Goursat and Furstenberg-Weiss type lemmas

I’m collecting in this blog post a number of simple group-theoretic lemmas, all of the following flavour: if is a subgroup of some product of groups, then one of three things has to happen:

• ( too small) is contained in some proper subgroup of , or the elements of are constrained to some sort of equation that the full group does not satisfy.
• ( too large) contains some non-trivial normal subgroup of , and as such actually arises by pullback from some subgroup of the quotient group .
• (Structure) There is some useful structural relationship between and the groups .

These sorts of lemmas show up often in ergodic theory, when the equidistribution of some orbit is governed by some unspecified subgroup of a product group , and one needs to know further information about this subgroup in order to take the analysis further. In some cases only two of the above three options are relevant. In the cases where is too “small” or too “large” one can reduce the groups to something smaller (either a subgroup or a quotient) and in applications one can often proceed in this case by some induction on the “size” of the groups (for instance, if these groups are Lie groups, one can often perform an induction on dimension), so it is often the structured case which is the most interesting case to deal with.

It is perhaps easiest to explain the flavour of these lemmas with some simple examples, starting with the case where we are just considering subgroups of a single group .

Lemma 1 Let be a subgroup of a group . Then exactly one of the following hold:

• (i) ( too small) There exists a non-trivial group homomorphism into a group such that for all .
• (ii) ( normally generates ) is generated as a group by the conjugates of .

Proof: Let be the group normally generated by , that is to say the group generated by the conjugates of . This is a normal subgroup of containing (indeed it is the smallest such normal subgroup). If is all of we are in option (ii); otherwise we can take to be the quotient group and to be the quotient map. Finally, if (i) holds, then all of the conjugates of lie in the kernel of , and so (ii) cannot hold.

Here is a “dual” to the above lemma:

Lemma 2 Let be a subgroup of a group . Then exactly one of the following hold:

• (i) ( too large) is the pullback of some subgroup of for some non-trivial normal subgroup of , where is the quotient map.
• (ii) ( is core-free) does not contain any non-trivial conjugacy class .

Proof: Let be the normal core of , that is to say the intersection of all the conjugates of . This is the largest normal subgroup of that is contained in . If is non-trivial, we can quotient it out and end up with option (i). If instead is trivial, then there is no non-trivial element that lies in the core, hence no non-trivial conjugacy class lies in and we are in option (ii). Finally, if (i) holds, then every conjugacy class of an element of is contained in and hence in , so (ii) cannot hold.

For subgroups of nilpotent groups, we have a nice dichotomy that detects properness of a subgroup through abelian representations:

Lemma 3 Let be a subgroup of a nilpotent group . Then exactly one of the following hold:

• (i) ( too small) There exists non-trivial group homomorphism into an abelian group such that for all .
• (ii) .

Informally: if is a variable ranging in a subgroup of a nilpotent group , then either is unconstrained (in the sense that it really ranges in all of ), or it obeys some abelian constraint .

Proof: By definition of nilpotency, the lower central series

eventually becomes trivial.

Since is a normal subgroup of , is also a subgroup of . Suppose first that is a proper subgroup of , then the quotient map is a non-trivial homomorphism to an abelian group that annihilates , and we are in option (i). Thus we may assume that , and thus

Note that modulo the normal group , commutes with , hence

and thus

We conclude that . One can continue this argument by induction to show that for every ; taking large enough we end up in option (ii). Finally, it is clear that (i) and (ii) cannot both hold.

Remark 4 When the group is locally compact and is closed, one can take the homomorphism in Lemma 3 to be continuous, and by using Pontryagin duality one can also take the target group to be the unit circle . Thus is now a character of . Similar considerations hold for some of the later lemmas in this post. Discrete versions of this above lemma, in which the group is replaced by some orbit of a polynomial map on a nilmanifold, were obtained by Leibman and are important in the equidistribution theory of nilmanifolds; see this paper of Ben Green and myself for further discussion.

Here is an analogue of Lemma 3 for special linear groups, due to Serre (IV-23):

Lemma 5 Let be a prime, and let be a closed subgroup of , where is the ring of -adic integers. Then exactly one of the following hold:

• (i) ( too small) There exists a proper subgroup of such that for all .
• (ii) .

Proof: It is a standard fact that the reduction of mod is , hence (i) and (ii) cannot both hold.

Suppose that (i) fails, then for every there exists such that , which we write as

We now claim inductively that for any and , there exists with ; taking limits as using the closed nature of will then place us in option (ii).

