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Singular semi-classical approximation on Liouville surfaces
Differential Geometry and its Applications 29 (2011) S125–S134
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Differential Geometry and its Applications www.elsevier.com/locate/difgeo
Singular semi-classical approximation on Liouville surfaces Kazuyoshi Kiyohara Department of Mathematics, Faculty of Science, Okayama University, Okayama 700-8530, Japan
a r t i c l e
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Article history: Available online 19 April 2011 Communicated by I. Koláˇr
The usual theory of semi-classical approximation for the laplacian on riemannian manifolds says that the energy levels of certain lagrangean submanifolds in the cotangent bundle provide approximate eigenvalues of the laplacian asymptotically. In this paper we consider a class of surfaces whose geodesic flows are completely integrable (Liouville surfaces defined over 2-sphere), and show the two results: One is the absence of the corresponding lagrangean submanifolds for certain eigenvalues; and the other is the existence of new approximate values, which are asymptotically finer along a certain direction even where the usual semi-classical approximate values exist. © 2011 Elsevier B.V. All rights reserved.
MSC: 35P20 53C22 Keywords: Semi-classical approximation Liouville surface Singular lagrangean Laplacian
1. Introduction The following fact is known as a “semi-classical approximation” for eigenvalues of the laplacian on a compact riemannian manifold, which is a higher-dimensional version of the WKB method (or the Liouville–Green transformation) for one-dimensional Schrödinger equations: Let M be a compact riemannian manifold and let 2E : T ∗ M → R be the square of the length function (thus E is the hamiltonian of the geodesic flow). Let L be a compact lagrangean submanifold of the cotangent bundle T ∗ M satisfying the following conditions: (1) L is on a level surface of E. (2) L has a volume element dμ such that ( L , dμ) is invariant by the geodesic flow. (3) Denote by α the canonical 1-form on T ∗ M and denote by m L ∈ H 1 ( L , Z) the Maslov class of L. Then the condition
1 2π
1
α − m L [γ ] ∈ Z γ
4
holds for any closed curve rule” for L.)
γ in L. (This condition is called “the Maslov quantization condition” or “the Bohr–Sommerfeld
If L satisfies the Maslov quantization condition, then the totality of the lagrangean submanifolds of the form t L (t > 0) satisfying the same condition is presented in the form L k = (bk + 1) L 0 (k = 0, 1, 2, . . .) for some primitive L 0 . Here, b is equal to 1 or 2 or 4, depending on the value of m L . Then we have ek := 2E | Lk = (bk + 1)2 e 0 , and the following theorem holds.
E-mail address: [email protected]. 0926-2245/$ – see front matter doi:10.1016/j.difgeo.2011.04.017
© 2011
Elsevier B.V. All rights reserved.
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K. Kiyohara / Differential Geometry and its Applications 29 (2011) S125–S134
Theorem 1. (See Weinstein [10].) There is a sequence λk (k = 0, 1, 2, . . .) of eigenvalues of the laplacian on M satisfying
√ λk = (bk + 1) e 0 + O k−1 .
Namely, we may say that for each lagrangean submanifold L satisfying the Maslov condition, the classical value 2E | L gives an approximate value of an eigenvalue of the laplacian (the actual approximation is asymptotic in nature though). If the geodesic flow of the manifold is completely integrable, then since there would be lots of such lagrangean submanifolds, one might expect that almost all eigenvalues would be approximated in this way. In this paper we consider this problem for Liouville surfaces defined over the two-spheres. The main results are the following: (1) There are some eigenvalues which do not correspond to the usual semi-classical values; (2) for those eigenvalues there are other semi-classical approximate values, which correspond to lagrangean submanifolds close to certain singular lagrangean subsets or to singular lagrangean subsets themselves. Semi-classical approximation for Liouville surface whose underlying manifold is a torus was studied in [6]. Many technical matters in this paper are similar to those in [6] (both rely on [7–9]), but the situation of Theorem 9 in Section 5, the main part in this paper, does not occur in the torus case. Another way to treat singular semi-classical approximation is given in [1]. A part of this paper was announced in [5]. This paper is organized as follows. First, in Section 2, we briefly review Liouville surfaces which are considered in this paper. In Section 3 we show that there is a natural second order operator which commute with the laplacian, and considering the simultaneous eigenfunctions, we obtain a natural labeling for eigenvalues of the laplacian. As a result, eigenvalues are labeled by the set Z0 × Z0 × {±1}, where Z0 denotes the set of nonnegative integers. In Section 4 we describe the usual semi-classical approximation in this setting, and finally in Sections 5–6 we give the singular semi-classical approximation and the proof. 2. Liouville surfaces In this section we briefly describe the basic properties of Liouville surfaces diffeomorphic to the two-spheres. A Liouville surface is, by definition, a two-dimensional riemannian manifold whose geodesic flow admits a nontrivial first integral which is a quadratic form on each cotangent space. The sphere of constant curvature and ellipsoids are typical examples. Here we give a constructive definition, which is a restricted version and is just treated in this paper. For the general theory of Liouville manifolds, see [3,2,4]. First, let us consider a torus
R = (R/α1 Z) × (R/α2 Z) = (x1 , x2 ) and let f 1 (x1 ), f 2 (x2 )
• • • • • •
(α1 , α2 > 0),
( f i ∈ C ∞ (R/αi Z)) be functions having the following properties:
1 − a f 1 (x1 ) 0 f 2 (x2 ) −a (a ∈ (0, 1) is a fixed number). α f i (xi ) = f i (−xi ) = f i ( 2i − xi ) for any xi ∈ R/αi Z (i = 1, 2). f 1 (0) = 0, f 1 (α1 /4) = 1 − a and f (x1 ) > 0 for 0 < x1 < α1 /4. f 2 (0) = 0, f 2 (α2 /4) = −a and f 2 (x2 ) < 0 for 0 < x2 < α2 /4. f i (0) = 0, f i (αi /4) = 0 (i = 1, 2).
√
The formal Taylor expansion of f 2 (x2 ) at x2 = 0 is obtained by substituting x1 = −1x2 to the formal Taylor expansion of f 1 (x1 ) at x1 = 0, i.e., if f 1 (x1 ) ∼ ak x2k (−1)k ak x2k 1 is the formal Taylor expansion at x1 = 0, then f 2 (x2 ) ∼ 2 at x2 = 0.
Note that the function f i has in fact the period the involution
αi /2 by the second condition. Let S be the quotient space of R divided by
(x1 , x2 ) → (−x1 , −x2 ). Then S is homeomorphic to the two-sphere and the quotient map R → S is a branched √ double covering of S with four ramification points. Taking the real part and the imaginary part of the function (x1 + −1x2 )2 as coordinate functions around the branch point x1 = x2 = 0 for example, one obtain a differentiable structure on S so that S is diffeomorphic to the two-sphere and the quotient map R → S is of C ∞ . Then, observing the Taylor expansion at the ramification points, one can easily verify that
g = f 1 (x1 ) − f 2 (x2 ) dx21 + dx22 , F=
1 f1 − f2
f 2 ξ12 + f 1 ξ22
(2.1)
represent a well-defined C ∞ riemannian metric on S and a C ∞ function on the cotangent bundle T ∗ S. Here (ξ1 , ξ2 ) denote the fiber coordinates corresponding to the base coordinates (x1 , x2 ). Denoting by E the energy function (the hamiltonian of the geodesic flow) with respect to the metric g, i.e.,
K. Kiyohara / Differential Geometry and its Applications 29 (2011) S125–S134
2E =
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ξ12 + ξ22 f 1 (x1 ) − f 2 (x2 )
(2.2)
we have the vanishing of the Poisson bracket { E , F } = 0, i.e., the function F is a first integral of the geodesic flow. Thus we have a Liouville surface ( S , F ). Proposition 2. The four branch points on S lie on a single closed geodesic (say C ) and the symmetry defined isometry of S which preserves the first integral F . Proof. The mapping (x1 , x2 ) → (x1 , −x2 ) on the torus induces a well-defined involution Since σ is an isometry of S, the fixed point set
C = (x1 , x2 ) ∈ S x1 = 0, α1 /2 or x2 = 0, α2 /2
σ with respect to C is a well-
σ on S and it preserves g and F .
is a totally geodesic submanifold, i.e., a closed geodesic. It is clear that C contains the four branch points (0, 0), (α1 /2, 0), (α1 /2, α2 /2), and (0, α2 /2). 2 3. Labeling for the eigenvalues Let us define two differential operators and on S by the formula
2 −1 ∂2 ∂ , + f 1 (x1 ) − f 2 (x2 ) ∂ x21 ∂ x22
−1 ∂2 ∂2 = f 2 (x2 ) 2 + f 1 (x1 ) 2 . f 1 (x1 ) − f 2 (x2 ) ∂ x1 ∂ x2
=
Then, as in the case of E and F , one can easily verify that those operators are well defined, self adjoint, and their principal symbols are 2E and F respectively. The following proposition is straightforward. Proposition 3. (1) The operator is the laplacian on S. (2) [ , ] = 0. (3) The involution σ leaves and invariant. In view of this proposition, one can consider simultaneous eigenfunctions with parity in easy.
