Global SL(2,R) ˜ representations of the Schrödinger equation with time-dependent potentials
aa r X i v : . [ m a t h . R T ] A p r GLOBAL ^ SL (2 , R ) REPRESENTATIONS OF THE SCHR ¨ODINGEREQUATION WITH TIME-DEPENDENT POTENTIALS
JOSE FRANCODEPARTMENT OF MATHEMATICSBAYLOR UNIVERSITYONE BEAR PLACE
Abstract.
We study the representation theory of the solution space of theone-dimensional Schr¨odinger equation with time-dependent potentials thatposses sl -symmetry. We give explicit local intertwining maps to multiplierrepresentations and show that the study of the solution space for potentials ofthe form V ( t, x ) = g ( t ) x + g ( t ) x + g ( t ) reduces to the study of the potentialfree case. We also show that the study of the time-dependent potentials of theform V ( t, x ) = λx − + g ( t ) x + g ( t ) reduces to the study of the potential V ( t, x ) = λx − . Therefore, we study the representation theory associated tosolutions of the Schr¨odinger equation with this potential. The subspace of so-lutions for which the action globalizes is constructed via nonstandard inductionoutside the semisimple category. Introduction
In the early seventies, the original prolongation algorithm of Sophus Lie wasused to classify the various time-independent potentials of the one-dimensionalSchr¨odinger equation, 2 iu t + u xx = 2 V ( t, x ) u that admit non-trivial, inequivalent Lie symmetries (c.f. [2], [6]). As local Lie grouprepresentations, it turns out that the representation theory associated to the spaceof solutions of the Schr¨odinger equation with the following potentials, V ( x ) = λ (1.1a) V ( x ) = λx (1.1b) V ( x ) = λx , (1.1c)where λ ∈ R is an arbitrary constant, reduces to the potential free case. Here,the symmetry algebra is isomorphic to g := sl (2 , R ) ⋉ h ( R ). More generally, it isknown that the time-dependent potential V ( t, x ) = g ( t ) x + g ( t ) x + g ( t ) has thesame symmetry Lie algebra g (c.f. [9]). Mathematics Subject Classification.
Key words and phrases.
Schr¨odinger equation, time-dependent potentials, Lie theory, repre-sentation theory, globalizations.
However, for the following potentials, V ( x ) = λx − (1.2a) V ( x ) = λ x + λ x − , (1.2b)where λ, λ i ∈ R are arbitrary constants, we will show that the problem will reduceto a study of an eigenvalue problem for, essentially, a Casimir element for sl (2 , R ).For these cases, the symmetry Lie algebra is isomorphic to sl (2 , R ) × R . Similarly,Truax showed that the time-dependent potential V ( t, x ) = λx − + g ( t ) x + g ( t )has the same symmetry Lie algebra.It is natural to use representation theory to study the solution space of thesedifferential operators. However, since the resulting actions are not global, the tech-niques of representation theory do not always apply. However, it is sometimespossible to look at special subspaces of solution that carry the structure of a globalrepresentation (c.f. [4], [8], [7]). For instance, in 2005, M. Sepanski and R. Stankedecomposed the solution space for the 1-dimensional potential free Schr¨odingerequation and studied it as a global Lie group representation in [7]. Recently, theyanalyzed the n -dimensional case for the potential free Schr¨odinger equation (c.f.[8]).In this paper, we give explicit local intertwining maps to multiplier represen-tations, showing that the study of the solution space for the potentials of theform V ( t, x ) = λx − + g ( t ) x + g ( t ), which includes (1.2b), reduces to thestudy of the potential (1.2a). For the sake of completeness, we show that thestudy of potentials (1.1) and the time-dependent potentials of the form V ( t, x ) = g ( t ) x + g ( t ) x + g ( t ), reduces to the study of the potential free case. Therefore,we study the representation theory associated to solutions of the Schr¨odinger equa-tion with potentials (1.2a). As in [8] and [7], the subspace of solutions for which theaction globalizes is constructed via nonstandard induction outside the semisimplecategory.A bit more precisely, we start with a parabolic-like subgroup P of the group G := ^ SL (2 , R ) ⋉ H where ^ SL (2 , R ) is the two-fold cover of SL (2 , R ) and H isthe three-dimensional Heisenberg group. Then, we look at the smoothly inducedrepresentation I ( q, r, s ) = Ind GP ( χ q,r,s )where χ q,r,s is a character on P with parameters r, s ∈ C and q ∈ Z (see Equation(2.1)).We show that in the non-compact version of I ( q, r, s ) for r = − / s = i/ f = (cid:18) r ( r + 2)2 + λ (cid:19) f where Ω is the Casimir element of sl (2 , R ). We show that the solution space in I ( q, r, s ) is non-empty iff λ = l ( l − / l ∈ Z ≥ .To each l ∈ Z ≥ we assign the triangular number λ = l ( l − /
2. This relationis one-to-one except for l = 0 and l = 1. Under this identification, we denote by H l the space of K -finite vectors of the solution space in I ( q, r, s ) of (1.3) for l ≥ λ = 0 we decompose the space of K -finite vectors of the solution space intotwo sl -invariant subspaces H ⊕ H (see Theorem 4). LOBAL ^ SL (2 , R ) REPRESENTATIONS OF THE SCHR ¨ODINGER EQUATION 3 We determine the structure of H l as an sl (2 , R )-module and show: Theorem 1.
Assume λ = l ( l − / for some l ∈ Z ≥ . If q = 0 or q = 2 then H l is irreducible as an sl -module. If q = 1 (respectively, q = 3 ), then H l has a uniquelowest (respectively, highest) weight submodule, H − l (respectively, H + l ). While the action of the Heisenberg group does not preserve H l we show that itdoes preserve the direct sum H := L ∞ l =0 H l . Also, write H ± for the direct sum ofhighest and lowest weight submodules with l ≥
2, whenever they exist. We prove:
Theorem 2. As g -modules: (1) If q = 0 or q = 2 , the composition series of H is ⊂ H ⊕ H ⊂ H. (2) If q = 1 , then the composition series of H is as follows ⊂ H − ⊕ H − ⊂ H ⊕ H ⊂ H ⊕ H ⊕ H − ⊂ H. (3) If q = 3 , then then the composition series of H is ⊂ H +0 ⊕ H +1 ⊂ H ⊕ H ⊂ H ⊕ H ⊕ H + ⊂ H. Notation
The Group.
