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CLASSIFICATION OF LINEAR DIFFERENTIAL OPERATORS

arXiv:funct-an/9307001v1 21 Jul 1993CLASSIFICATION OF LINEAR DIFFERENTIAL OPERATORSWITH AN INVARIANT SUBSPACE OF MONOMIALSGERHARD POST AND ALEXANDER TURBINERJuly 9, 1993Abstract. A complete classification of linear differential operators possessingfinite-dimensional invariant subspace with a basis of monomials is presented.1.

IntroductionOne of the old-standing problems in the theory of special functions is the classi-fication of all linear differential operators that admit an infinite sequence of orthog-onal eigenvectors in the form of polynomials. See [2] for an overview.

In 1929 S.Bochner [1] had solved this problem for second order differential operators on thecomplex (real) line. Therefore we name this problem the “Bochner problem”.The main purpose of the present paper is to study a more general problem,which we name (following [4]) the “generalized Bochner problem”.

We ask for aclassification of all linear differential operators, which possess a finite-dimensionalinvariant subspace of polynomials. It turns out this problem is rather sophisticated.However, things get much more tangible when, instead of invariant subspace inpolynomials, we require that the (finite-dimensional) invariant subspace has a basisof monomials.

This problem is solved completely (Section 3). In [4] a particularcase of this problem was solved, namely if the invariant subspace is the linear spaceof polynomials of degree not higher than some fixed integer.

The classification ofthe linear operators possessing such an invariant subspace was given through theuniversal enveloping algebra of the algebra sl2 taken in a special representation byfirst-order differential operators.The results of Section 3 have an impact to the general problem. This is pre-sented in Section 4.

In Section 5, we give the explicit expressions for the second-order differential operators T2, possessing a finite-dimensional invariant subspacein monomials. These operators are of great interest for finding explicit solutions tothe Schr¨odinger equation(−d2dx2 + V (x))Ψ(x) = λΨ(x)(1.1)since the eigenvalue problem for the operator T2, T2ϕ = λϕ can be reduced to theSchr¨odinger equation by a change of the variable, x′ = x′(x), and introducing a newA.T.

is supported in part by CAST grant, US National Academy of Sciences1

2GERHARD POST AND ALEXANDER TURBINER(gauge-transformed) function, Ψ = ϕ exp(−a(x)). Hence the operators describedin Section 5 lead to a special class of quasi-exactly-solvable Schr¨odinger equationscomplementary those described already [4].Finally we make some concluding remarks in Section 6 concerning the case wherethe powers of x are not natural numbers.Though in this paper we work over the complex numbers, the main results holdfor real numbers as well.2.

GeneralitiesHere we state some general results concerning differential operators which leave afinite-dimensional subspace of C[x] invariant. The results of this section can easilybe extended to more variables.

Let D denote the algebra of linear (finite-order)differential operators on C[x] with polynomial coefficients. We denote the symbolddx by ∂.Let V be a finite-dimensional subspace in C[x] of dimension n, and let T :C[x] →C[x] be a linear operator.

It is easy to show that any linear operator T canbe written as an infinite order linear differential operator:T =∞Xi=0Pi∂i,Pi ∈C[x].This can be proved easily by an inductive construction:P0 = T (1),P1 = T (x) −xP0, . .

. .It follows immediately that the action of T on V can be represented by a finite-orderdifferential operator, since V ⊂IP k (where IP k denotes the space of polynomials ofdegree not higher than k, and k sufficiently large), and hence for Tk:Tk =kXi=0Pi∂i,we have that Tk(v) = T (v) for all v ∈V .So the following proposition is immediate:Proposition 2.1.

Let V ⊂C[x] be a subspace of dimension n, and let DV denotethe algebra of differential operators that leave V invariant (i.e. Tk ∈DV ⇒Tk(V ) ⊂V ).

Then DV is isomorphic to the semi-direct product of End(V ) and I, whereEnd(V ) is the algebra of linear operators on V , and the ideal I is the algebra ofdifferential operators which annihilate V .□We can paraphrase this proposition in the following way. As representation onV , DV is the full n × n-matrix algebra, and the kernel of this representation is aninfinite dimensional ideal in DV .Another consequence of this observation is, that for any element v ∈C[x] andλ ∈C, we can find a differential operator T such that T v = λv.In this paper, we mainly discuss the case that V is graded.Now C[x] is agraded algebra by putting deg(xk) = k. Let us take V a graded subspace.