σ . The following proposition is
Proposition 4. For fixed λ, μ ∈ R and = ±1, the vector space of functions u = u (x1 , x2 ) on S satisfying
u = λu ,
σ ∗u = u
u = μu ,
is of at most one dimensional. Moreover, such u has the form u = u 1 (x1 )u 2 (x2 ) and each function u i ∈ C ∞ (R/αi Z) of one variable satisfies the following ordinary differential equation
u 1 (x1 ) = −λ f 1 (x1 ) − κ u 1 (x1 ),
u 2 (x2 ) = −λ κ − f 2 (x2 ) u 2 (x2 ),
(3.1) (3.2)
κ = μ/λ, and the parity condition u 1 (−x1 ) = u 1 (x1 ),
u 2 (−x2 ) = u 2 (x2 ).
(3.3)
To this simultaneous eigenfunction u = u 1 (x1 )u 2 (x2 ) we associate a pair of integers ( p , q) by
# 0 < x1 < α1 /2 u 1 (x1 ) = 0 = p ,
# 0 < x2 < α2 /2 u 2 (x2 ) = 0 = q.
(3.4) (3.5)
By the standard Sturm–Liouville argument we have the following theorem (see also Theorem 4.1 in [6]). Let Z0 denote the set of nonnegative integers.
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Theorem 5. To each element ( p , q, ) ∈ Z0 × Z0 × {±1}, there are unique λ and μ = λκ and functions u 1 (x1 ) and u 2 (x2 ) satisfying (3.1)–(3.5). In particular, there is a one–one correspondence between the set of the eigenvalues of the laplacian on S and the set Z0 × Z0 × {±1}, multiplicities being counted. In view of the above correspondence, we denote by
λ( p , q, ),
κ ( p , q, )
the values of λ and κ corresponding to ( p , q, ) ∈ Z0 × Z0 × {±1}. The only exception is the case where p = q = 0 and = 1; in this case λ = μ = 0 and κ is not defined. For example, in the case where S is the sphere of constant curvature 1, we have
λ( p , q, ) =
( p + q)( p + q + 1)
if = 1,
( p + q + 1)( p + q + 2) if = −1.
The values of κ depend on a and they are probably not explicitly known (in this case, f i are elliptic functions and u i are Lamé’s functions). The following proposition is also obtained by the standard Sturm–Liouville theory. Proposition 6. For any ( p , q, ) ∈ Z0 × Z0 × {±1}, we have
C 1 ( p + q)
λ( p , q, ) C 2 ( p + q),
−a < κ ( p , q, ) < 1 − a,
( p , q, ) = (0, 0, −1),
(3.6)
( p , q, ) = (0, 0, 1),
(3.7)
where C 1 and C 2 are positive constants which do not depend on p, q. 4. Lagrangean tori satisfying the Maslov quantization condition (usual semi-classical approximation) A point of the image of the mapping (2E , F ) : T ∗ S −{0} → R2 is regular if the value of F /2E does not equal to 1 − a, 0, −a (note that the range of F /2E is equal to [−a, 1 − a]), and in this case, the inverse image of a point is a disjoint union of two tori, which are mapped from one to the other by (−1)-multiplication. When F /2E is 1 − a or −a, the inverse image is a disjoint union of two circles, and if F /2E = 0, then the inverse image is a union of two tori which are transversally intersected in a level surface of E on two circles. To simplify the subsequent descriptions, we put α1 /4
A 1 (κ ) =
f 1 (x) − κ dx,
(4.1)
κ − f 2 (x) dx.
(4.2)
x1 (κ )
α2 /4
A 2 (κ ) =
x2 (κ )
Here x1 (κ ) denotes the unique x ∈ [0, α1 /4] such that f 1 (x) = κ if κ 0, and x1 (κ ) = 0 if unique value in [0, α2 /4] satisfying f 2 (x2 (κ )) = κ , and x2 (κ ) = 0 if κ > 0.
κ < 0. Also, x2 (κ ) is, if κ 0, the
Proposition 7. (1) The lagrangean torus given by 2E = λ+ , F /2E = κ+ ∈ (0, 1 − a) satisfies the Maslov quantization condition if and only if there are integers p 0, q˜ > 0 such that
λ+ A 1 (κ+ ) =
π 2
p+
1
2
,
π λ+ A 2 (κ+ ) = q˜ . 2
(4.3)
(2) The lagrangean torus given by 2E = λ− , F /2E = κ− ∈ (−a, 0) satisfies the Maslov quantization condition if and only if there are p˜ > 0, q 0 such that
λ− A 1 (κ− ) =
π 2
p˜ ,
λ− A 2 (κ− ) =
π 2
q+
1 2
.
(4.4)
Proof. Let γi (i = 1, 2) be generators (closed curves) of H 1 ( L , Z), L being a lagrangean torus defined by 2E = λ+ , F /2E = κ+ , such that the projection of γ1 (resp. γ2 ) does not depend on x2 (resp. x1 ). Then by (2.1) and (2.2) we have ξi2 = λ+ | f i (xi ) − κ+ | (i = 1, 2) on L. Also, the projection image of γ1 by the bundle projection π : T ∗ S → S crosses the boundary of π ( L ) two times, while π (γ2 ) does not intersect the boundary of π ( L ). Therefore we have
K. Kiyohara / Differential Geometry and its Applications 29 (2011) S125–S134
m L [γi ] =
α = 4 λ+ A i (κ+ ), which implies (1). (2) is similar.
2 (i = 1),
(i = 2),
0
γi
2
Since A 1 (κ ) is decreasing and A 2 (κ ) is increasing in above proposition satisfy the inequalities
A 1 (0)
(1)
A 2 (0)
>
p+ q˜
1 2
(2)
,
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A 1 (0) A 2 (0)
<
κ respectively, it follows that the integers in (1) and (2) of the
p˜ q+
(4.5)
1 2
respectively. Now, substituting
q˜ = q + (1 − )/2,
p˜ = p + (1 − )/2
to the above formulas, we define “semi-classical values”
λ± = λ± ( p , q, ),
κ± = κ ± ( p , q , )
by the formulas (4.3) and (4.4) respectively for ( p , q, ) satisfying (4.5). Then we have the following theorem, which corresponds to the “usual” semi-classical approximation. Theorem 8. There are constants C 1 , C 2 > 0 such that for any ( p , q, ) ∈ Z0 × Z0 × {±1} satisfying
A 1 (0) A 2 (0)
p + 1 /2
>
resp.
q + (1 − )/2
A 1 (0) A 2 (0)
<
p + (1 − )/2 q + 1 /2
,
p + q C1,
the inequality
√ | λ − λ+ |
√
resp. | λ −
C2
κ+ (1 − a − κ+ )( p + q) λ− |
C2
−κ− (a + κ− )( p + q)
hold, where λ = λ( p , q, ) and λ± = λ± ( p , q, ). For example, putting
( pk , qk ) = (2kp + p + k, 2kq + q) (k = 0, 1, 2, . . .), then we have
κ+ ( pk , qk , 1) = κ+ ( p , q, 1),
λ+ ( pk , qk , 1) = (2k + 1) λ+ ( p , q, 1),
and obtain the situation stated in Introduction. However, observe that this theorem contains much finer information. In particular, it implies that the rate of approximation is getting worse when κ+ , or κ , tends to zero, i.e., the corresponding lagrangean submanifold is getting close to the singular one defined by F /2E = 0. The proof of the theorem is obtained by applying the usual WKB method (by means of trigonometric and Airy functions) to each Eq. (3.1) and (3.2); see [7], Chaps. 6 and 11. Similar results are obtained in the torus case [6] by using Weber function, see pp. 35–37, the cases IIa and IIIa in [6]. As a consequence, one would recognize the following two problems: First, there are no lagrangean submanifolds approximating the value λ( p , q, 1) for ( p , q) in the band region
p q + 1 /2
A 1 (0) A 2 (0)
p + 1 /2 q
,
and there are two lagrangean submanifolds approximating the value λ( p , q, −1) for ( p , q) in the band region
p + 1 /2 q+1
A 1 (0) A 2 (0)
p+1 q + 1 /2
;
and secondly, the approximation becomes ineffective when κ± tends to zero.
