ ∈ L ( R + ) and ψ (0) = 0. Assumptions 2.1and 2.2 should be slightly modified. Namely, we assume that the equation v ( x ) = λ has only one solution x + = x + ( λ ) and v ′ ( x + ) > , x + ) is a potential well.We suppose that the limit of v ( x ) as x → v ′ ( x ) and v ′′ ( x ) are bounded in a neighborhood of x = 0. Then the results of Theorem 2.5 onthe solution u + ( x ) of equation (1.1) are true for all x ≥
0. In particular, it followsfrom formula (2.19) that u + (0; λ, ~ ) = π / ~ / ( λ − v (0)) − / sin (cid:0) ~ − Z x + ( λ )0 ( λ − v ( x )) / dx + π/ (cid:1) + O ( ~ / ) . (5.10)Since ψ ( x ; λ, ~ ) = c + u + ( x ; λ, ~ ), this yields the quantization condition Z x + ( λ )0 ( λ − v ( x )) / dx = π ~ ( n + 3 /
4) + O ( ~ )where n = n ( λ, ~ ) is an integer.Consider now the boundary condition ψ ′ (0) = bψ (0), b = ¯ b . It follows from(2.21) that u ′ + (0; λ, ~ ) = − π / ~ − / ( λ − v (0)) / cos (cid:0) ~ − Z x + ( λ )0 ( λ − v ( x )) / dx + π/ (cid:1) + O ( ~ / ) . Comparing this formula with (5.10), we see that the value of u + (0; λ, ~ ) is inessentialso that the quantization condition looks like Z x + ( λ )0 ( λ − v ( x )) / dx = π ~ ( n + 1 /
4) + O ( ~ ) . It does not depend on b .Other results of Section 4 can also be naturally extended to the problem on thehalf-axis.I thank D. Robert for a discussion of papers on microlocal analysis. References [1] V. M. Babich and V. S. Buldyrev,
Asymptotic methods in diffraction problems of short-lengthwaves , Nauka, 1972 (Russian).[2] M. V. Fedoryuk,
Asymptotic methods for linear ordinary differential equations , Nauka, 1983(Russian).[3] M. V. Fedoryuk and V. P. Maslov,
Semi-classical approximation in quantum mechanics ,Amsterdam, Reidel, 1981.[4] B. Helffer, A. Martinez et D. Robert, Ergodicit´e et limite semi-classique, Comm. Math. Phys., , 313-326, 1987. [5] B. Helffer et D. Robert, Puits de potentiels g´en´eralis´es, Ann. Institut H. Poincar´e, phys.th´eor., , No 3, 291-331, 1984.[6] L. D. Landau and E. M. Lifshitz, Classical mechanics , Pergamon Press, 1960.[7] F. W. J. Olver,
Asymptotics and special functions , Academic Press, 1974.[8] B. Simon, Semiclassical analysis of low lying eigenvalues, II, Tunneling, Annals of Math., , 89-118, 1984.
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