Let G = SL (2 , R ) and let H denote the three dimensionalHeisenberg group with product,( v , v , v )( v ′ , v ′ , v ′ ) = ( v + v ′ , v + v ′ , v + v ′ + v v ′ − v v ′ ) . Following the realization of the two-fold cover of G in [5], define the complexupper half plane D := { z ∈ C | Im z > } and let G act on D by fractional lineartransformations, that is, if g = (cid:0) a bc d (cid:1) ∈ G and z ∈ D , then g.z = az + bcz + d . Define d : G × D → C by d ( g, z ) := cz + d . Then there are exactly two smoothsquare roots of d ( g, z ) for each g ∈ G and z ∈ D . The double cover can be realizedas: f G = { ( g, ǫ ) | g ∈ SL (2 , R ) and smooth ǫ : D → C such that ǫ ( z ) = d ( g, z ) for z ∈ D (cid:9) with the product defined by( g , ǫ ( z ))( g , ǫ ( z )) = ( g g , ǫ ( g .z ) ǫ ( z )) . Finally, the symmetry group that we are interested in, is G := f G ⋉ H . Here e G projects to G and acts on H by the standard action on the first two coordinatesand leaves the third fixed. JOSE FRANCO
Parabolic Subgroup and Induced Representations.
As in [8], we con-sider the parabolic subalgebra of lower triangular matrices q ⊂ sl (2 , R ) with Lang-lands decomposition m ⊕ a ⊕ n . If exp e G : sl (2 , R ) → f G denotes the exponentialmap then: A := exp f G ( a ) = { ( (cid:0) t t − (cid:1) , z e − t/ ) | t ∈ R ≥ } N := exp f G ( n ) = { (( t ) , z | t ∈ R } N := exp f G ( n ) = { (( t ) , z
7→ √ tz + 1) | t ∈ R } . Let k := { (cid:0) θ − θ (cid:1) : θ ∈ R } then K := exp f G ( k ) = { ( (cid:0) cos θ sin θ − sin θ cos θ (cid:1) , z
7→ √ cos θ − z sin θ ) | θ ∈ R } where √· denotes the principal square root in C . Writing M for the centralizer of A in K then M = { m j := ( (cid:0) − − (cid:1) j , z → i − j ) | j = 0 , , , } . Let W ⊂ H be given by W = { (0 , v, w ) | v, w ∈ R } ∼ = R and let X := { ( x, , | x ∈ R } . Let us write w for the Lie algebra of W . Then P = M AN ⋉ W is the analogueof a parabolic subgroup in G corresponding to p := q ⋉ w .For later use, we notice that an element in g = (cid:2) ( (cid:0) a bc d (cid:1) , z ǫ ( z )) , ( u, v, w ) (cid:3) ∈ G is in the image of the mapping P × ( N × X ) → G given by ( p, n ) pn , if a = 0.This induces a decomposition of such g into its P and N × X components, (cid:2) ( (cid:0) a bc d (cid:1) , z ǫ ( z )) , ( u, v, w ) (cid:3) = (cid:2) ( (cid:0) a c a − (cid:1) , z ǫ ( z + b/a )) , (0 , v, w + ( u + bv/a ) v ) (cid:3) · (cid:2) ( (cid:0) b/a (cid:1) , z , ( u + bv/a, , (cid:3) . On the open dense set where a = 0 let p : G → P and n : G → N × X be theprojections from the previous decomposition.It is well known that the character group on A is isomorphic to the additivegroup C so any character on A can be indexed by a constant r ∈ C and defined by χ r (cid:0) ( (cid:0) t t − (cid:1) , z e − t/ ) (cid:1) = t r for t >
0. A character on M is parametrized by q ∈ Z and defined by χ q ( m j ) = i jq .A character on W can be parametrized by s ∈ C and defined by, χ s (cid:0) (0 , v, w ) (cid:1) = e sw . Finally, any character on P that is trivial on N is parametrized by a triplet ( q, r, s )where s, r ∈ C and q ∈ Z and defined by(2.1) χ q,r,s (cid:16) (( − j (cid:0) a c a − (cid:1) , z i − j e − a/ √ acz + 1) , (0 , v, w ) (cid:17) = i jq | a | r e sw . The representation space induced by χ q,r,s will be denoted by I ( q, r, s ) and definedby I ( q, r, s ) := { φ : G → C | φ ∈ C ∞ and φ ( gp ) = χ − q,r,s ( p ) φ ( g ) for g ∈ G, p ∈ P } the G -action on I ( q, r, s ) is given by ( g .φ )( g ) = φ ( g − g ). LOBAL ^ SL (2 , R ) REPRESENTATIONS OF THE SCHR ¨ODINGER EQUATION 5 Non-compact Picture.
Since H = XW then G = ( N × X ) P a.e. with N × X isomorphic to R via ( t, x ) N t,x := [(( t ) , z , ( x, , N , thisrestriction induces an injection of I ( q, r, s ) into C ∞ ( R ) which is identified as I ′ ( q, r, s ) = { f ∈ C ∞ ( R ) | f ( t, x ) = φ ( N t,x ) for some φ ∈ I ( q, r, s ) } . This space is endowed with the corresponding action so that the map φ f where f ( t, x ) = φ ( N t,x ), becomes intertwining. Thus I ( q, r, s ) ∼ = I ′ ( q, r, s ) as G -modules. As in the semisimple case, we will call this the non-compact picture. Proposition 1.
Let f ∈ I ′ ( q, r, s ) , ( g, ǫ ) ∈ f G , and ( u, v, w ) ∈ H . Then, (( g, ǫ ) .f )( t, x ) = ( a − ct ) r − q/ ǫ ( g − . ( t + z )) e − scx a − ct f (cid:18) dt − ba − ct , xa − ct (cid:19) (2.2a) (( u, v, w ) .f )( t, x ) = e − s ( uv − vx − tv + w ) f ( t, x − u − tv ) . (2.2b) Proof.