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5. Approximation in the region where κ → 0 as λ → ∞ (singular semi-classical approximation)
α ∈ R and = ±1,
π α e 1 1 + arg Γ . D (α , ) = log + iα + arctan e −π α − 2 |α | 2 2 2 4
We put, for
As is easily seen, we have
D (−α , ) = − D (α , ). When α is near 0, then the first term is dominant, and when We would like to define new semi-classical values
λ0 = λ0 ( p , q, ),
α → ∞, then D (α , ) tends to − π /8 by the Stirling formula.
κ0 = κ 0 ( p , q , )
for certain ( p , q, ) ∈ Z0 × Z0 × {±1} by the formulas
√
λ0 κ 0 π 1 , p+ − √ , = 2 c0 2 2 4
√
λ0 κ 0 π 1 , λ0 A 2 (κ0 ) − D q+ − √ , =
λ0 A 1 (κ0 ) + D
2 c0
where c 0 =
2
2
(5.1) (5.2)
4
1 f (0). 2 1
Theorem 9. There are positive constants C 1 > 0 (large), C 2 , C 3 , C 4 > 0 (small) such that for any positive integers p , q satisfying
p + q C1,
A 2 (0) p − A 1 (0)q C 2 log( p + q),
(5.3)
the following statements hold: (1) (2) (3)
There are unique λ0 = λ0 ( p , q, ) and κ0 = κ0 ( p , q, ) satisfying the above (5.1), (5.2) and √ λ| κ | √ √ C3. | λ − λ0 | C 4 ( p + q)−2/3 .
√
λ 0 | κ0 | C 3 .
Here λ = λ( p , q, ), κ = κ ( p , q, ). We have a corollary. Note that
A 1 (κ ) + A 2 (κ ) = a0 + a1 κ + O when
κ2
κ → 0.
Corollary 10. If a1 = 0, then for ( p , q, ) satisfying the assumption of Theorem 9, we have
π ( p + q + 1 ) √ C 2 λ− 2 , A 1 (0) + A 2 (0) ( p + q + 1/2)2/3
where C is a constant independent of ( p , q).
√
The proof is straightforward. Namely, in this case we have simpler approximate values than λ0 . The condition a1 = 0 being only a one-dimensional constraint for the functions f 1 and f 2 , there are of course many cases satisfying it. The simplest one is the sphere of constant curvature; in this case A 1 + A 2 is constant. 6. Proof of Theorem 9 For the proof of Theorem 9 we use Weber’s parabolic cylinder functions to approximate the solutions of the differential equations (3.1), (3.2). Following [8,9], we first transform Eqs. (3.1) and (3.2): Define the change of variables xi → ζi ([−αi /4, αi /4] → [−ζ˜i , ζ˜i ]) (i = 1, 2) by
f i (xi ) − κ dxi = ζ 2 − κ1 , i d ζi
where
κ1 and ζ˜i are given by
x i = 0 ↔ ζ i = 0,
K. Kiyohara / Differential Geometry and its Applications 29 (2011) S125–S134
x0
√
κ − f 1 (x) dx =
κ1
0
ζ˜1 f 1 (x) − κ dx = ζ 2 − κ1 dζ, √
x0
α2 /4
κ − f 2 (x) dx =
0
κ1
ζ˜2
κ1 + ζ 2 d ζ
0
κ 0 (x0 is the unique zero of f 1 (x) − κ in [0, α1 /4]), and x0
√
−κ1
f 2 (x) − κ dx =
0
−κ1 − ζ 2 dζ,
0
α1 /4
f 1 (x) − κ dx =
0
ζ˜1
ζ 2 − κ1 dζ,
0
ζ˜2
α2 /4
κ − f 2 (x) dx = √
x0
for
κ1 − ζ 2 dζ,
0
α1 /4
for
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κ1 + ζ 2 d ζ
−κ1
κ < 0 (x0 is the unique zero of f 2 (x) − κ in [0, α2 /4]). Thus κ1 and x0 are functions of κ . We easily have κ 1 κ1 = √ + O κ 2 , c 0 = f 1 (0). c0
2
(6.1)
Denoting by
w i (κ1 , ζi ) =
dxi
− 12 ui ,
d ζi
we have
w i = −λ ζi2 − κ1 + ψi (κ1 , ζi ) w i
(i = 1, 2),
(6.2)
where
ψi (κ1 , ζi ) =
dxi
12
d2 dζi2
d ζi
dxi
− 12
d ζi
.
By [8] we know that ψi (κ1 , ζi ) are continuous on a neighborhood of κ1 = 0 and |ζi | ζ˜i . In the subsequent argument we only consider the case i = 1; the case i = 2 is similar by exchanging κ and −κ . Now we summarize various facts on Weber’s (modified) parabolic cylinder function and its application to approximation theory which are needed here. Put
W c (α , x) =
√ π α /4 π i /8 ix2 /4 2e e e Γ ( 12
+ iα
)1/2 |Γ ( 12
+ iα
∞ )|1/2
√
e −x(1−i )t /
2 − 12 +i α −t 2 /2
t
e
dt .
0
Then w = W c satisfies the differential equation
2 x w =− −α w
(6.3)
4
and so are its real and imaginary parts. We have Proposition 11. There is a unique real solution w = W (α , x) of (6.3) such that
W c (α , x) = k(α )−1/2 W (α , x) + ik(α )1/2 W (α , −x),
k(α ) =
e 2π α + 1 − e π α .
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√ The function W (α , x) is called Weber’s modified parabolic cylinder function. Clearly, the function w (ζ ) = W ( 12 λκ1 , √ ± 2ζ λ1/4 ) satisfies
w = −λ ζ 2 − κ1 w . We need the following result of Olver. Theorem 12. (See [9].) Eq. (6.2) has a solution w (λ, κ1 , ζ ) such that
w (λ, κ1 , ζ ) = W where
1√ 2
√
λκ1 , 2 ζ λ
1/4
+ (λ, κ1 , ζ ),
−1 1 √ √ l ( 2 λκ1 ) | (λ, κ1 , ζ )| 1√ 1/4 E λκ1 , 2 ζ λ exp √ Vζ,ζ˜i ( F ) − 1 , √ √ 2 M ( 12 λκ1 , 2ζ λ1/4 ) 2λ1/4
2
l(b) = sup Ω(x) M (b, x)
ζ˜1
x∈R
Vζ,ζ˜i ( F ) =
,
ζ
|ψ(κ1 , t )| dt , √ Ω( 2t λ1/4 )
and Ω(x) is any continuous function on R, positive on x = 0, and Ω(x)/|x| is bounded at |x| = ∞. Here, the auxiliary functions M (α , x) and E (α , x) are defined as follows:
⎧ 2 −1 2 1/2 ⎪ (x −σ (α )), ⎨ {k(α ) W (α , x) + k(α ) W (α , −x) } 1 / 2 M (α , x) = {2W (α , x) W (α , −x)} (−σ (α ) x σ (α )), ⎪ ⎩ − 1 2 2 1/2 {k(α ) W (α , x) + k(α ) W (α , −x) } (σ (α ) x), ⎧ 1/2 ⎪ (x −σ (α )), ⎨ k(α ) E (α , x) = { W (α , −x)/ W (α , x)}1/2 (−σ (α ) x σ (α )), ⎪ ⎩ k(α )−1/2 (σ (α ) x)
and
σ (α ) is the smallest positive x satisfying k(α )−1/2 W (α , x) = k(α )1/2 W (α , −x).