The case is already handled, so now suppose . If , we see from the case that we can write where and . Thus to establish the claim it suffices to do so under the additional hypothesis that .

First suppose that for some with . By the case, we can find of the form for some . Raising to the power and using and , we note that , giving the claim in this case.

Any matrix of trace zero with coefficients in is a linear combination of , , and is thus a sum of matrices that square to zero. Hence, if is of the form , then for some matrix of trace zero, and thus one can write (up to errors) as the finite product of matrices of the form with . By the previous arguments, such such matrix lies in up to errors, and hence does also. This completes the proof of the case.

Now suppose and the claim has already been proven for . Arguing as before, it suffices to close the induction under the additional hypothesis that , thus we may write . By induction hypothesis, we may find with . But then , and we are done.

We note a generalisation of Lemma 3 that involves two groups rather than just one:

Lemma 6 Let be a subgroup of a product of two nilpotent groups . Then exactly one of the following hold:

• (i) ( too small) There exists group homomorphisms , into an abelian group , with non-trivial, such that for all , where is the projection of to .
• (ii) for some subgroup of .

Proof: Consider the group . This is a subgroup of . If it is all of , then must be a Cartesian product and option (ii) holds. So suppose that this group is a proper subgroup of . Applying Lemma 3, we obtain a non-trivial group homomorphism into an abelian group such that whenever . For any in the projection of to , there is thus a unique quantity such that whenever . One easily checcks that is a homomorphism, so we are in option (i).

Finally, it is clear that (i) and (ii) cannot both hold, since (i) places a non-trivial constraint on the second component of an element of for any fixed choice of .

We also note a similar variant of Lemma 5, which is Lemme 10 of this paper of Serre:

Lemma 7 Let be a prime, and let be a closed subgroup of . Then exactly one of the following hold:

• (i) ( too small) There exists a proper subgroup of such that for all .
• (ii) .

Proof: As in the proof of Lemma 5, (i) and (ii) cannot both hold. Suppose that (i) does not hold, then for any there exists such that . Similarly, there exists with . Taking commutators of and , we can find with . Continuing to take commutators with and extracting a limit (using compactness and the closed nature of ), we can find with . Thus, the closed subgroup of does not obey conclusion (i) of Lemma 5, and must therefore obey conclusion (ii); that is to say, contains . Similarly contains ; multiplying, we end up in conclusion (ii).

The most famous result of this type is of course the Goursat lemma, which we phrase here in a somewhat idiosyncratic manner to conform to the pattern of the other lemmas in this post:

Lemma 8 (Goursat lemma) Let be a subgroup of a product of two groups . Then one of the following hold:

• (i) ( too small) is contained in for some subgroups , of respectively, with either or (or both).
• (ii) ( too large) There exist normal subgroups of respectively, not both trivial, such that arises from a subgroup of , where is the quotient map.
• (iii) (Isomorphism) There is a group isomorphism such that is the graph of . In particular, and are isomorphic.

Here we almost have a trichotomy, because option (iii) is incompatible with both option (i) and option (ii). However, it is possible for options (i) and (ii) to simultaneously hold.

Proof: If either of the projections , from to the factor groups (thus and fail to be surjective, then we are in option (i). Thus we may assume that these maps are surjective.

Next, if either of the maps , fail to be injective, then at least one of the kernels , is non-trivial. We can then descend down to the quotient and end up in option (ii).

The only remaining case is when the group homomorphisms are both bijections, hence are group isomorphisms. If we set we end up in case (iii).

We can combine the Goursat lemma with Lemma 3 to obtain a variant:

Corollary 9 (Nilpotent Goursat lemma) Let be a subgroup of a product of two nilpotent groups . Then one of the following hold:

• (i) ( too small) There exists and a non-trivial group homomorphism such that for all .
• (ii) ( too large) There exist normal subgroups of respectively, not both trivial, such that arises from a subgroup of .
• (iii) (Isomorphism) There is a group isomorphism such that is the graph of . In particular, and are isomorphic.

Proof: If Lemma 8(i) holds, then by applying Lemma 3 we arrive at our current option (i). The other options are unchanged from Lemma 8, giving the claim.

Now we present a lemma involving three groups that is known in ergodic theory contexts as the “Furstenberg-Weiss argument”, as an argument of this type arose in this paper of Furstenberg and Weiss, though perhaps it also implicitly appears in other contexts also. It has the remarkable feature of being able to enforce the abelian nature of one of the groups once the other options of the lemma are excluded.