This result is proved in a more general setting in [8]. (cid:3)
Corollary 1.
The action of (cid:0) a bc − a (cid:1) ∈ sl (2 , R ) on I ′ ( q, r, s ) is given by the differ-ential operator (2.3) ( ct − a ) x∂ x + ( ct − at − b ) ∂ t + ( ra − csx − rct ) . An element ( u, v, w ) ∈ h acts on I ′ ( q, r, s ) by the differential operator ( tv − u ) ∂ x + s ( w − vx ) . Proof.
It follows from differentiating the group actions on I ′ ( q, r, s ). (cid:3) Casimir Elements.
Write (cid:3) = 2 i∂ t + ∂ x for the potential free Schr¨odinger operator. By equation (2.3), the standard sl -triple { h, e ± } acts by h = − x∂ x − t∂ t + r (2.4) e + = − ∂ t (2.5) e − = tx∂ x + t ∂ t − ( sx + rt )(2.6)on the non-compact picture. LetΩ = 1 / h − h + 2 e + e − be the Casimir element in the enveloping algebra of sl (2 , R ) and defineΩ ′ = 2Ω − r ( r + 2) . Corollary 2. On I ′ ( q, r, s ) , Ω acts by Ω = 12 (cid:0) sx ∂ t + x ∂ x − (1 + 2 r ) x∂ x + r ( r + 2) (cid:1) . In particular, for r = − / and s = i/ , Ω ′ acts by Ω ′ − λ = x ( (cid:3) − λ/x ) so that, ker(Ω ′ − λ ) = ker( (cid:3) − λ/x ) . JOSE FRANCO
Proof.
A straightforward calculation using the actions of the standard sl -tripleand the definition of the Casimir element gives the desired result. (cid:3) As a consequence of Corollary 2, we are interested in the study of ker(Ω ′ − λ ).We begin with a result on the invariance under f G and the subgroup { (0 , , w ) | w ∈ R } ⊂ H . Proposition 2.
The subspace ker(Ω ′ − λ ) in I ′ ( q, r, s ) is invariant under theaction of f G and under the action of the subgroup { (0 , , w ) | w ∈ R } of H . Thespace is not left invariant by the complement of { (0 , , w ) | w ∈ R } in H .Proof. Since Ω is in the center of the enveloping algebra of g , the f G -invariance isclear. Let ( u, v, w ) ∈ H . Using the action on Corollary 1 we can calculate[ (cid:3) − λ/x , ( u, v, w )] = [2 i∂ t + ∂ x − λ/x , ( tv − u ) ∂ x + i/ w − vx )]= [2 i∂ t , ( tv − u ) ∂ x ] + [ ∂ x , i/ w − vx )] − λ/x , ( tv − u ) ∂ x ]= 2 iv∂ x − iv∂ x + 4 λ ( tv − u ) x = 4 λ ( tv − u ) x . (cid:3) Though, all of H does not leave ker(Ω ′ − λ ) invariant, it will play an importantrole in linking together different e G -invariant kernels.3. Multiplier Representations
In this section, we consider the potential V ( t, x ) = g ( t ) x + g ( t ) x + g ( t ). Weconstruct local intertwining maps from the non-compact picture, I ′ ( q, r, s ) to amultiplier representation. This will allow us to reduce the study to the potentialfree case.It was shown in [9] that the only time-dependent potentials having full sl -symmetry are potentials of the form V ( t, x ) = g ( t ) x + g ( t ) x + g ( t ) + λ/x with λ · g ( t ) = 0. If λ = 0, the symmetry Lie algebra is isomorphic to g = sl ⋉ h andit is isomorphic to sl (2 , R ) × R otherwise.When λ = 0 we show that there exists a local intertwining isomorphism betweenthe solution space of the potential free Schr¨odinger equation and the solution spaceof the Schr¨odinger equation with this time-dependent potential in I ′ ( q, r, s ). Noticethat the cases where g i ( t ) = λ and g j ( t ) = 0 for i = j are the time-independentpotentials listed in (1.1).When λ = 0 we show that, in the same multiplier representation, the studyreduces to the eigenvalue problem that is the main concern of this paper. There-fore, this completes the study of all non-trivial time-dependent and independentpotentials having at least the sl -symmetry.To write down the algebra generators explicitly, fix two real, nontrivial, linearlyindependent solutions of b ′′ + 2 g b = 0. These solutions, χ and χ , are normalizedso that W ( χ , χ ) = 1. With χ and χ , we define ϕ j = χ j for j ∈ { , } and ϕ = 2 χ χ . Let C j ( t ) = R t χ j g for j ∈ { , } , A l = − χ l C l LOBAL ^ SL (2 , R ) REPRESENTATIONS OF THE SCHR ¨ODINGER EQUATION 7 for l ∈ { , } , and A = − ( χ C + χ C ) . It was shown in [9] that the differential operators:(3.1) L j = ( − j +1 ( ϕ j ∂ t + (cid:0) ϕ ′ j x + A j (cid:1) ∂ x + B j )for 1 ≤ j ≤
3, generate an algebra isomorphic to sl (2 , R ), where B j = − iϕ ′′ j x − i A ′ j x + 14 ϕ ′ j + ig ϕ j + i D j , D l = − C l for l ∈ { , } , and D = −C C . The following bracket relations hold:[ L , L ] = − L , [ L , L ] = 2 L , and [ L , L ] = L . Note that, in order to have astandard sl -triple, our definition of L differs in sign with respect to the definitionin [9].In order to define the appropriate multiplier representation space we start bydefining a change of variables γ : R → R by γ ( t, x ) := (cid:18)Z t χ , χ ( t ) x + Z t C χ (cid:19) . In the following, we assume that the required integrability conditions are satisfied.We define ν : N × X → C by ν ( N γ ( t,x ) ) = e R t B u ) χ u ) + (cid:0) χ u ) ( ϕ ′ ( u ) x + A ( u )) (cid:1) du and extend it to a map on an open dense subset of G by ν ( g ) = ν ( n ( g )) . Let f ∈ I ′ ( q, r, s ) and define the map f ˜ f by(3.2) ˜ f ( t, x ) = e R t B u ) χ u ) + (cid:0) χ u ) ( ϕ ′ ( u ) x + A ( u )) (cid:1) du f ( γ ( t, x )) . The space I ′ ( q, r, s ) µ is defined as the image of I ′ ( q, r, s ) under this map (the reasonfor the subscript µ will become evident below). This space is provided with thestructure of a G -module that makes the map intertwining.Next we construct the multiplier representation. We start by defining the mul-tiplier µ ( g , g ) = ν ( g − g ) ν ( g − ) − . For φ ∈ I ( q, r, s ) define, on an open dense set of G/P , the map˜ φ ( gP ) = µ ( g − , I ) − φ ( g ) . We define I ( q, r, s ) µ as the image of I ( q, r, s ) under the map φ → ˜ φ .Finally, the intertwining map from I ( q, r, s ) µ to I ′ ( q, r, s ) µ is given by˜ φ ˜ f whenever ˜ f ( t, x ) = ˜ φ ( N γ ( t,x ) P ) . Group action on I ′ ( q, r, s ) µ . In this section we calculate the local action of G on I ′ ( q, r, s ) µ and we show that the study of the solution space for this generalpotential reduces to the study of the kernel of the differential operator Ω ′ as in thepotential free case. For notational convenience, define ρ ( t, x ) = ν ( N γ ( t,x ) ) . JOSE FRANCO
Proposition 3.