We put Ω(x) = |x|1/3 . Then as indicated in [8], Sections 9.2 and 10.2, we have
l(b) = O (1) as b → ±∞, and Vζ,ζ˜ ( F ) C λ−1/12 uniformly in ζ for some constant C . Thus, putting i
w c (λ, κ1 , ζ ) = k
−1/2
1√ 2
λκ1
w (λ, κ1 , ζ ) + ik
1√ 2
1/2 λκ1
w (λ, κ1 , −ζ ),
we have
√ √ √ √ w c (λ, κ1 , ζ ) − W c 1 λκ1 , 2ζ λ1/4 = M 1 λκ1 , 2ζ λ1/4 O λ−1/3 . 2 2
Define phase functions Θ and θ by
W c (α , x) = W c (α , x) e i Θ(α ,x) ,
−1/2
Θ(α , 0) = Θ0 (α ) = arg k(α )
w c (λ, κ1 , ζ ) = w c (λ, κ1 , ζ ) e i θ (λ,κ1 ,ζ ) , 1/2
+ ik(α )
∈ (0, π /4),
θ(λ, κ1 , 0) = Θ0
1√ 2
λκ1 .
Note that 2Θ0 (α ) = arctan(e −π α ). Since | W c | M, we therefore obtain
θ(λ, κ1 , ζ ) = Θ
1√ 2
√
λκ1 , 2 ζ λ
1/4
+ O λ−1/3 .
(6.4)
K. Kiyohara / Differential Geometry and its Applications 29 (2011) S125–S134
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By [8], Section 8.2, we have
1
π
4
4
Θ(α , x) = x2 − α log x +
+
1
arg Γ
2
1 2
1 + α2 + iα + O 2
(6.5)
x
as |α /x| → 0. Also, we have
ζ
1
t 2 − κ1 dt =
√
2
ζ2 −
κ1 4
−
κ1 2
log √
log 2 ζ − κ1 + O κ12 , 2 |κ 1 |
(6.6)
κ˜ 1
where κ˜ 1 = max{0, κ1 }. By (6.4), (6.5), and (6.6), we thus have
√ λ t 2 − κ1 dt + √ ζ
θ(λ, κ1 , ζ ) =
κ˜ 1
+
π 4
+
1 2
arg Γ
1 2
√
λκ1
log √
λ|κ1 |
4
√
+
2e
i λκ1
+O
2
√
λκ12 + O λ−1/3 .
(6.7)
We put
w + (λ, κ1 , ζ ) = k(α )1/2 − ik(α )−1/2 w c (λ, κ1 , ζ ) , w − (λ, κ1 , ζ ) = k(α )−1/2 − ik(α )1/2 w c (λ, κ1 , ζ ) ,
α=
1√ 2
λκ1 .
Then w + and w − are even and odd functions in ζ respectively, which satisfy w + (λ, κ1 , 0) > 0 and w − (λ, κ1 , 0) > 0. Thus the functions w + and w − are written as the product of positive functions and cos θ+ and sin θ− respectively, where
π
1√
+ Θ0 λκ1 + θ(λ, κ1 , ζ ), 2
1√ λκ1 + θ(λ, κ1 , ζ ). θ− (λ, κ1 , ζ ) = −Θ0
θ+ (λ, κ1 , ζ ) = −
2
(6.8)
2
Now let u 1,± (λ, κ , x) be even and odd periodic solution of Eq. (3.1) which satisfies (3.4). We may assume u 1,± are transformed to w ± in the way described at the beginning of this section. Take a point x0 near the midpoint of x0 and α1 /4 so that x0 is the p -th zero of u 1,+ (λ, κ , x), x > 0. Let ζ˜1 be the corresponding value of ζ to x0 . Then we have
π θ+ λ, κ1 , ζ˜1 = + p − 1 π . 2
Then combined with (6.7), (6.8), and the equality ˜
x0
f 1 (x) − κ dx =
ζ1
√
x˜ 0
x˜ 0 = max{0, x0 },
ζ 2 − κ1 dζ,
κ˜ 1
we have
x0
√ 1 π λ f 1 (x) − κ dx + D (α , 1) = p − π + + O,
2
x˜ 0
8
α=
1√ 2
λκ1 ,
where O -term is given as
O
√
λκ12 + O λ−1/3 .
Using the usual semi-classical approximation on the interval [x0 , α1 /2 − x0 ], we have
√
α1/2−x0
λ x0
f 1 (x) − κ dx = π p + 1 + O λ−1/2 ,
(6.9)
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K. Kiyohara / Differential Geometry and its Applications 29 (2011) S125–S134
where p is the number of zeros of u 1,+ in the open interval (x0 , α1 /2 − x0 ). Since 2p + p = p, we therefore obtain
√
λ A 1 (κ ) + D
√ λκ1 2
π 1 + O. ,1 = p+ 2
4
By the similar argument for the odd solution u 1,− , we also have
√
λ A 1 (κ ) + D
√ λκ1 2
π 3 + O. , −1 = p+ 2
4
The case of Eq. (3.2) is similar, and we have
√
λ A 2 (κ ) − D
√ λκ1 2
π 1 +O , = q+ − 2
2
4
( = ±1).
In view of (6.1) we have
D
√ λκ1 2
√ √ √ √ λκ , = D √ , + O λκ 2 + O λκ 2 log( λκ ) .
2 c0
Summarizing those results, we obtain the following theorem. Theorem 13. λ = λ( p , q, ) and κ = κ ( p , q, ) satisfy the following formulas
√
λκ π 1 + O, p+ − √ , = 2 c0 2 2 4
√
√ λκ π 1 + O, λ A 2 (κ ) − D √ , = q+ −
√
λ A 1 (κ ) + D
2 c0
2
2
4
where = ±1 and
O=O
√
√ √ λκ 2 + O λκ 2 log( λκ ) + O λ−1/3 .
Concerning the O -term, we note the following three cases:
√
λκ = O (1) ⇒ O = O λ−1/3 , √ √ ⇒ O = O λ−1/6 log λ , λκ = O λ1/6 √ √ ⇒ O = O λ−1/3 log λ . λκ = O λ1/12 Using the first case, we obtain Theorem 9. Using the other two cases, we have similar results. For the connection with the region where the usual semi-classical application is effective, the second case would be preferable. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
Y. Colin de Verdière, B. Parisse, Singular Bohr–Sommerfeld rules, Comm. Math. Phys. 205 (1999) 459–500. M. Igarashi, K. Kiyohara, K. Sugahara, Noncompact Liouville surfaces, J. Math. Soc. Japan 45 (1993) 459–479. K. Kiyohara, Compact Liouville surfaces, J. Math. Soc. Japan 43 (1991) 555–591. K. Kiyohara, Two classes of Riemannian manifolds whose geodesic flows are integrable, Mem. Amer. Math. Soc. 130 (619) (1997). K. Kiyohara, Semi-classical approximations on Liouville surfaces, Suuriken Koukyuuroku 1119 (1999) 35–47 (in Japanese). D.V. Kosygin, A.A. Minasov, Ya.G. Sinai, Statistical properties of Laplace–Beltrami operators on Liouville surfaces, Uspekhi Mat. Nauk 48 (1993) 3–130 (English transl. Russian Math. Surveys 48 (1993) 1–142). F.W.J. Olver, Asymptotics and Special Functions, Academic Press, 1974. F.W.J. Olver, Second order linear differential equations with two turning points, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 278 (1975) 137–174. F.W.J. Olver, Improved error bounds for second-order differential equations with two turning points, J. Res. Nat. Bur. Stand. B 80 (1976) 437–440. A. Weinstein, On Maslov’s quantization condition, in: Lecture Notes in Math., vol. 459, Springer-Verlag, Berlin, New York, 1975, pp. 341–372.