Thismeans exactly that V has a basis of monomials.Hence we assume that V =⟨xi1, xi2, . .

. , xin⟩, which we abbreviate to V = ⟨xI⟩, I = {i1, i2, .

. .

, in}.

DIFFERENTIAL OPERATORS AND INVARIANT MONOMIALS3The fact that C[x] is graded has as a consequence that End(C[x]) is also graded,putting deg(T ) = m, if T (xi) ∈⟨xi+m⟩for all i. Moreover for graded V it followsthat DV is also graded.

So, in order to describe the structure of DV , it is sufficientto describe the homogeneous components of DV . This is the concern of the nextsection.3.

Algebras leaving a space of monomials invariantAs before, we are interested in finite-order linear differential operators T , suchthat T (V ) ⊂V . We assume T to be graded of degree m and order k, which meansthatT =kXi=0cixi+m∂i.Here ck ̸= 0 and m ∈Z.

Moreover ci = 0 for i + m < 0. In particular, we see thatif the degree of T is negative, say deg(T ) = −m, m > 0, then the order of T is atleast m.The following lemma plays a crucial role in our classification.Lemma 3.1.

Let T be a differential operator of degree m and order k.(1) Suppose m ≥0. Then there exist numbers α1, .

. .

, αk ∈C such thatT = ckxm(x∂−α1)(x∂−α2) . .

. (x∂−αk).

(2) Suppose m < 0, hence k ≥−m. Then there exist numbers α1, .

. .

, αk+m ∈C such thatT = ck∂−m(x∂−α1)(x∂−α2) . .

. (x∂−αk+m).Proof.

If m ≥0 it is clear that any differential operator of degree m is of the formT = xmkPi=0cixi∂i. Similarly, if m < 0, T can be put in the form T = ∂−mkPi=0˜cixi∂iwith ˜ck = ck.

So it suffices to prove the lemma for m = 0. But then we haveT (xα) = Pk(α)xα, where Pk is a k-th order polynomial.

So Pk(α) = ck(α−α1)(α−α2) · · · (α −αk) for some αi ∈C. Now((x∂−α1) · · · (x∂−αk))(xα) = (α −α1)(α −α2) · · · (α −αk),From this it follows that T = ck(x∂−α1)(x∂−α2) · · · (x∂−αk), since the repre-sentation of D on C[x] is faithful.It is easy to see that this factorization is unique up to the ordering of the factorsx∂−αi.

However, a certain ordering of the factors has no meaning, since thesefactors commute.Above representation of T given by lemma 3.1 is very convenient, when westudy the operators T which leave V = ⟨xi1, . .

. , xin⟩invariant, i.e.

T ∈DV . Weintroduce the following notation.

For I = {i1, i2, . .

. , in} and m ∈Z, we putI(m) = {i ∈I | i + m ≥0 and i + m ̸∈I}.If deg(T ) = m and T ∈DV , then it is clear that with necessity T (xi) = 0 fori ∈I(m), since T (xi) = c · xi+m, but xi+m ̸∈V , so c = 0.This leads to thefollowing

4GERHARD POST AND ALEXANDER TURBINERTheorem 3.2. Let V = ⟨xi1, .

. .

, xin⟩and T a finite-order differential operatorsuch that deg(T ) = m. Suppose I(m) = {α1, . .

. , αk}.

Then T ∈DV , if and only ifT = eT · (x∂−α1)(x∂−α2) · · · (x∂−αk),where eT is some differential operator of degree m.Proof. The if-part is trivial.

So assume that T ∈DV , and suppose it has order s.According to lemma 3.1, for m ≥0, T can be represented in the formT = c · xm(x∂−β1)(x∂−β2) · · · (x∂−βs)(c ∈C, c ̸= 0).We need that T (xi) = 0 for i ∈I(m). On the other hand, we haveT (xi) = c(i −β1)(i −β2) · · · (i −βs)xi+m.Hence it follows that {α1, .

. .

, αk} ⊂{β1, . .

. , βs}.