Fix g = (cid:0) a bc d (cid:1) ∈ G and let ( g, ǫ ) ∈ f G . Define Θ( t ) = R t χ and Ξ( t, x ) = χ ( t ) x + R t C χ . Then (3.3) (( g, ǫ ) . ˜ f )( t, x ) = ρ ( t, x ) ρ ◦ γ − (cid:16) d Θ − ba − c Θ , Ξ a − c Θ (cid:17) ( a − c Θ) r − q/ ǫ ( g − . (Θ + z )) e − sc Ξ2 a − c Θ ˜ f ◦ γ − (cid:18) d Θ − ba − c Θ , Ξ a − c Θ (cid:19) . For ( u, v, w ) ∈ H ( R )(3.4) (( u, v, w ) . ˜ f )( t, x ) = ρ ( t, x ) ρ ◦ γ − (Θ , Ξ − u − v Θ) e − s ( uv − v Ξ − v Θ+ w ) ˜ f ◦ γ − (Θ , Ξ − u − v Θ) . Proof.
This is a straightforward calculation. It follows directly from using theisomorphism determined by Equation (3.2) on the actions computed in Proposition1. (cid:3)
Differentiating these actions, we recover the generators of the algebra of symme-try operators found by Truax in [9]. We start with some useful calculations
Lemma 1.
The functions χ ( t ) and χ ( t ) satisfy (1) χ = χ Z t χ , (2) A = (cid:18)Z t χ (cid:19) A − χ (cid:18)Z t χ (cid:19) Z t C χ , (3) ϕ ′ = (cid:18)Z t χ (cid:19) ϕ ′ + 2 Z t χ . Proof.
This lemma follows from the definitions together with the fact that χ χ ′ − χ χ ′ = 1 and the chain rule. (cid:3) Corollary 3.
For r = − / and s = i/ the standard sl -basis { h, e + , e − } actson I ′ ( q, r, s ) µ by the differential operators { L , L , L } respectively. An element ( u, v, w ) ∈ H ( R ) acts on the same space, by ( uχ − vχ ) ∂ x − i ( uχ ′ + vχ ′ ) x + i ( u C + v C − sw ) Proof.
All these calculations are similar. We only provide the details for the actionof e − since it is the most involved. Let γ − ( t, x ) = (Θ − ( t ) , Ψ( t, x )). Using (3.3),we obtain((( c ) , ǫ ) . ˜ f )( t, x ) = ρ ( t, x ) ρ (cid:16) Θ − (cid:16) Θ1 − c Θ (cid:17) , Ψ (cid:16) Ξ1 − c Θ (cid:17)(cid:17) (1 − c Θ) r − q/ ǫ ( g − . (Θ + z )) e − sc Ξ21 − c Θ ˜ f (cid:18) Θ − (cid:18) Θ1 − c Θ (cid:19) , Ψ (cid:18) Ξ1 − c Θ (cid:19)(cid:19) . LOBAL ^ SL (2 , R ) REPRESENTATIONS OF THE SCHR ¨ODINGER EQUATION 9 We next take ddc (cid:12)(cid:12) c =0 to obtain the action of e − . For the coefficient of ∂ t we obtain − Θ ( t ) ∂ Θ − dt (cid:12)(cid:12)(cid:12)(cid:12) t =Θ = χ ( t ) (cid:18)Z t χ (cid:19) = χ ( t ) = ϕ ( t ) . The first equality above follows from differentiating, the second equality fromLemma 1, and the third equality from the definition of ϕ .For the coefficient of ∂ x we obtainΘ( t )Ξ( t, x ) ∂ Ψ ∂x + Θ( t ) ∂ Ψ ∂t = Θ( t )Ξ( t, x ) χ + Θ( t ) ( 12 ϕ ′ x + A ) = x Θ + 12 x ( ϕ ′ − A = 12 ϕ ′ x + A . The first equality above follows from differentiating, the second from computingthe partial derivatives of the inverse function, and the third by Lemma 1.Lastly, we compute the multiplication term. To this end, we start by computingthe following expression B Θ − i − r Θ = B + ix ϕ ′ Θ χ + ix (Θ C χ − χ ′ Θ Z t C χ ) − i Θ C Z t C χ . Since the multiplication term is given by Θ B − i Ξ − r Θ + ρ ( t,x ) ∂ρ∂x ∂ Ψ ∂x Ξ( t, x )Θ =Θ B + i Θ( − χ ′ x + C )(1 /χ x + R C /χ ) = B , the operator corresponding to e − is L . (cid:3) Using this action of the sl -triple, a straightforward but long calculation givesthe following corollary. Corollary 4.