Contents lists available at ScienceDirect
Differential Geometry and its Applications www.elsevier.com/locate/difgeo
Singular semi-classical approximation on Liouville surfaces Kazuyoshi Kiyohara Department of Mathematics, Faculty of Science, Okayama University, Okayama 700-8530, Japan
a r t i c l e
i n f o
a b s t r a c t
Article history: Available online 19 April 2011 Communicated by I. Koláˇr
The usual theory of semi-classical approximation for the laplacian on riemannian manifolds says that the energy levels of certain lagrangean submanifolds in the cotangent bundle provide approximate eigenvalues of the laplacian asymptotically. In this paper we consider a class of surfaces whose geodesic flows are completely integrable (Liouville surfaces defined over 2-sphere), and show the two results: One is the absence of the corresponding lagrangean submanifolds for certain eigenvalues; and the other is the existence of new approximate values, which are asymptotically finer along a certain direction even where the usual semi-classical approximate values exist. © 2011 Elsevier B.V. All rights reserved.
MSC: 35P20 53C22 Keywords: Semi-classical approximation Liouville surface Singular lagrangean Laplacian
1. Introduction The following fact is known as a “semi-classical approximation” for eigenvalues of the laplacian on a compact riemannian manifold, which is a higher-dimensional version of the WKB method (or the Liouville–Green transformation) for one-dimensional Schrödinger equations: Let M be a compact riemannian manifold and let 2E : T ∗ M → R be the square of the length function (thus E is the hamiltonian of the geodesic flow). Let L be a compact lagrangean submanifold of the cotangent bundle T ∗ M satisfying the following conditions: (1) L is on a level surface of E. (2) L has a volume element dμ such that ( L , dμ) is invariant by the geodesic flow. (3) Denote by α the canonical 1-form on T ∗ M and denote by m L ∈ H 1 ( L , Z) the Maslov class of L. Then the condition
1 2π
1
α − m L [γ ] ∈ Z γ
4
holds for any closed curve rule” for L.)
γ in L. (This condition is called “the Maslov quantization condition” or “the Bohr–Sommerfeld
If L satisfies the Maslov quantization condition, then the totality of the lagrangean submanifolds of the form t L (t > 0) satisfying the same condition is presented in the form L k = (bk + 1) L 0 (k = 0, 1, 2, . . .) for some primitive L 0 . Here, b is equal to 1 or 2 or 4, depending on the value of m L . Then we have ek := 2E | Lk = (bk + 1)2 e 0 , and the following theorem holds.
E-mail address: [email protected]. 0926-2245/$ – see front matter doi:10.1016/j.difgeo.2011.04.017
© 2011
Elsevier B.V. All rights reserved.
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K. Kiyohara / Differential Geometry and its Applications 29 (2011) S125–S134
Theorem 1. (See Weinstein [10].) There is a sequence λk (k = 0, 1, 2, . . .) of eigenvalues of the laplacian on M satisfying
√ λk = (bk + 1) e 0 + O k−1 .
Namely, we may say that for each lagrangean submanifold L satisfying the Maslov condition, the classical value 2E | L gives an approximate value of an eigenvalue of the laplacian (the actual approximation is asymptotic in nature though). If the geodesic flow of the manifold is completely integrable, then since there would be lots of such lagrangean submanifolds, one might expect that almost all eigenvalues would be approximated in this way. In this paper we consider this problem for Liouville surfaces defined over the two-spheres. The main results are the following: (1) There are some eigenvalues which do not correspond to the usual semi-classical values; (2) for those eigenvalues there are other semi-classical approximate values, which correspond to lagrangean submanifolds close to certain singular lagrangean subsets or to singular lagrangean subsets themselves. Semi-classical approximation for Liouville surface whose underlying manifold is a torus was studied in [6]. Many technical matters in this paper are similar to those in [6] (both rely on [7–9]), but the situation of Theorem 9 in Section 5, the main part in this paper, does not occur in the torus case. Another way to treat singular semi-classical approximation is given in [1]. A part of this paper was announced in [5]. This paper is organized as follows. First, in Section 2, we briefly review Liouville surfaces which are considered in this paper. In Section 3 we show that there is a natural second order operator which commute with the laplacian, and considering the simultaneous eigenfunctions, we obtain a natural labeling for eigenvalues of the laplacian. As a result, eigenvalues are labeled by the set Z0 × Z0 × {±1}, where Z0 denotes the set of nonnegative integers. In Section 4 we describe the usual semi-classical approximation in this setting, and finally in Sections 5–6 we give the singular semi-classical approximation and the proof. 2. Liouville surfaces In this section we briefly describe the basic properties of Liouville surfaces diffeomorphic to the two-spheres. A Liouville surface is, by definition, a two-dimensional riemannian manifold whose geodesic flow admits a nontrivial first integral which is a quadratic form on each cotangent space. The sphere of constant curvature and ellipsoids are typical examples. Here we give a constructive definition, which is a restricted version and is just treated in this paper. For the general theory of Liouville manifolds, see [3,2,4]. First, let us consider a torus
R = (R/α1 Z) × (R/α2 Z) = (x1 , x2 ) and let f 1 (x1 ), f 2 (x2 )
• • • • • •
(α1 , α2 > 0),
( f i ∈ C ∞ (R/αi Z)) be functions having the following properties:
1 − a f 1 (x1 ) 0 f 2 (x2 ) −a (a ∈ (0, 1) is a fixed number). α f i (xi ) = f i (−xi ) = f i ( 2i − xi ) for any xi ∈ R/αi Z (i = 1, 2). f 1 (0) = 0, f 1 (α1 /4) = 1 − a and f (x1 ) > 0 for 0 < x1 < α1 /4. f 2 (0) = 0, f 2 (α2 /4) = −a and f 2 (x2 ) < 0 for 0 < x2 < α2 /4. f i (0) = 0, f i (αi /4) = 0 (i = 1, 2).
√
The formal Taylor expansion of f 2 (x2 ) at x2 = 0 is obtained by substituting x1 = −1x2 to the formal Taylor expansion of f 1 (x1 ) at x1 = 0, i.e., if f 1 (x1 ) ∼ ak x2k (−1)k ak x2k 1 is the formal Taylor expansion at x1 = 0, then f 2 (x2 ) ∼ 2 at x2 = 0.
Note that the function f i has in fact the period the involution
αi /2 by the second condition. Let S be the quotient space of R divided by
(x1 , x2 ) → (−x1 , −x2 ). Then S is homeomorphic to the two-sphere and the quotient map R → S is a branched √ double covering of S with four ramification points. Taking the real part and the imaginary part of the function (x1 + −1x2 )2 as coordinate functions around the branch point x1 = x2 = 0 for example, one obtain a differentiable structure on S so that S is diffeomorphic to the two-sphere and the quotient map R → S is of C ∞ . Then, observing the Taylor expansion at the ramification points, one can easily verify that
g = f 1 (x1 ) − f 2 (x2 ) dx21 + dx22 , F=
1 f1 − f2
f 2 ξ12 + f 1 ξ22
(2.1)
represent a well-defined C ∞ riemannian metric on S and a C ∞ function on the cotangent bundle T ∗ S. Here (ξ1 , ξ2 ) denote the fiber coordinates corresponding to the base coordinates (x1 , x2 ). Denoting by E the energy function (the hamiltonian of the geodesic flow) with respect to the metric g, i.e.,
K. Kiyohara / Differential Geometry and its Applications 29 (2011) S125–S134
2E =
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ξ12 + ξ22 f 1 (x1 ) − f 2 (x2 )
(2.2)
we have the vanishing of the Poisson bracket { E , F } = 0, i.e., the function F is a first integral of the geodesic flow. Thus we have a Liouville surface ( S , F ). Proposition 2. The four branch points on S lie on a single closed geodesic (say C ) and the symmetry defined isometry of S which preserves the first integral F . Proof. The mapping (x1 , x2 ) → (x1 , −x2 ) on the torus induces a well-defined involution Since σ is an isometry of S, the fixed point set
C = (x1 , x2 ) ∈ S x1 = 0, α1 /2 or x2 = 0, α2 /2
σ with respect to C is a well-
σ on S and it preserves g and F .
is a totally geodesic submanifold, i.e., a closed geodesic. It is clear that C contains the four branch points (0, 0), (α1 /2, 0), (α1 /2, α2 /2), and (0, α2 /2). 2 3. Labeling for the eigenvalues Let us define two differential operators and on S by the formula
2 −1 ∂2 ∂ , + f 1 (x1 ) − f 2 (x2 ) ∂ x21 ∂ x22
−1 ∂2 ∂2 = f 2 (x2 ) 2 + f 1 (x1 ) 2 . f 1 (x1 ) − f 2 (x2 ) ∂ x1 ∂ x2
=
Then, as in the case of E and F , one can easily verify that those operators are well defined, self adjoint, and their principal symbols are 2E and F respectively. The following proposition is straightforward. Proposition 3. (1) The operator is the laplacian on S. (2) [ , ] = 0. (3) The involution σ leaves and invariant. In view of this proposition, one can consider simultaneous eigenfunctions with parity in easy.