Lemma 10 (Furstenberg-Weiss lemma) Let be a subgroup of a product of three groups . Then one of the following hold:

• (i) ( too small) There is some proper subgroup of and some such that whenever and .
• (ii) ( too large) There exists a non-trivial normal subgroup of with abelian, such that arises from a subgroup of , where is the quotient map.
• (iii) is abelian.

Proof: If the group is a proper subgroup of , then we are in option (i) (with ), so we may assume that

Similarly we may assume that

Now let be any two elements of . By the above assumptions, we can find such that

and

Taking commutators to eliminate the terms, we conclude that

Thus the group contains every commutator , and thus contains the entire group generated by these commutators. If fails to be abelian, then is a non-trivial normal subgroup of , and now arises from in the obvious fashion, placing one in option (ii). Hence the only remaining case is when is abelian, giving us option (iii).

As before, we can combine this with previous lemmas to obtain a variant in the nilpotent case:

Lemma 11 (Nilpotent Furstenberg-Weiss lemma) Let be a subgroup of a product of three nilpotent groups . Then one of the following hold:

• (i) ( too small) There exists and group homomorphisms , for some abelian group , with non-trivial, such that whenever , where is the projection of to .
• (ii) ( too large) There exists a non-trivial normal subgroup of , such that arises from a subgroup of .
• (iii) is abelian.

Informally, this lemma asserts that if is a variable ranging in some subgroup , then either (i) there is a non-trivial abelian equation that constrains in terms of either or ; (ii) is not fully determined by and ; or (iii) is abelian.

Proof: Applying Lemma 10, we are already done if conclusions (ii) or (iii) of that lemma hold, so suppose instead that conclusion (i) holds for say . Then the group is not of the form , since it only contains those with . Applying Lemma 6, we obtain group homomorphisms , into an abelian group , with non-trivial, such that whenever , placing us in option (i).

The Furstenberg-Weiss argument is often used (though not precisely in this form) to establish that certain key structure groups arising in ergodic theory are abelian; see for instance Proposition 6.3(1) of this paper of Host and Kra for an example.

One can get more structural control on in the Furstenberg-Weiss lemma in option (iii) if one also broadens options (i) and (ii):

Lemma 12 (Variant of Furstenberg-Weiss lemma) Let be a subgroup of a product of three groups . Then one of the following hold:

• (i) ( too small) There is some proper subgroup of for some such that whenever . (In other words, the projection of to is not surjective.)
• (ii) ( too large) There exists a normal of respectively, not all trivial, such that arises from a subgroup of , where is the quotient map.
• (iii) are abelian and isomorphic. Furthermore, there exist isomorphisms , , to an abelian group such that

  

The ability to encode an abelian additive relation in terms of group-theoretic properties is vaguely reminiscent of the group configuration theorem.

Proof: We apply Lemma 10. Option (i) of that lemma implies option (i) of the current lemma, and similarly for option (ii), so we may assume without loss of generality that is abelian. By permuting we may also assume that are abelian, and will use additive notation for these groups.

We may assume that the projections of to and are surjective, else we are in option (i). The group is then a normal subgroup of ; we may assume it is trivial, otherwise we can quotient it out and be in option (ii). Thus can be expressed as a graph for some map . As is a group, must be a homomorphism, and we can write it as for some homomorphisms , . Thus elements of obey the constraint .

If or fails to be injective, then we can quotient out by their kernels and end up in option (ii). If fails to be surjective, then the projection of to also fails to be surjective (since for , is now constrained to lie in the range of ) and we are in option (i). Simiarly if fails to be surjective. Thus we may assume that the homomorphisms are bijective and thus group isomorphisms. Setting to the identity, we arrive at option (iii).

Combining this lemma with Lemma 3, we obtain a nilpotent version:

Corollary 13 (Variant of nilpotent Furstenberg-Weiss lemma) Let be a subgroup of a product of three groups . Then one of the following hold:

• (i) ( too small) There are homomorphisms , to some abelian group for some , with not both trivial, such that whenever .
• (ii) ( too large) There exists a normal of respectively, not all trivial, such that arises from a subgroup of , where is the quotient map.
• (iii) are abelian and isomorphic. Furthermore, there exist isomorphisms , , to an abelian group such that

  

Here is another variant of the Furstenberg-Weiss lemma, attributed to Serre by Ribet (see Lemma 3.3):

Lemma 14 (Serre’s lemma) Let be a subgroup of a finite product of groups with . Then one of the following hold:

• (i) ( too small) There is some proper subgroup of for some such that whenever .
• (ii) ( too large) One has .
• (iii) One of the has a non-trivial abelian quotient .