After rearranging the β’s wefindT = c · xm(x∂−β1)(x∂−β2) · · · (x∂−βs−k)(x∂−α1) · · · (x∂−αk).So for m ≥0 the proposition is proved; eT = c·xm(x∂−β1)(x∂−β2) · · · (x∂−βs−k).For m < 0 the proof is similar.REMARK. From the previous proposition it follows that the order of T ∈DVwith deg(T ) = m is at least k, where k is the number of elements in I(m).

In fact,up to a scalar coefficient, the element of order k is unique:T = xm(x∂−α1) · · · (x∂−αk),where {α1, . .

. , αk} = I(m).If m = 0, I(m) = ∅, and we have T = 1.

So all differential operators of degree 0are in DV ; these elements form a commutative subalgebra of DV generated by x∂.4. The case of polynomial subspacesIn section 3, we performed the classification of differential operators with aninvariant subspace V that has a basis of monomials.

If V has no basis of monomials,DV is not be graded, but only filtered. This causes a major difficulty.

However,considering the corresponding grading, we still can deduce some properties of DVin this case.So, let V have a basis of the formxi1 + c11xi1−1 + . .

. ,xi2 + c21xi2−1 + .

. .

,. .

.,xin + cn1xin−1 + . .

.We can assume that all ij are different. The graded space V (g) associated to V is⟨xi1, .

. .

, xin⟩.Let T ∈D of order k be of the formT =mXi=−kT (i)with T (i) of degree i, and T (m) ̸= 0. (An operator of order k has degree −k orhigher, the term of degree −k being a multiple of ∂k).

We call T (m) the associatedgraded operator. Now it is easy to prove

DIFFERENTIAL OPERATORS AND INVARIANT MONOMIALS5Theorem 4.1. Let V, V (g), T and T (m) be as above.If T ∈DV , then T (m) ∈DV (g).Proof.

Suppose T ∈DV , and consider T (xij + cj1xij−1 + . .

. ).

If m = 0 there isnothing to prove, so consider m ̸= 0. Let I = {i1, i2, .

. .

, in}. If ij ∈I(m) thenT (m)(xij) should be 0, since no term of T can cancel this term.

Hence we findexactly that T (m)(xij ) = 0 for all ij ∈I(m), i.e. T (m) ∈DV (g).From this we derive an easy corollary:If Tk ∈D of order k possesses an infinite number of (linearly independent) eigen-vectors, then T = P0i=−k T (i), degree of T (i) = i and T (0) ̸= 0.A similar reasoning can be performed for the part of T with minimal degree, butthis seems to give not much information.5.

Classification of second order differential operators5.1. Generic situation.

Now we proceed to 2-nd order differential operators T2,which admit a finite-dimensional invariant space of polynomials with a basis ofmonomials. We are interested in this problem in connection with finding explicitsolutions to the Schr¨odinger equation (1.1).

As mentioned in the Introduction, thisinvolves some transformations see [3]. These transformations are of 2 types: the firstis a change of basis, and the second is called “gauge” transformation, which amountsto changing T to gT g−1, where g is a non-zero function.

To delete some ambiguityin our spaces of monomials, we impose 2 conditions on V = {xi1, xi2, . .

. , xin}:(1) We assume that 1 ∈V (it removes an ambiguity resulting from gaugetransformations).

(2) We assume that gcd(i1, i2, . .

. , in) = 1, i.e.

that the powers have no com-mon factor (it removes an ambiguity resulting from changes of variable).For the classification of differential operators in DV these assumptions do not makemuch difference. If T2 = P T (i), where the degree of T (i) is i, these two assumptionseffect only the terms T (−1) and T (−2).So let us start the classification.

Suppose T2 ∈DV , T2 = Pm T (m) where T (m)has degree m and order less or equal to 2. If T (m) is non-zero, then according toproposition 3.2, I(m) contains 0,1 or 2 elements.

We distinguish these 4 cases:(1) For all m > 0, I(m) contains more than 2 elements. So T2 contains noterms of positive degree.

Hence T2 preserves ⟨1, x, x2, . .

. , xn⟩for all n.This type of operators is already studied in [4] called there exactly-solvableoperators), and we do not repeat it here.

(2) I(m) is empty. This can only happen, if m = 0.

But this case is trivial,since all operators of degree 0 are in DV . Hence degree 0 contributes to T2the operator α1x2∂2 + α2x∂+ α3.