For the parameters r = − / and s = i/ , the Casimir element actson I ( q, r, s ) µ by Ω = 12 (cid:2) ( x − χ C + χ C ) ( (cid:3) − g ( t ) x + g ( t ) x + g ( t ))) − / (cid:3) . In particular, ker Ω ′ = ker (cid:0) (cid:3) − g ( t ) x + g ( t ) x + g ( t )) (cid:1) in I ′ ( q, r, s ) µ .If λ = 0 then g ( t ) ≡ and Ω ′ acts by Ω ′ = x ( (cid:3) − g ( t ) x + g ( t ))) . Thus ker(Ω ′ − λ ) = ker (cid:0) (cid:3) − g ( t ) x + g ( t ) + λ/x (cid:1) in I ′ ( q, r, s ) µ . This shows that when λ = 0, at least locally, the time-dependent cases analyzedreduce to the potential free case. It also shows that when λ = 0 the study reducesto the study of the eigenvalue problem of Ω ′ . (See Corollary 2.)4. The Compact Picture and K -types The group f G has Iwasawa decomposition f G = KAN and the product inducesa diffeomorphism G ∼ = ( K × X ) × ( AN ⋉ W ). Since ( AN ⋉ W ) ⊂ P , an element φ ∈ I ( q, r, s ) is completely determined by its restriction to K × X . Moreover, since( K × X ) ∩ P = M we have that the restriction of φ ∈ I ( q, r, s ) (which we will stilldenote by φ ) satisfies φ ( gm ) = χ q,r,s ( m ) − φ ( g ) for g ∈ K × X and m ∈ M .There exists an isomorphism K × X ∼ = S × R given by ( θ, y ) [( g θ , ǫ θ ) , ( y, , π -periodic with respect to θ . Thus we can identify φ ∈ I ( q, r, s ) with a map F : S × R → C , φ F iff φ ([( g θ , ǫ θ ) , ( y, , F ( θ, y ). Then F ∈ C ∞ ( S × R ) and F ( θ + 4 π, y ) = F ( θ, y ).The function F inherits from φ additional “parity” identities. By the definition, ǫ θ + πj ( z ) = cos( θ + πj ) − z sin( θ + πj ) = ( − j ǫ θ ( z ). We then get F ( θ + πj, ( − j y ) = φ ([( g θ + πj , ǫ θ + πj ) , (( − j y, , φ ([( g θ , ǫ θ ) , ( y, , m j )= χ q,r,s ( m j ) − φ ([( g θ , ǫ θ ) , ( y, , i − jq F ( θ, y ) . Define(4.1) I ′′ ( q, r, s ) = { F ∈ C ∞ ( R ) | F ( θ + jπ, ( − j y ) = i − jq F ( θ, y ) } . Then the map φ F is a vector space isomorphism between I ( q, r, s ) and I ′′ ( q, r, s ).The space I ′′ ( q, r, s ) inherits a unique G -module structure, so that this map becomesan intertwining map. We call this the compact picture, as in the semisimple case,though K × X is not compact here.In turn, the isomorphism T induces an isomorphism between I ′ ( q, r, s ) and I ′′ ( q, r, s ) which we will write out explicitly. We begin with the following decom-position:[( g θ , ǫ θ ) , ( y, , θ ) , z , ( y sec θ, , · [( (cid:16) / cos θ − sin θ cos θ (cid:17) , ǫ θ ) , (0 , − y tan θ, y tan θ )] . Since F ( θ, y ) = φ ([( g θ , ǫ θ ) , ( y, , F ( θ, y ) = χ q,r,s ([( (cid:16) / cos θ − sin θ cos θ (cid:17) , ǫ θ ) , (0 , − y tan θ, y tan θ )]) − · φ ([(( θ ) , z , ( y sec θ, , θ ) − r e − sy tan θ f (tan θ, y sec θ )for f ∈ I ( q, r, s ) and θ ∈ ( − π/ , π/ F ∈ I ′′ ( q, r, s ), this expressioncan be extended smoothly to any θ ∈ R by using the fact that F ( θ + jπ, y ) = i − jq F ( θ, ( − j y ) and continuity to get to the integer multiples of π/
2. Then wedefine the isomorphism T : I ′ ( q, r, s ) → I ′′ ( q, r, s ) by T ( f ) = F . The inverse to thismap can be calculated and it is:(4.3) f ( t, x ) = (1 + t ) − r/ e stx t F (arctan t, x (1 + t ) − / ) . Under this isomorphism, via the chain rule, we obtain ∂ t ↔
12 ( − y sin 2 θ∂ y + cos θ∂ θ + 2 sy cos 2 θ − / r sin 2 θ )(4.4a) ∂ x ↔ sy sin θ + cos θ∂ y . (4.4b)This will enable us to transfer the actions of the algebra from the non-compactpicture, I ′ ( q, r, s ), to the compact picture, I ′′ ( q, r, s ).Define a standard basis of sl ( C ) given by κ = i ( e − − e + )and η ± = 1 / h ± i ( e + + e − )) . LOBAL ^ SL (2 , R ) REPRESENTATIONS OF THE SCHR ¨ODINGER EQUATION 11 Applying equations (4.4) to the action in Corollary 1, it can be shown that the sl -triple just defined acts on I ′′ ( q, r, s ) by the differential operators κ = i∂ θ (4.5) η ± = 12 e ∓ iθ (cid:0) y∂ y ∓ i∂ θ − (1 / ± isy ) (cid:1) . (4.6) Proposition 4. If Ω ′′ denotes the differential operator by which the central element Ω ′ acts on I ′′ ( q, r, s ) then Ω ′′ = y (cid:18) s∂ θ + 4 s y + ∂ y + 1 + 2 ry ∂ y (cid:19) . Proof.
Under the isomorphism I ′ ( q, r, s ) ∼ = I ′′ ( q, r, s ) we obtain the following ex-pressions:4 sx ∂ t y ( − sy tan θ∂ y + 4 s∂ θ + 8 s y − s y sec θ + 2 sr tan θ ) x ∂ x y (4 s y tan θ + ∂ y + 4 sy tan θ∂ y + 2 s tan θ )(1 + 2 r ) x∂ x (1 + 2 r ) y (2 s tan θ + 1 /y∂ y ) . Adding them we get 1 y Ω ′′ = 4 s∂ θ + 4 s y + ∂ y + 1 + 2 ry ∂ y . (cid:3) Lemma 2.