σ . The following proposition is
Proposition 4. For fixed λ, μ ∈ R and = ±1, the vector space of functions u = u (x1 , x2 ) on S satisfying
u = λu ,
σ ∗u = u
u = μu ,
is of at most one dimensional. Moreover, such u has the form u = u 1 (x1 )u 2 (x2 ) and each function u i ∈ C ∞ (R/αi Z) of one variable satisfies the following ordinary differential equation
u 1 (x1 ) = −λ f 1 (x1 ) − κ u 1 (x1 ),
u 2 (x2 ) = −λ κ − f 2 (x2 ) u 2 (x2 ),
(3.1) (3.2)
κ = μ/λ, and the parity condition u 1 (−x1 ) = u 1 (x1 ),
u 2 (−x2 ) = u 2 (x2 ).
(3.3)
To this simultaneous eigenfunction u = u 1 (x1 )u 2 (x2 ) we associate a pair of integers ( p , q) by
# 0 < x1 < α1 /2 u 1 (x1 ) = 0 = p ,
# 0 < x2 < α2 /2 u 2 (x2 ) = 0 = q.
(3.4) (3.5)
By the standard Sturm–Liouville argument we have the following theorem (see also Theorem 4.1 in [6]). Let Z0 denote the set of nonnegative integers.
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K. Kiyohara / Differential Geometry and its Applications 29 (2011) S125–S134
Theorem 5. To each element ( p , q, ) ∈ Z0 × Z0 × {±1}, there are unique λ and μ = λκ and functions u 1 (x1 ) and u 2 (x2 ) satisfying (3.1)–(3.5). In particular, there is a one–one correspondence between the set of the eigenvalues of the laplacian on S and the set Z0 × Z0 × {±1}, multiplicities being counted. In view of the above correspondence, we denote by
λ( p , q, ),
κ ( p , q, )
the values of λ and κ corresponding to ( p , q, ) ∈ Z0 × Z0 × {±1}. The only exception is the case where p = q = 0 and = 1; in this case λ = μ = 0 and κ is not defined. For example, in the case where S is the sphere of constant curvature 1, we have
λ( p , q, ) =
( p + q)( p + q + 1)
if = 1,
( p + q + 1)( p + q + 2) if = −1.
The values of κ depend on a and they are probably not explicitly known (in this case, f i are elliptic functions and u i are Lamé’s functions). The following proposition is also obtained by the standard Sturm–Liouville theory. Proposition 6. For any ( p , q, ) ∈ Z0 × Z0 × {±1}, we have
C 1 ( p + q)
λ( p , q, ) C 2 ( p + q),
−a < κ ( p , q, ) < 1 − a,
( p , q, ) = (0, 0, −1),
(3.6)
( p , q, ) = (0, 0, 1),
(3.7)
where C 1 and C 2 are positive constants which do not depend on p, q. 4. Lagrangean tori satisfying the Maslov quantization condition (usual semi-classical approximation) A point of the image of the mapping (2E , F ) : T ∗ S −{0} → R2 is regular if the value of F /2E does not equal to 1 − a, 0, −a (note that the range of F /2E is equal to [−a, 1 − a]), and in this case, the inverse image of a point is a disjoint union of two tori, which are mapped from one to the other by (−1)-multiplication. When F /2E is 1 − a or −a, the inverse image is a disjoint union of two circles, and if F /2E = 0, then the inverse image is a union of two tori which are transversally intersected in a level surface of E on two circles. To simplify the subsequent descriptions, we put α1 /4
A 1 (κ ) =
f 1 (x) − κ dx,
(4.1)
κ − f 2 (x) dx.
(4.2)
x1 (κ )
α2 /4
A 2 (κ ) =
x2 (κ )
Here x1 (κ ) denotes the unique x ∈ [0, α1 /4] such that f 1 (x) = κ if κ 0, and x1 (κ ) = 0 if unique value in [0, α2 /4] satisfying f 2 (x2 (κ )) = κ , and x2 (κ ) = 0 if κ > 0.
κ < 0. Also, x2 (κ ) is, if κ 0, the
Proposition 7. (1) The lagrangean torus given by 2E = λ+ , F /2E = κ+ ∈ (0, 1 − a) satisfies the Maslov quantization condition if and only if there are integers p 0, q˜ > 0 such that
λ+ A 1 (κ+ ) =
π 2
p+
1
2
,
π λ+ A 2 (κ+ ) = q˜ . 2
(4.3)
(2) The lagrangean torus given by 2E = λ− , F /2E = κ− ∈ (−a, 0) satisfies the Maslov quantization condition if and only if there are p˜ > 0, q 0 such that
λ− A 1 (κ− ) =
π 2
p˜ ,
λ− A 2 (κ− ) =
π 2
q+
1 2
.
(4.4)
Proof. Let γi (i = 1, 2) be generators (closed curves) of H 1 ( L , Z), L being a lagrangean torus defined by 2E = λ+ , F /2E = κ+ , such that the projection of γ1 (resp. γ2 ) does not depend on x2 (resp. x1 ). Then by (2.1) and (2.2) we have ξi2 = λ+ | f i (xi ) − κ+ | (i = 1, 2) on L. Also, the projection image of γ1 by the bundle projection π : T ∗ S → S crosses the boundary of π ( L ) two times, while π (γ2 ) does not intersect the boundary of π ( L ). Therefore we have
K. Kiyohara / Differential Geometry and its Applications 29 (2011) S125–S134
m L [γi ] =
α = 4 λ+ A i (κ+ ), which implies (1). (2) is similar.
2 (i = 1),
(i = 2),
0
γi
2
Since A 1 (κ ) is decreasing and A 2 (κ ) is increasing in above proposition satisfy the inequalities
A 1 (0)
(1)
A 2 (0)
>
p+ q˜
1 2
(2)
,
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A 1 (0) A 2 (0)
<
κ respectively, it follows that the integers in (1) and (2) of the
p˜ q+
(4.5)
1 2
respectively. Now, substituting
q˜ = q + (1 − )/2,
p˜ = p + (1 − )/2
to the above formulas, we define “semi-classical values”
λ± = λ± ( p , q, ),
κ± = κ ± ( p , q , )
by the formulas (4.3) and (4.4) respectively for ( p , q, ) satisfying (4.5). Then we have the following theorem, which corresponds to the “usual” semi-classical approximation. Theorem 8. There are constants C 1 , C 2 > 0 such that for any ( p , q, ) ∈ Z0 × Z0 × {±1} satisfying
A 1 (0) A 2 (0)
p + 1 /2
>
resp.
q + (1 − )/2
A 1 (0) A 2 (0)
<
p + (1 − )/2 q + 1 /2
,
p + q C1,
the inequality
√ | λ − λ+ |
√
resp. | λ −
C2
κ+ (1 − a − κ+ )( p + q) λ− |
C2
−κ− (a + κ− )( p + q)
hold, where λ = λ( p , q, ) and λ± = λ± ( p , q, ). For example, putting
( pk , qk ) = (2kp + p + k, 2kq + q) (k = 0, 1, 2, . . .), then we have
κ+ ( pk , qk , 1) = κ+ ( p , q, 1),
λ+ ( pk , qk , 1) = (2k + 1) λ+ ( p , q, 1),
and obtain the situation stated in Introduction. However, observe that this theorem contains much finer information. In particular, it implies that the rate of approximation is getting worse when κ+ , or κ , tends to zero, i.e., the corresponding lagrangean submanifold is getting close to the singular one defined by F /2E = 0. The proof of the theorem is obtained by applying the usual WKB method (by means of trigonometric and Airy functions) to each Eq. (3.1) and (3.2); see [7], Chaps. 6 and 11. Similar results are obtained in the torus case [6] by using Weber function, see pp. 35–37, the cases IIa and IIIa in [6]. As a consequence, one would recognize the following two problems: First, there are no lagrangean submanifolds approximating the value λ( p , q, 1) for ( p , q) in the band region
p q + 1 /2
A 1 (0) A 2 (0)
p + 1 /2 q
,
and there are two lagrangean submanifolds approximating the value λ( p , q, −1) for ( p , q) in the band region
p + 1 /2 q+1
A 1 (0) A 2 (0)
p+1 q + 1 /2
;
and secondly, the approximation becomes ineffective when κ± tends to zero.