Proof: The claim is trivial for (and we don’t need (iii) in this case), so suppose that . We can assume that each is a perfect group, , otherwise we can quotient out by the commutator and arrive in option (iii). Similarly, we may assume that all the projections of to , are surjective, otherwise we are in option (i).

We now claim that for any and any , one can find with for and . For this follows from the surjectivity of the projection of to . Now suppose inductively that and the claim has already been proven for . Since is perfect, it suffices to establish this claim for of the form for some . By induction hypothesis, we can find with for and . By surjectivity of the projection of to , one can find with and . Taking commutators of these two elements, we obtain the claim.

Setting , we conclude that contains . Similarly for permutations. Multiplying these together we see that contains all of , and we are in option (ii).

Is there a non-analytic function with all differences analytic?

I was asked the following interesting question from a bright high school student I am working with, to which I did not immediately know the answer:

Question 1 Does there exist a smooth function which is not real analytic, but such that all the differences are real analytic for every ?

The hypothesis implies that the Newton quotients are real analytic for every . If analyticity was preserved by smooth limits, this would imply that is real analytic, which would make real analytic. However, we are not assuming any uniformity in the analyticity of the Newton quotients, so this simple argument does not seem to resolve the question immediately.

In the case that is periodic, say periodic with period , one can answer the question in the negative by Fourier series. Perform a Fourier expansion . If is not real analytic, then there is a sequence going to infinity such that as . From the Borel-Cantelli lemma one can then find a real number such that (say) for infinitely many , hence for infinitely many . Thus the Fourier coefficients of do not decay exponentially and hence this function is not analytic, a contradiction.

I was not able to quickly resolve the non-periodic case, but I thought perhaps this might be a good problem to crowdsource, so I invite readers to contribute their thoughts on this problem here. In the spirit of the polymath projects, I would encourage comments that contain thoughts that fall short of a complete solution, in the event that some other reader may be able to take the thought further.

Boosting the van der Corput inequality using the tensor power trick

In this previous blog post I noted the following easy application of Cauchy-Schwarz:

Lemma 1 (Van der Corput inequality) Let be unit vectors in a Hilbert space . Then

Proof: The left-hand side may be written as for some unit complex numbers . By Cauchy-Schwarz we have

and the claim now follows from the triangle inequality.

As a corollary, correlation becomes transitive in a statistical sense (even though it is not transitive in an absolute sense):

Corollary 2 (Statistical transitivity of correlation) Let be unit vectors in a Hilbert space such that for all and some . Then we have for at least of the pairs .

Proof: From the lemma, we have

The contribution of those with is at most , and all the remaining summands are at most , giving the claim.

One drawback with this corollary is that it does not tell us which pairs correlate. In particular, if the vector also correlates with a separate collection of unit vectors, the pairs for which correlate may have no intersection whatsoever with the pairs in which correlate (except of course on the diagonal where they must correlate).

While working on an ongoing research project, I recently found that there is a very simple way to get around the latter problem by exploiting the tensor power trick:

Corollary 3 (Simultaneous statistical transitivity of correlation) Let be unit vectors in a Hilbert space for and such that for all , and some . Then there are at least pairs such that . In particular (by Cauchy-Schwarz) we have for all .

Proof: Apply Corollary 2 to the unit vectors and , in the tensor power Hilbert space .

It is surprisingly difficult to obtain even a qualitative version of the above conclusion (namely, if correlates with all of the , then there are many pairs for which correlates with for all simultaneously) without some version of the tensor power trick. For instance, even the powerful Szemerédi regularity lemma, when applied to the set of pairs for which one has correlation of , for a single , does not seem to be sufficient. However, there is a reformulation of the argument using the Schur product theorem as a substitute for (or really, a disguised version of) the tensor power trick. For simplicity of notation let us just work with real Hilbert spaces to illustrate the argument. We start with the identity

where is the orthogonal projection to the complement of . This implies a Gram matrix inequality

for each where denotes the claim that is positive semi-definite. By the Schur product theorem, we conclude that

and hence for a suitable choice of signs ,

One now argues as in the proof of Corollary 2.