This part we call trivial, and is alwayspresent in T2. (3) I(m) contains one element.

Let us assume here that i1 > i2 > · · · > in.Then we have ij = ij−1 + m so that ij = (n −j)m (since we assumed that1 ∈V , so in = 0). But we also assumed that the ij have no common factor,so it follows that m = 1, and hence V = ⟨1, x, x2, .

. .

, xn−1⟩. Hence we arein the case that is extensively discussed in [4].

6GERHARD POST AND ALEXANDER TURBINER(4) I(m) contains 2 elements, m > 0, and no I(l) for l > 0 contains one element.Suppose i1 and i2 are the 2 elements in I(m). Then the set {i1, i2, .

. .

, in}is of the following specific form:i3 = i1 −m, i5 = i3 −m, . .

. , i2r−1 = i2r−3 −mandi4 = i2 −m, i6 = i4 −m, .

. .

, i2s = i2s−2 −mWe call i1, i3, . .

. , i2r−1 and i2, i4, .

. .

, i2s chains with step m and length rand s, respectively. In general, with no special relation, by which we meanthat I can be split into two chains in exactly one way, the most generalsecond-order operator in DV isT2 = α1xm(x∂−i1)(x∂−i2) + α2x2∂2 + α3x∂+ α4and T2 contains an extra term in two cases:(a) If m=1, (and therefore i2s=0), we get an extra termα5∂(x∂−i2r−1).

(b) If m=2 and {i2s, i2r−1} = {0, 1}, we get an extra term α5∂2.All this can easily be proved, by examining the possibilities for which I(m)can have 0,1 or 2 elements.5.2. Special subspaces.

For existence of non-trivial second order operators inDV , it is necessary that I can be split into 2 chains. The form of T2 above isunder the assumption that I can be split into two chains in exactly one way.There are cases, where I can be split in more than one way.As an exampleconsider I = {0, 1, 2, .

. .

, 98, 100}.Then I = {0, 1, 2, . .

. , 98} ∪{100} or I ={0, 2, 4, .

. .

, 98, 100} ∪{1, 3, 5, . .

. , 97}.

Consequently the space V admits a moregeneral second-order operator. Here we describe all special cases, which fall into 4groups:Case A.

The dimension of V is 3, so I = {0, m, m + l}, l ̸= m. ThenT2 = α1xl+m(x∂−m)(x∂−l −m) + α2xl+1∂(x∂−l −m)+ α3xm(x∂−m)(x∂−l −m) + α4x2∂2 + α5x∂+ α6T2 gets a certain extra terms in the following cases:(a) m = 1, l > 2. Extra term α7∂(x∂−l −m).

(b) m = 1, l = 2. Extra terms α7∂(x∂−3) + α8∂2.

(c) l = 1 (hence m > 1). Extra term α7∂(x∂−m).Case B.

The dimension of V is 4, and I is “symmetric”, i.e. I = {0, m, m+l, 2m+l},l ̸= m. ThenT2 = α1xl+m(x∂−2m −l)(x∂−l −m) + α2xm(x∂−2m −l)(x∂−m)+ α3x2∂2 + α4x∂+ α5T2 gets an extra term only if(a) m = 1.

Extra term α6∂(x∂−m −l).

DIFFERENTIAL OPERATORS AND INVARIANT MONOMIALS7Case C. I has one runner ahead at distance 2, i.e. I = {0, 1, 2, .

. .

, n −2, n}. ThenT2 takes the formT2 = α1x2(x∂−n)(x∂−n + 3) + α2x(x∂−n)(x∂−n + 2)+ α3x2∂2 + α4x∂+ α5+ α6∂(x∂−n) + α7∂2Case D. I has one runner left behind at distance 2, i.e.

I = {0, 2, 3, 4, . .

. , n}.

ThenT2 is of the formT2 = α1x2(x∂−n)(x∂−n + 1) + α2x2∂(x∂−n)+ α3x2∂2 + α4x∂+ α5+ α6∂(x∂−2)These special cases exhaust the list of all exceptional cases, which do not belongto the general classification given above. Let us prove this.

It is clear that wecan assume that the dimension is 5 or more, since the dimensions 3 and 4 areconsidered above. We know that I can be split into two chains of step, say, m, soi3 = i1−m, i5 = i3−m, .

. .