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5. Approximation in the region where κ → 0 as λ → ∞ (singular semi-classical approximation)
α ∈ R and = ±1,
π α e 1 1 + arg Γ . D (α , ) = log + iα + arctan e −π α − 2 |α | 2 2 2 4
We put, for
As is easily seen, we have
D (−α , ) = − D (α , ). When α is near 0, then the first term is dominant, and when We would like to define new semi-classical values
λ0 = λ0 ( p , q, ),
α → ∞, then D (α , ) tends to − π /8 by the Stirling formula.
κ0 = κ 0 ( p , q , )
for certain ( p , q, ) ∈ Z0 × Z0 × {±1} by the formulas
√
λ0 κ 0 π 1 , p+ − √ , = 2 c0 2 2 4
√
λ0 κ 0 π 1 , λ0 A 2 (κ0 ) − D q+ − √ , =
λ0 A 1 (κ0 ) + D
2 c0
where c 0 =
2
2
(5.1) (5.2)
4
1 f (0). 2 1
Theorem 9. There are positive constants C 1 > 0 (large), C 2 , C 3 , C 4 > 0 (small) such that for any positive integers p , q satisfying
p + q C1,
A 2 (0) p − A 1 (0)q C 2 log( p + q),
(5.3)
the following statements hold: (1) (2) (3)
There are unique λ0 = λ0 ( p , q, ) and κ0 = κ0 ( p , q, ) satisfying the above (5.1), (5.2) and √ λ| κ | √ √ C3. | λ − λ0 | C 4 ( p + q)−2/3 .
√
λ 0 | κ0 | C 3 .
Here λ = λ( p , q, ), κ = κ ( p , q, ). We have a corollary. Note that
A 1 (κ ) + A 2 (κ ) = a0 + a1 κ + O when
κ2
κ → 0.
Corollary 10. If a1 = 0, then for ( p , q, ) satisfying the assumption of Theorem 9, we have
π ( p + q + 1 ) √ C 2 λ− 2 , A 1 (0) + A 2 (0) ( p + q + 1/2)2/3
where C is a constant independent of ( p , q).
√
The proof is straightforward. Namely, in this case we have simpler approximate values than λ0 . The condition a1 = 0 being only a one-dimensional constraint for the functions f 1 and f 2 , there are of course many cases satisfying it. The simplest one is the sphere of constant curvature; in this case A 1 + A 2 is constant. 6. Proof of Theorem 9 For the proof of Theorem 9 we use Weber’s parabolic cylinder functions to approximate the solutions of the differential equations (3.1), (3.2). Following [8,9], we first transform Eqs. (3.1) and (3.2): Define the change of variables xi → ζi ([−αi /4, αi /4] → [−ζ˜i , ζ˜i ]) (i = 1, 2) by
f i (xi ) − κ dxi = ζ 2 − κ1 , i d ζi
where
κ1 and ζ˜i are given by
x i = 0 ↔ ζ i = 0,
K. Kiyohara / Differential Geometry and its Applications 29 (2011) S125–S134
x0
√
κ − f 1 (x) dx =
κ1
0
ζ˜1 f 1 (x) − κ dx = ζ 2 − κ1 dζ, √
x0
α2 /4
κ − f 2 (x) dx =
0
κ1
ζ˜2
κ1 + ζ 2 d ζ
0
κ 0 (x0 is the unique zero of f 1 (x) − κ in [0, α1 /4]), and x0
√
−κ1
f 2 (x) − κ dx =
0
−κ1 − ζ 2 dζ,
0
α1 /4
f 1 (x) − κ dx =
0
ζ˜1
ζ 2 − κ1 dζ,
0
ζ˜2
α2 /4
κ − f 2 (x) dx = √
x0
for
κ1 − ζ 2 dζ,
0
α1 /4
for
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κ1 + ζ 2 d ζ
−κ1
κ < 0 (x0 is the unique zero of f 2 (x) − κ in [0, α2 /4]). Thus κ1 and x0 are functions of κ . We easily have κ 1 κ1 = √ + O κ 2 , c 0 = f 1 (0). c0
2
(6.1)
Denoting by
w i (κ1 , ζi ) =
dxi
− 12 ui ,
d ζi
we have
w i = −λ ζi2 − κ1 + ψi (κ1 , ζi ) w i
(i = 1, 2),
(6.2)
where
ψi (κ1 , ζi ) =
dxi
12
d2 dζi2
d ζi
dxi
− 12
d ζi
.
By [8] we know that ψi (κ1 , ζi ) are continuous on a neighborhood of κ1 = 0 and |ζi | ζ˜i . In the subsequent argument we only consider the case i = 1; the case i = 2 is similar by exchanging κ and −κ . Now we summarize various facts on Weber’s (modified) parabolic cylinder function and its application to approximation theory which are needed here. Put
W c (α , x) =
√ π α /4 π i /8 ix2 /4 2e e e Γ ( 12
+ iα
)1/2 |Γ ( 12
+ iα
∞ )|1/2
√
e −x(1−i )t /
2 − 12 +i α −t 2 /2
t
e
dt .
0
Then w = W c satisfies the differential equation
2 x w =− −α w
(6.3)
4
and so are its real and imaginary parts. We have Proposition 11. There is a unique real solution w = W (α , x) of (6.3) such that
W c (α , x) = k(α )−1/2 W (α , x) + ik(α )1/2 W (α , −x),
k(α ) =
e 2π α + 1 − e π α .
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K. Kiyohara / Differential Geometry and its Applications 29 (2011) S125–S134
√ The function W (α , x) is called Weber’s modified parabolic cylinder function. Clearly, the function w (ζ ) = W ( 12 λκ1 , √ ± 2ζ λ1/4 ) satisfies
w = −λ ζ 2 − κ1 w . We need the following result of Olver. Theorem 12. (See [9].) Eq. (6.2) has a solution w (λ, κ1 , ζ ) such that
w (λ, κ1 , ζ ) = W where
1√ 2
√
λκ1 , 2 ζ λ
1/4
+ (λ, κ1 , ζ ),
−1 1 √ √ l ( 2 λκ1 ) | (λ, κ1 , ζ )| 1√ 1/4 E λκ1 , 2 ζ λ exp √ Vζ,ζ˜i ( F ) − 1 , √ √ 2 M ( 12 λκ1 , 2ζ λ1/4 ) 2λ1/4
2
l(b) = sup Ω(x) M (b, x)
ζ˜1
x∈R
Vζ,ζ˜i ( F ) =
,
ζ
|ψ(κ1 , t )| dt , √ Ω( 2t λ1/4 )
and Ω(x) is any continuous function on R, positive on x = 0, and Ω(x)/|x| is bounded at |x| = ∞. Here, the auxiliary functions M (α , x) and E (α , x) are defined as follows:
⎧ 2 −1 2 1/2 ⎪ (x −σ (α )), ⎨ {k(α ) W (α , x) + k(α ) W (α , −x) } 1 / 2 M (α , x) = {2W (α , x) W (α , −x)} (−σ (α ) x σ (α )), ⎪ ⎩ − 1 2 2 1/2 {k(α ) W (α , x) + k(α ) W (α , −x) } (σ (α ) x), ⎧ 1/2 ⎪ (x −σ (α )), ⎨ k(α ) E (α , x) = { W (α , −x)/ W (α , x)}1/2 (−σ (α ) x σ (α )), ⎪ ⎩ k(α )−1/2 (σ (α ) x)
and
σ (α ) is the smallest positive x satisfying k(α )−1/2 W (α , x) = k(α )1/2 W (α , −x).