A separate application of tensor powers to amplify correlations was also noted in this previous blog post giving a cheap version of the Kabatjanskii-Levenstein bound, but this seems to not be directly related to this current application.

The inclusion-exclusion principle for commuting projections

The (classical) Möbius function is the unique function that obeys the classical Möbius inversion formula:

Proposition 1 (Classical Möbius inversion) Let be functions from the natural numbers to an additive group . Then the following two claims are equivalent:

• (i) for all .
• (ii) for all .

There is a generalisation of this formula to (finite) posets, due to Hall, in which one sums over chains in the poset:

Proposition 2 (Poset Möbius inversion) Let be a finite poset, and let be functions from that poset to an additive group . Then the following two claims are equivalent:

• (i) for all , where is understood to range in .
• (ii) for all , where in the inner sum are understood to range in with the indicated ordering.

(Note from the finite nature of that the inner sum in (ii) is vacuous for all but finitely many .)

Comparing Proposition 2 with Proposition 1, it is natural to refer to the function as the Möbius function of the poset; the condition (ii) can then be written as

Proof: If (i) holds, then we have

for any . Iterating this we obtain (ii). Conversely, from (ii) and separating out the term, and grouping all the other terms based on the value of , we obtain (1), and hence (i).

In fact it is not completely necessary that the poset be finite; an inspection of the proof shows that it suffices that every element of the poset has only finitely many predecessors .

It is not difficult to see that Proposition 2 includes Proposition 1 as a special case, after verifying the combinatorial fact that the quantity

is equal to when divides , and vanishes otherwise.

I recently discovered that Proposition 2 can also lead to a useful variant of the inclusion-exclusion principle. The classical version of this principle can be phrased in terms of indicator functions: if are subsets of some set , then

In particular, if there is a finite measure on for which are all measurable, we have

One drawback of this formula is that there are exponentially many terms on the right-hand side: of them, in fact. However, in many cases of interest there are “collisions” between the intersections (for instance, perhaps many of the pairwise intersections agree), in which case there is an opportunity to collect terms and hopefully achieve some cancellation. It turns out that it is possible to use Proposition 2 to do this, in which one only needs to sum over chains in the resulting poset of intersections:

Proposition 3 (Hall-type inclusion-exclusion principle) Let be subsets of some set , and let be the finite poset formed by intersections of some of the (with the convention that is the empty intersection), ordered by set inclusion. Then for any , one has

where are understood to range in . In particular (setting to be the empty intersection) if the are all proper subsets of then we have

In particular, if there is a finite measure on for which are all measurable, we have

Using the Möbius function on the poset , one can write these formulae as

and

Proof: It suffices to establish (2) (to derive (3) from (2) observe that all the are contained in one of the , so the effect of may be absorbed into ). Applying Proposition 2, this is equivalent to the assertion that

for all . But this amounts to the assertion that for each , there is precisely one in with the property that and for any in , namely one can take to be the intersection of all in such that contains .

Example 4 If with , and are all distinct, then we have for any finite measure on that makes measurable that

due to the four chains , , , of length one, and the three chains , , of length two. Note that this expansion just has six terms in it, as opposed to the given by the usual inclusion-exclusion formula, though of course one can reduce the number of terms by combining the factors. This may not seem particularly impressive, especially if one views the term as really being three terms instead of one, but if we add a fourth set with for all , the formula now becomes

and we begin to see more cancellation as we now have just seven terms (or ten if we count as four terms) instead of terms.

Example 5 (Variant of Legendre sieve) If are natural numbers, and is some sequence of complex numbers with only finitely many terms non-zero, then by applying the above proposition to the sets and with equal to counting measure weighted by the we obtain a variant of the Legendre sieve

where range over the set formed by taking least common multiples of the (with the understanding that the empty least common multiple is ), and denotes the assertion that divides but is strictly less than . I am curious to know of this version of the Legendre sieve already appears in the literature (and similarly for the other applications of Proposition 2 given here).

If the poset has bounded depth then the number of terms in Proposition 3 can end up being just polynomially large in rather than exponentially large. Indeed, if all chains in have length at most then the number of terms here is at most . (The examples (4), (5) are ones in which the depth is equal to two.) I hope to report in a later post on how this version of inclusion-exclusion with polynomially many terms can be useful in an application.