, i2r−1 and i4 = i2−m, i6 = i4−m, . .

. , i2s.

Supposethat I can also be split in two chains of step l. We may assume that l > m. Nowwe distinguish three cases:(1) r = 1, so i3 is not present.We have either I(l) ⊃{i1, i2} or I(l) ⊃{i1, i4}. Therefore we consider twosubcases:a. I(l) ⊃{i1, i2}, so l ̸= i1 −i2.

Since | I(l) |= 2, we have i4 ̸∈I(l), soi1 −i4 = l. But then i1 −i6 ̸= l, and therefore i2 −i6 = 2m = l.This is not allowed, since i1 −i4 = 2m implies that i1 −i2 = m.b. I(l) ⊃{i1, i4}, so i1 −i2 = l. But also i6 ̸∈I(l), and this is onlypossible if i2 −i6 = l = 2m.

Therefore this configuration leads tocase C.(2) s = 1, so i4 is not present.Clearly I(l) ⊃{i1, i3}. Moreover i2 < i3 since otherwise also i2 ∈I(l).

Butthen i5 ̸∈I(l) implies that i5 + l = i1, and hence l = 2m. Again usingi2 ̸∈I(l) implies that i2 + l = i2r−1.

Therefore this configuration leads tocase D.(3) r > 1 and s > 1, so i3 and i4 are both present.Always I(l) ⊃{i1, i3}, and necessarily i2 < i3, since otherwise also i2 ∈I(l).Again we have two subcases:a. i5 is present. We need i5 + l = i1, so l = 2m.

Like in case 2 above,it follows that i2 = i2r−1 −2m. But then i4 ∈I(l).b.

i6 is present, but not i5. We need either i1 −i2 = l or i3 −i2 = l.If i1 −i2 = l, then i6 + l = i2 is the only possibility, so l = 2m, andhence i3−i2 = m. This is forbidden because then we have one chain.If i3 −i2 = l, then i4 + l ̸∈{i1, i3, i2}, so i4 ∈I(l).

So this is alsoimpossible.

8GERHARD POST AND ALEXANDER TURBINER6. ConclusionIn the previous sections we considered the case that the powers of x are naturalnumbers.

From algebraic point of view, this is not a crucial restriction. One couldtake the powers to be all integer, rational, real or complex numbers, or any otherabelian subgroup of C. (If one takes C, one has to define a suitable total ordering onC to be able to consider positive and negative).

This has as a main advantage thatsuch algebras of “polynomials” admit the isomorphism x 7→1/x. Moreover, themap xk 7→xk+l is a linear isomorphism; it is the gauge transformation discussedbefore.An even more special case is that the set of allowed powers form a field.

In thiscase, the change of basis x 7→xm is an isomorphism.We shortly discuss the case that the powers of x are real numbers. The algebraof “generalized polynomials”, which we denote (suggestively) by C[xα] is as a linearspaceMα∈RC · xαand the multiplication is given byxβ · xγ = xβ+γβ, γ ∈RLet D denote the algebra of differential operators with coefficients in C[xα] now.The degree of T ∈D can be any real number.

For T ∈D, deg(T ) = m we haveT =kXi=0cixi+m∂i.with ci ∈C, and ck ̸= 0. Lemma 3.1 for this operator T now looks like:Lemma 6.1.

There exist numbers α1, . .

. , αk ∈C such thatT = ckxm(x∂−α1)(x∂−α2) .

. .

(x∂−αk).In particular, the case that m < 0 dissappears, or, better, ∂−m factorizes:∂s = x−s(x∂−(s −1))(x∂−(s −2)) · · · (x∂−1)x∂(s > 0)We denote I = {i1, i2, . .

. , in}, where ij ∈R, j = 1, .

. .

, n, and define for m ∈RI(m) = {i ∈I | i + m ̸∈I}.Let V = ⟨xi1, xi2, . .

. , xin⟩and let DV denote the algebra of operators in D thatleave V invariant.

One can easily check that Theorem 3.2 holds literally. Due tothe remarks at the beginning of this section, we have the following.Proposition 6.2.

(1) The gauge transformation T 7→xlT x−l gives an isomorphism between DVand DW , with V = ⟨xi1, xi1, . .

. , xin⟩and W = ⟨xi1−l, xi2−l, .

. .