We put Ω(x) = |x|1/3 . Then as indicated in [8], Sections 9.2 and 10.2, we have
l(b) = O (1) as b → ±∞, and Vζ,ζ˜ ( F ) C λ−1/12 uniformly in ζ for some constant C . Thus, putting i
w c (λ, κ1 , ζ ) = k
−1/2
1√ 2
λκ1
w (λ, κ1 , ζ ) + ik
1√ 2
1/2 λκ1
w (λ, κ1 , −ζ ),
we have
√ √ √ √ w c (λ, κ1 , ζ ) − W c 1 λκ1 , 2ζ λ1/4 = M 1 λκ1 , 2ζ λ1/4 O λ−1/3 . 2 2
Define phase functions Θ and θ by
W c (α , x) = W c (α , x) e i Θ(α ,x) ,
−1/2
Θ(α , 0) = Θ0 (α ) = arg k(α )
w c (λ, κ1 , ζ ) = w c (λ, κ1 , ζ ) e i θ (λ,κ1 ,ζ ) , 1/2
+ ik(α )
∈ (0, π /4),
θ(λ, κ1 , 0) = Θ0
1√ 2
λκ1 .
Note that 2Θ0 (α ) = arctan(e −π α ). Since | W c | M, we therefore obtain
θ(λ, κ1 , ζ ) = Θ
1√ 2
√
λκ1 , 2 ζ λ
1/4
+ O λ−1/3 .
(6.4)
K. Kiyohara / Differential Geometry and its Applications 29 (2011) S125–S134
S133
By [8], Section 8.2, we have
1
π
4
4
Θ(α , x) = x2 − α log x +
+
1
arg Γ
2
1 2
1 + α2 + iα + O 2
(6.5)
x
as |α /x| → 0. Also, we have
ζ
1
t 2 − κ1 dt =
√
2
ζ2 −
κ1 4
−
κ1 2
log √
log 2 ζ − κ1 + O κ12 , 2 |κ 1 |
(6.6)
κ˜ 1
where κ˜ 1 = max{0, κ1 }. By (6.4), (6.5), and (6.6), we thus have
√ λ t 2 − κ1 dt + √ ζ
θ(λ, κ1 , ζ ) =
κ˜ 1
+
π 4
+
1 2
arg Γ
1 2
√
λκ1
log √
λ|κ1 |
4
√
+
2e
i λκ1
+O
2
√
λκ12 + O λ−1/3 .
(6.7)
We put
w + (λ, κ1 , ζ ) = k(α )1/2 − ik(α )−1/2 w c (λ, κ1 , ζ ) , w − (λ, κ1 , ζ ) = k(α )−1/2 − ik(α )1/2 w c (λ, κ1 , ζ ) ,
α=
1√ 2
λκ1 .
Then w + and w − are even and odd functions in ζ respectively, which satisfy w + (λ, κ1 , 0) > 0 and w − (λ, κ1 , 0) > 0. Thus the functions w + and w − are written as the product of positive functions and cos θ+ and sin θ− respectively, where
π
1√
+ Θ0 λκ1 + θ(λ, κ1 , ζ ), 2
1√ λκ1 + θ(λ, κ1 , ζ ). θ− (λ, κ1 , ζ ) = −Θ0
θ+ (λ, κ1 , ζ ) = −
2
(6.8)
2
Now let u 1,± (λ, κ , x) be even and odd periodic solution of Eq. (3.1) which satisfies (3.4). We may assume u 1,± are transformed to w ± in the way described at the beginning of this section. Take a point x0 near the midpoint of x0 and α1 /4 so that x0 is the p -th zero of u 1,+ (λ, κ , x), x > 0. Let ζ˜1 be the corresponding value of ζ to x0 . Then we have
π θ+ λ, κ1 , ζ˜1 = + p − 1 π . 2
Then combined with (6.7), (6.8), and the equality ˜
x0
f 1 (x) − κ dx =
ζ1
√
x˜ 0
x˜ 0 = max{0, x0 },
ζ 2 − κ1 dζ,
κ˜ 1
we have
x0
√ 1 π λ f 1 (x) − κ dx + D (α , 1) = p − π + + O,
2
x˜ 0
8
α=
1√ 2
λκ1 ,
where O -term is given as
O
√
λκ12 + O λ−1/3 .
Using the usual semi-classical approximation on the interval [x0 , α1 /2 − x0 ], we have
√
α1/2−x0
λ x0
f 1 (x) − κ dx = π p + 1 + O λ−1/2 ,
(6.9)
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K. Kiyohara / Differential Geometry and its Applications 29 (2011) S125–S134
where p is the number of zeros of u 1,+ in the open interval (x0 , α1 /2 − x0 ). Since 2p + p = p, we therefore obtain
√
λ A 1 (κ ) + D
√ λκ1 2
π 1 + O. ,1 = p+ 2
4
By the similar argument for the odd solution u 1,− , we also have
√
λ A 1 (κ ) + D
√ λκ1 2
π 3 + O. , −1 = p+ 2
4
The case of Eq. (3.2) is similar, and we have
√
λ A 2 (κ ) − D
√ λκ1 2
π 1 +O , = q+ − 2
2
4
( = ±1).
In view of (6.1) we have
D
√ λκ1 2
√ √ √ √ λκ , = D √ , + O λκ 2 + O λκ 2 log( λκ ) .
2 c0
Summarizing those results, we obtain the following theorem. Theorem 13. λ = λ( p , q, ) and κ = κ ( p , q, ) satisfy the following formulas
√
λκ π 1 + O, p+ − √ , = 2 c0 2 2 4
√
√ λκ π 1 + O, λ A 2 (κ ) − D √ , = q+ −
√
λ A 1 (κ ) + D
2 c0
2
2
4
where = ±1 and
O=O
√
√ √ λκ 2 + O λκ 2 log( λκ ) + O λ−1/3 .
Concerning the O -term, we note the following three cases:
√
λκ = O (1) ⇒ O = O λ−1/3 , √ √ ⇒ O = O λ−1/6 log λ , λκ = O λ1/6 √ √ ⇒ O = O λ−1/3 log λ . λκ = O λ1/12 Using the first case, we obtain Theorem 9. Using the other two cases, we have similar results. For the connection with the region where the usual semi-classical application is effective, the second case would be preferable. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
Y. Colin de Verdière, B. Parisse, Singular Bohr–Sommerfeld rules, Comm. Math. Phys. 205 (1999) 459–500. M. Igarashi, K. Kiyohara, K. Sugahara, Noncompact Liouville surfaces, J. Math. Soc. Japan 45 (1993) 459–479. K. Kiyohara, Compact Liouville surfaces, J. Math. Soc. Japan 43 (1991) 555–591. K. Kiyohara, Two classes of Riemannian manifolds whose geodesic flows are integrable, Mem. Amer. Math. Soc. 130 (619) (1997). K. Kiyohara, Semi-classical approximations on Liouville surfaces, Suuriken Koukyuuroku 1119 (1999) 35–47 (in Japanese). D.V. Kosygin, A.A. Minasov, Ya.G. Sinai, Statistical properties of Laplace–Beltrami operators on Liouville surfaces, Uspekhi Mat. Nauk 48 (1993) 3–130 (English transl. Russian Math. Surveys 48 (1993) 1–142). F.W.J. Olver, Asymptotics and Special Functions, Academic Press, 1974. F.W.J. Olver, Second order linear differential equations with two turning points, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 278 (1975) 137–174. F.W.J. Olver, Improved error bounds for second-order differential equations with two turning points, J. Res. Nat. Bur. Stand. B 80 (1976) 437–440. A. Weinstein, On Maslov’s quantization condition, in: Lecture Notes in Math., vol. 459, Springer-Verlag, Berlin, New York, 1975, pp. 341–372.