Actually in our application we need an abstraction of the above formula, in which the indicator functions are replaced by more abstract idempotents:

Proposition 6 (Hall-type inclusion-exclusion principle for idempotents) Let be pairwise commuting elements of some ring with identity, which are all idempotent (thus for ). Let be the finite poset formed by products of the (with the convention that is the empty product), ordered by declaring when (note that all the elements of are idempotent so this is a partial ordering). Then for any , one has

where are understood to range in . In particular (setting ) if all the are not equal to then we have

Morally speaking this proposition is equivalent to the previous one after applying a “spectral theorem” to simultaneously diagonalise all of the , but it is quicker to just adapt the previous proof to establish this proposition directly. Using the Möbius function for , we can rewrite these formulae as

and

Proof: Again it suffices to verify (6). Using Proposition 2 as before, it suffices to show that

for all (all sums and products are understood to range in ). We can expand

where ranges over all subsets of that contain . For such an , if we write , then is the greatest lower bound of , and we observe that vanishes whenever fails to contain some with . Thus the only that give non-zero contributions to (8) are the intervals of the form for some (which then forms the greatest lower bound for that interval), and the claim (7) follows (after noting that for any ).

Job advertisement: Director of UCLA Endowed Olga Radko Math Circle (ORMC)

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The Academic Administrator is required to teach three mathematics courses in the undergraduate curriculum per academic year as assigned by the Department. This is also intended to help with the recruitment of UCLA students as docents and instructors for the ORMC.

As the director of ORMC, the Academic Administrator will have primary responsibility for all aspects of ORMC operations:

• Determining the structure of ORMC, including the number and levels of groups
• Recruiting, training and supervising instructors, docents, and postdoctoral fellows associated with the ORMC
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• Working with the Mathematics Department finance office to ensure timely payment of stipends and wages to ORMC instructors and docents, as appropriate
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246B, Notes 4: The Riemann zeta function and the prime number theorem

Previous set of notes: Notes 3. Next set of notes: 246C Notes 1.

One of the great classical triumphs of complex analysis was in providing the first complete proof (by Hadamard and de la Vallée Poussin in 1896) of arguably the most important theorem in analytic number theory, the prime number theorem:

Theorem 1 (Prime number theorem) Let denote the number of primes less than a given real number . Then

(or in asymptotic notation, as ).

(Actually, it turns out to be slightly more natural to replace the approximation in the prime number theorem by the logarithmic integral , which turns out to be a more precise approximation, but we will not stress this point here.)

The complex-analytic proof of this theorem hinges on the study of a key meromorphic function related to the prime numbers, the Riemann zeta function . Initially, it is only defined on the half-plane :

Definition 2 (Riemann zeta function, preliminary definition) Let be such that . Then we define

Note that the series is locally uniformly convergent in the half-plane , so in particular is holomorphic on this region. In previous notes we have already evaluated some special values of this function:

However, it turns out that the zeroes (and pole) of this function are of far greater importance to analytic number theory, particularly with regards to the study of the prime numbers.

The Riemann zeta function has several remarkable properties, some of which we summarise here:

Theorem 3 (Basic properties of the Riemann zeta function)

• (i) (Euler product formula) For any with , we have

  

  where the product is absolutely convergent (and locally uniform in ) and is over the prime numbers .
• (ii) (Trivial zero-free region) has no zeroes in the region .
• (iii) (Meromorphic continuation) has a unique meromorphic continuation to the complex plane (which by abuse of notation we also call ), with a simple pole at and no other poles. Furthermore, the Riemann xi function

  

  is an entire function of order (after removing all singularities). The function is an entire function of order one after removing the singularity at .
• (iv) (Functional equation) After applying the meromorphic continuation from (iii), we have

  

  for all (excluding poles). Equivalently, we have

  

  for all . (The equivalence between the (5) and (6) is a routine consequence of the Euler reflection formula and the Legendre duplication formula, see Exercises 26 and 31 of Notes 1.)

Proof: We just prove (i) and (ii) for now, leaving (iii) and (iv) for later sections.

The claim (i) is an encoding of the fundamental theorem of arithmetic, which asserts that every natural number is uniquely representable as a product over primes, where the are natural numbers, all but finitely many of which are zero. Writing this representation as , we see that

whenever , , and consists of all the natural numbers of the form for some . Sending and to infinity, we conclude from monotone convergence and the geometric series formula that

whenever is real, and then from dominated convergence we see that the same formula holds for complex with as well. Local uniform convergence then follows from the product form of the Weierstrass -test (Exercise 19 of Notes 1).