Let ( g θ ′ , ǫ θ ′ ) ∈ K and F ∈ I ′′ ( q, r, s ) then ( g θ ′ , ǫ θ ′ ) .F ( θ, y ) = F ( θ − θ ′ , y ) Proof. ( g θ ′ , ǫ θ ′ ) .F ( θ, y ) = φ ([( g θ ′ , ǫ θ ′ ) − ( g θ , ǫ θ ) , ( y, , φ ([( g θ − θ ′ , ǫ θ − θ ′ ) , ( y, , F ( θ − θ ′ , y ) (cid:3) There exists an isomorphism K ∼ = S given by ( g θ , ǫ θ ) e iθ/ . Therefore, thecharacters on K are of the form χ Km ( g θ , ǫ θ ) = e − imθ/ for m ∈ Z . Using Lemma2, a weight vector F m ∈ I ′′ ( q, r, s ) of weight m , for the action of K , satisfies( g θ ′ , ǫ θ ′ ) .F m ( θ, y ) = F m ( θ − θ ′ , y ) = e − imθ ′ / F m ( θ, y ). Setting θ = 0 and θ ′ = − θ we obtain F m ( θ, y ) = e − imθ/ F m (0 , y ). Let ˜ F m ( y ) := F m (0 , y ) so that a weightvector is of the form F m ( θ, y ) = e − imθ/ ˜ F m ( y ) . Lemma 3.
Fix m ∈ Z and ˜ F ∈ C ∞ ( R ) . Then F ( θ, y ) = e − imθ/ ˜ F ( y ) is annihilatedby Ω ′′ − λ if and only if ˜ F ( y ) is annihilated by the differential operator D = y ∂ y − (2 λ − my + y ) Proof.
Explicitly calculating the action of Ω ′′ − λ on F ( θ, y ) = e − imθ/ ˜ F ( y ), oneobtains that (Ω ′′ − λ ) F = e − imθ/ D ˜ F . (cid:3)
Proposition 5.
There exist a K -finite vector of weight m in ker(Ω ′′ − λ ) ⊂ I ′′ ( q, r, s ) iff (4.7) l = 12 (1 + √ λ ) is a positive integer (equivalently, λ = l ( l − / for l ∈ Z > ) and m ≡ l + q mod 4 . In this case, if λ = 0 , there exists a unique (up to scalar multiples) weightvector of weight m in ker(Ω ′′ − λ ) ⊂ I ′′ ( q, r, s ) given by (4.8) F m ( θ, y ) = e − imθ/ e − y / y l F (cid:18) l − m , l + 12 , y (cid:19) Proof.
By Lemma 3, for F m to be in ker(Ω ′′ − λ ) ⊂ I ′′ ( q, r, s ), it is necessary that D ˜ F m = 0. Because of the form of D , it respects the decomposition of ˜ F m in termsof its even and odd components. Moreover, each of the components is determinedby its value on R + . Working first with y ≥ F m ( y ) = e − y / H ( y )for some smooth function H . Then, the condition D ˜ F m = 0 is equivalent to(4.9) (cid:0) y ∂ y + (2 y − y ) ∂ y + (( m − y − λ )) H ( y ) = 0 . Following [3], the Frobenius method for this equation yields a solution spanned bytwo linearly independent solutions. The indicial roots for this equation are l = 12 (1 − √ λ )and l = 12 (1 + √ λ ) . Then, the first linearly independent solution is of the form H ( y ) = y l (1 + ∞ X j =1 c j ( l ) y j )for some c j ( l ) ∈ R . This function extends to a smooth function on R only if l ∈ Z ≥ iff λ = 0 or λ is a triangular number (i.e., λ = k ( k − / k ∈ Z > ).If λ = 0, the difference between the indicial roots is an odd integer (i.e., √ λ ),and the second solution is of the form(4.10) H ( y ) = aH ( y ) ln | y | + y l (1 + ∞ X j =1 c j ( l ) y j )for some a, c j ( l ) ∈ R . Since l < H is not continuous at y = 0.Let l = l and write ˜ F m ( y ) = e − y / y l L ( y ). Applying the differential operator D to a function of the form e − y / y l L ( y ) we obtain the differential equation(4.11) 4 y L ′′ ( y ) + 2(1 + 2 l − y ) L ′ ( y ) − (1 + 2 l − m ) L ( y ) = 0 . Recall that the confluent hypergeometric differential equation is( z∂ z + ( b − z ) ∂ z − a ) F ( a, b, z ) = 0(c.f. [1]). This equation has well known solutions in the form of confluent hyperge-ometric functions of the first and second kind. However, the smoothness conditionrequired by being in I ′′ ( q, r, s ) shows that the unique solution to (4.11) corresponds LOBAL ^ SL (2 , R ) REPRESENTATIONS OF THE SCHR ¨ODINGER EQUATION 13 to a multiple of the confluent hypergeometric function of the first kind. We maytherefore take L ( y ) = F (cid:0) l − m , l + , y (cid:1) .Finally, a simple calculation using the required parity condition on elements in I ′′ ( q, r, s ) from Equation (4.1) applied to F m ( θ, y ) reduces to e − imπj/ ( − jl = i − jq which is equivalent to m − l ≡ q mod 4.So far, we have established the theorem for non-negative values of y . Extend ˜ F to R by ˜ F m ( y ) = e − y / y l F (cid:0) l − m , l + , y (cid:1) which is even or odd dependingon the parity of l . Since D ˜ F m ( y ) = 0 for y ≥ D is even, D ˜ F m ( y ) = 0 for y ∈ R . Moreover, F m is manifestly smooth and the unique extension to all R .If λ = 0 then l = 0 or l = 1, which corresponds to the potential free case andagain, it is known that there exists a unique solution for each l . The solutionscorrespond to the even ( l = 0) and the odd ( l = 1) solutions found there. (c.f.,[8]) (cid:3) Notice that we have set up a correspondence between the set of eigenvalues withnon-empty eigenspace in I ′′ ( q, r, s ) and Z ≥ via λ = l ( l − /
2. This correspondencewill be one-to-one (except for λ = 0 where it is two-to-one). For λ = 0 the corre-sponding parameter l can be recovered by l = (1 + √ λ ). For the potentialfree case, λ = 0, we have associated the parameters l = 0 and l = 1.For use in the following section, we record the following properties of the con-gruent hypergeometric function (c.f. [1]) d n dz n F ( a, b, z ) = ( a ) n ( b ) n F ( a + n, b + n, z )(4.12a) b F ( a, b, z ) − b F ( a − , b, z ) − z F ( a, b + 1 , z ) = 0(4.12b) b (1 − b + z ) F ( a, b, z ) + b ( b − F ( a − , b − , z ) − az F ( a + 1 , b + 1 , z ) = 0(4.12c) ( a − z ) F ( a, b, z ) + ( b − a ) F ( a − , b, z )(1 − b ) F ( a, b − , z ) = 0(4.12d) ( a − b + 1) F ( a, b, z ) − a F ( a + 1 , b, z ) + ( b − F ( a, b − , z ) = 0(4.12e) 5. Structure of ker(Ω ′′ − λ ) ⊂ I ′′ ( q, r, s )In this section we will study the structure of ker(Ω ′′ − λ ) K as an sl -module. Proposition 6.