, xin−l⟩. (2) The change of basis x′ = xm induces an isomorphism between DV and DW ,where V = ⟨xi1, xi1, .

. .

, xin⟩and W = ⟨xi1/m, xi2/m, . .

. , xin/m⟩.

DIFFERENTIAL OPERATORS AND INVARIANT MONOMIALS9Proof. (1) Part (1) is obvious.

(2) By the change of basis x′ = xm we have thatx∂= m x′∂′,∂′ =ddx′ .Hence the operator T = ˜T ·(x∂−α1)(x∂−α2) · · · (x∂−αk), where deg( ˜T) = lis mapped to T ′ = ˜T ′ · (mx′∂′ −α1)(mx′∂′ −α2) · · · (mx′∂′ −αk) withdeg( ˜T ′) = l/m. This leads directly to the proof of statement (2) of theProposition.□The classification of second order operators T2 ∈DV is similar as before.

A dif-ference is that |I(m)| = |I(−m)|, so that there is always a sort of symmetry in T2.Here we give one example, the generic 2-chain case. Suppose |I(m)| = 2, so I hasthe structure I = {i1, i3, .

. .

, i2r−1} ∪{i2, i4, . .

. , i2s} withi3 = i1 −m, i5 = i3 −m, .

. .

, i2r−1 = i2r−3 −mandi4 = i2 −m, i6 = i4 −m, . .

. , i2s = i2s−2 −mIf for all l > 0, l ̸= m there holds |I(l)| > 2, then the most general 2nd order operatorT2 ∈DV takes the form:T2 = α1xm(x∂−i1)(x∂−i2) + α2x2∂2 + α3x∂+ α4 + α5x−m(x∂−i2r−1)(x∂−i2s)Note the symmetry in T2: there are as many terms of degree m as terms of degree−m.Final remark.In the case that V = ⟨1, x, x2, .

. .

, xn⟩, the algebra DV isessentially generated by {∂, 2x∂−n, x2∂−nx}, see [4]. These 3 elements form theLie algebra sl2(C), and hence DV is essentially some representation of the universalenveloping algebra of sl2(C).

We were not able to find a similar structure in DVfor general V , even for the simplest case V = ⟨1, x, x3⟩. Particularly, one can showthat for the space V = ⟨1, x, x3⟩the algebra DV is an infinite-dimensional, finite-generated algebra.

It is defined by 11 generators, which are differential operators offirst-, second- and third order, and the commutator of any two of them is expressibleas an ordered cubic polynomial in these generators.AcknowledgementThe second author (A.T.) wishes to thank M.A. Shubin for useful discussion.

10GERHARD POST AND ALEXANDER TURBINERReferences1. S. Bochner, ¨Uber Sturm-Liouvillesche Polynomsysteme, Math.

Z. 29, p. 730-736 (1929).2.

L.L. Littlejohn, Orthogonal polynomial solutions to ordinary differential equations, in “Orthog-onal polynomials and their applications”, eds.

M. Alfaro e.a., Lecture Notes in Mathematics1329, Springer-Verlag, Berlin (1970).3. A.V.

Turbiner, Quasi-exactly solvable problems and sl(2) algebra, Comm. Math.

Phys. 118, p.467-474 (1988).4.

A.V. Turbiner, Lie-algebras and polynomials, Journ.

Phys. A25 L1087-L1093 (1992);Lie-algebraic approach to the theory of polynomial solutions.

I. Ordinary differential equationsand finite-difference equations in one variable, preprint CPT-91/P.2679 (1991);Lie algebras and linear operators with invariant subspace, preprint I.H.E.S./P/92/95 (Decem-ber 1992 and June 1993 (corrected version)), to appear in Lie Algebras, Cohomologies and NewFindings in Quantum Mechanics, AMS Contemporary Math, N. Kamran and P. Olver (eds.

),AMS, 1993Gerhard Post, Department of Applied Mathematics, University of Twente, P.O. Box217, 7500 AE Enschede, The NetherlandsE-mail address: post@math.utwente.nlAlexander Turbiner, Institute for Theoretic and Experimental Physics,117259 Moscow, Russia (on leave in absence)Current address: Alexander Turbiner, Mathematics Department, Case Western Reserve Uni-versity, Cleveland OH 44106E-mail address: turbiner@vxcern.cern.ch

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