With l = (1 + √ λ ) as in Proposition 5 and m ≡ l + q mod 4 , let Ψ m,l ( θ, y ) = e − imθ/ e − y / y l F (cid:18) l − m , l + 12 , y (cid:19) . The sl -triple { κ, η ± } acts on Ψ m,l by κ. Ψ m,l = m m,l (5.1) η ± . Ψ m,l = − l + 1 ± m m ± ,l (5.2) Lowest weight vectors occur if m ≡ l + 1 mod 4 and the lowest weight vector is ofthe form e − (2 l +1) iθ e − y y l . Highest weight vectors occur if m ≡ − l − and the highest weight vectoris of the form e (2 l +1) iθ e y y l . Proof.
In Equations (4.5) and (4.6), we wrote down the action of the sl -triple { κ, η ± } . The stated action of κ follows by inspection. Directly applying the differ-ential operator η + gives η + . Ψ m,l ( θ, y ) = e − i ( m ± θ/ e − y / y l − − l + m l ) ((1 + 2 l ) · F (cid:16) l − m , l + 12 , y (cid:17) + 2 y F (cid:16) l − m , l + 32 , y (cid:17) ) . Applying (4.12c) with a = l − m and b = l + to the action of η + we obtain η + . Ψ m,l ( θ, y ) = − e − i ( m ± θ/ e − y / y l ((4 l − · F (cid:16) − l − m , l − , y (cid:17) − (3 − l + m ) F (cid:16) l − m , l + 12 , y (cid:17) ) . Using (4.12b) we obtain the desired result. For η − , we similarly apply (4.12b) with a = l − m and b = l + to obtain the desired result.The assertion about the highest and lowest weights follow from the action of η ± as differential operators; since the weight vectors that are annihilated by each ofthese are the ones correspondent to the weights ∓ (2 l + 1) respectively. Directlyevaluating and observing that F ( a, a, z ) = e z and F (0 , b, z ) = 1, the givenexpressions are obtained. (cid:3) Definition 1.
Let H l = ker(Ω ′′ − λ ) K denote the K -finite vectors in ker(Ω ′′ − λ ) ⊂ I ′′ ( q, r, s ) . For k ∈ Z ≥ define (5.3) H k = span C { Ψ m,k : m ≡ k + q mod 4 for m ∈ Z } . For q ≡ and k ∈ Z ≥ define (5.4) H + k = span C { Ψ m,k : m ≥ k + 1 and m ≡ k + 1 mod 4 for m ∈ Z } . For q ≡ − and k ∈ Z ≥ define (5.5) H − k = span { Ψ m,k : m ≤ − (2 k + 1) and m ≡ − (2 k + 1) mod 4 for m ∈ Z } . Lemma 4. If q ≡ , then H + l is the unique irreducible sl -submodule of H l . If q ≡ − , then H − l is the unique irreducible sl -submodule of H l .Proof. From Equation (5.2), the representation is irreducible whenever ± (2 l + 1) = m for any m ≡ l + q mod 4, this occurs when q ∈ Z . We can have 2 l + 1 = m forsome m ≡ l + q mod 4 iff q ≡ q ≡ − H − l (resp. H + l ) is clearly the unique irreducible submodule of H l . (cid:3) LOBAL ^ SL (2 , R ) REPRESENTATIONS OF THE SCHR ¨ODINGER EQUATION 15 Theorem 3.
Given q ∈ Z and l = (1 + √ λ ) , then as sl -modules: (1) If q ≡ or q ≡ then H l = ker(Ω ′′ − λ ) K is irreducible asan sl -module. (2) If q ≡ − , then H − l is he only irreducible submodule and the com-position series for ker(Ω ′′ − λ ) K is given by ⊂ H − l ⊂ H l (3) If q ≡ , then H + l is the only irreducible submodule and the compo-sition series of ker(Ω ′′ − λ ) K is given by sl -submodule ⊂ H + l ⊂ H l Proof.
Follows from Lemma 4. (cid:3) Heisenberg action and connections with other kernels
In this section we will examine the action of the Heisenberg algebra. This willallow us to join all the (non-zero) eigenspaces together in one representation.Recall that the element ( u, v, ∈ h ( R ) acts on I ′ ( q, r, s ) by ( tv − u ) ∂ x − svx so, under the isomorphism (4.2), the elements E ∓ := (1 , ± i, ∈ h ( C )act by the differential operators ∓ e ± iθ ( ± ∂ y − isy ) Proposition 7.
Let m ∈ Z and k ∈ Z ≥ . Then, E − . Ψ m,k = (1 + 2 k − m )( k − k − k + 1) Ψ m − ,k +1 − k Ψ m − ,k − and E + . Ψ m,k = (1 + 2 k + m )( k − k −
1) Ψ m +2 ,k +1 − k Ψ m +2 ,k − . Proof.
Combining (4.12b) and (4.12e) with a + 1 instead of a , one obtains(6.1) F ( a, b, z ) = F ( a, b − , z ) − azb ( b − F ( a + 1 , b + 1 , z ) . Using Equation (4.12e) with b + 1 in place of b and combining it with (4.12c), oneobtains(6.2) F ( a, b, z ) = − F ( a − , b − , z ) + b − ab − z F ( a, b + 1 , z ) . Using Equation (4.12a) we can compute the action of E ± directly. Let a = k − m and b = k + 1 /
2. Then it is straightforward to see E − . Ψ m,k = − e − i ( m +2) θ/ − y / y k − (cid:0) ( − b ) F ( a, b, y )+ 4 ay /b F ( a + 1 , b + 1 , y ) (cid:1) . Applying (6.1), one gets the first equation.
A similar calculation using Equation (4.12a) shows E + . Ψ m,k = 12 e − i ( m − θ/ − y / y k − (cid:0) (1 − b + 4 y ) F ( a, b, y ) − ay /b F ( a + 1 , b + 1 , y ) (cid:1) . An application of (4.12c) gives E + . Ψ m,k = − e − i ( m − θ/ − y / y k − (cid:0) b − F ( a − , b − , y )+ (3 − b ) F ( a, b, y ) (cid:1) and substituting in the expression in (6.2) gives the desired result. (cid:3) From Equation (4.7), it follows that if the eigenvalue λ corresponds to the pa-rameter l , then λ + l + 1 corresponds to the parameter l + 1 and λ − l correspondsto the parameter l − H . Corollary 5. If m = 2 k + 1 , the action of E ± on the lowest weight is given by E − .ψ k +1 ,k = − k Ψ k − ,k − and E + .ψ k +1 ,k = (2 k + 1)( k − k − k +3 ,k +1 − k Ψ k +3 ,k − . If m = − (2 k + 1) , the action of E ± on the lowest weight is given by E − .ψ − (2 k +1) ,k = 2( k − k − − k − ,k +1 − k Ψ − k − ,k − and E + .ψ − (2 k +1) ,k = − k Ψ − k +1 ,k − . Proof.
This follows directly from the previous proposition. (cid:3)
We now will show how Proposition 7 and Corollary 5 imply that the action of h ties together the kernels indexed by k , in a g -module. Recall, the cases where k = 0 and k = 1 correspond to the potential free case. Definition 2.
Let (6.3) H = M l ∈ Z ≥ H l . Whenever the spaces are defined, let (6.4) H ± = M l ∈ Z ≥ H ± l . Theorem 4.
Let q ∈ Z and k ∈ Z ≥ . With respect to the action of g : (1) If q = 0 or q = 2 , the composition series of H is ⊂ H ⊕ H ⊂ H. LOBAL ^ SL (2 , R ) REPRESENTATIONS OF THE SCHR ¨ODINGER EQUATION 17 (2) If q ≡ − , then the composition series of g -submodules of H is asfollows ⊂ H − ⊕ H − ⊂ H ⊕ H ⊂ H ⊕ H ⊕ H − ⊂ H. (3) If q ≡ , then then the composition series of g -submodules of H is ⊂ H +0 ⊕ H +1 ⊂ H ⊕ H ⊂ H ⊕ H ⊕ H + ⊂ H. Proof.
Let q ≡ q ≡ E ± sends elements in H only to H and the action of E ± sends elements in H only to H . Under this assumption on q , each H k is irreducible under the sl action, thus H ⊕ H is irreducible under the g action. Now we look at the quotient H/ ( H ⊕ H ). Let π : H → H/ ( H ⊕ H ) be the natural projection. Let H k denotethe image of H k under π , then the image of H under π can be decomposed as adirect sum as H = L j H k j as an sl -module. Proposition 7 implies that E ± .H k j has a non-zero component in H k j − and in H k j +1 , for k j ≥
2. Since the H k j − and H k j +1 are inequivalent sl -representations, then E ± .H k j generates H k j − ⊕ H k j +1 under the action of sl (2 , R ). Irreducibility follows easily from this.Since, the proofs of (2) and (3) are essentially identical, we will only look at theproof of (3). Irreducibility of H +0 ⊕ H +1 under g follows from irreducibility under sl and from the actions on the lowest weights described in Corollary 5.Next we look at the quotient ( H ⊕ H ) / ( H +0 ⊕ H +1 ). For j ∈ { , } , write H j for the image of H j under the natural projection H ⊕ H → ( H ⊕ H ) / ( H +0 ⊕ H +1 ). Then, H gets sent to H and H gets sent to H by the action of theHeisenberg algebra. This, together with irreducibility of H and H under sl ,gives irreducibility under g .Finally, we look at the quotient ( H ⊕ H ⊕ H + ) / ( H ⊕ H ). Write L k ≥ H + k forthe image of H ⊕ H ⊕ H + under the natural projection. The Heisenberg algebraacts as before, and E ± .H + k j has a component in H + k j − and in H + k j +1 , for k j ≥ H k for k ≥ (cid:3) Part (2) of Theorem 4 can be seen pictorially as follows: m bbbbbb bbbbbbbb bbbbbbb bbbbbbbb bbbbbbb bbbbbbbb bbbbbbb bbbbbbbb ........................ ... ... ... ... ... ... ... ... ...... ... ... ... ... ... ... ... ... bbbbbbbb ... λ = 0 E + E − E + E − λ > b E + E − ( ( ( ( ( ( ( ( ( k Here, the sl -triple leaves each vertical string invariant, acting by rising and loweringoperators and by multiplication by a constant. The operator E + moves from the K -type corresponding to m and k to a linear combination of elements of the K -types corresponding to m + 2 and k ±
1, for k ≥
2. Similarly E − sends m to m − k to k ±
1. However, for k = 0 and k = 1, the action leaves invariant the directsum H ⊕ H (c.f. Proposition 7). Each vertical strip has a lowest weight, which isdistinguished with an inverted bracket. The action of E ± also respects the lowestweight structure as stated in Corollary 5. References [1] Milton Abramowitz and Irene A. Stegun.
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