The claim (ii) is immediate from (i) since the Euler product is absolutely convergent and all terms are non-zero.

We remark that by sending to in Theorem 3(i) we conclude that

and from the divergence of the harmonic series we then conclude Euler’s theorem . This can be viewed as a weak version of the prime number theorem, and already illustrates the potential applicability of the Riemann zeta function to control the distribution of the prime numbers.

The meromorphic continuation (iii) of the zeta function is initially surprising, but can be interpreted either as a manifestation of the extremely regular spacing of the natural numbers occurring in the sum (1), or as a consequence of various integral representations of (or slight modifications thereof). We will focus in this set of notes on a particular representation of as essentially the Mellin transform of the theta function that briefly appeared in previous notes, and the functional equation (iv) can then be viewed as a consequence of the modularity of that theta function. This in turn was established using the Poisson summation formula, so one can view the functional equation as ultimately being a manifestation of Poisson summation. (For a direct proof of the functional equation via Poisson summation, see these notes.)

Henceforth we work with the meromorphic continuation of . The functional equation (iv), when combined with special values of such as (2), gives some additional values of outside of its initial domain , most famously

If one formally compares this formula with (1), one arrives at the infamous identity

although this identity has to be interpreted in a suitable non-classical sense in order for it to be rigorous (see this previous blog post for further discussion).

From Theorem 3 and the non-vanishing nature of , we see that has simple zeroes (known as trivial zeroes) at the negative even integers , and all other zeroes (the non-trivial zeroes) inside the critical strip . (The non-trivial zeroes are conjectured to all be simple, but this is hopelessly far from being proven at present.) As we shall see shortly, these latter zeroes turn out to be closely related to the distribution of the primes. The functional equation tells us that if is a non-trivial zero then so is ; also, we have the identity

for all by (1), hence for all (except the pole at ) by meromorphic continuation. Thus if is a non-trivial zero then so is . We conclude that the set of non-trivial zeroes is symmetric by reflection by both the real axis and the critical line . We have the following infamous conjecture:

Conjecture 4 (Riemann hypothesis) All the non-trivial zeroes of lie on the critical line .

This conjecture would have many implications in analytic number theory, particularly with regard to the distribution of the primes. Of course, it is far from proven at present, but the partial results we have towards this conjecture are still sufficient to establish results such as the prime number theorem.

Return now to the original region where . To take more advantage of the Euler product formula (3), we take complex logarithms to conclude that

for suitable branches of the complex logarithm, and then on taking derivatives (using for instance the generalised Cauchy integral formula and Fubini’s theorem to justify the interchange of summation and derivative) we see that

From the geometric series formula we have

and so (by another application of Fubini’s theorem) we have the identity

for , where the von Mangoldt function is defined to equal whenever is a power of a prime for some , and otherwise. The contribution of the higher prime powers is negligible in practice, and as a first approximation one can think of the von Mangoldt function as the indicator function of the primes, weighted by the logarithm function.

The series and that show up in the above formulae are examples of Dirichlet series, which are a convenient device to transform various sequences of arithmetic interest into holomorphic or meromorphic functions. Here are some more examples:

Exercise 5 (Standard Dirichlet series) Let be a complex number with .

• (i) Show that .
• (ii) Show that , where is the divisor function of (the number of divisors of ).
• (iii) Show that , where is the Möbius function, defined to equal when is the product of distinct primes for some , and otherwise.
• (iv) Show that , where is the Liouville function, defined to equal when is the product of (not necessarily distinct) primes for some .
• (v) Show that , where is the holomorphic branch of the logarithm that is real for , and with the convention that vanishes for .
• (vi) Use the fundamental theorem of arithmetic to show that the von Mangoldt function is the unique function such that

  

  for every positive integer . Use this and (i) to provide an alternate proof of the identity (8). Thus we see that (8) is really just another encoding of the fundamental theorem of arithmetic.

Given the appearance of the von Mangoldt function , it is natural to reformulate the prime number theorem in terms of this function:

Theorem 6 (Prime number theorem, von Mangoldt form) One has

(or in asymptotic notation, as ).

Let us see how Theorem 6 implies Theorem 1. Firstly, for any , we can write

The sum is non-zero for only values of , and is of size , thus

Since , we conclude from Theorem 6 that

as . Next, observe from the fundamental theorem of calculus that

Multiplying by and summing over all primes , we conclude that

From Theorem 6 we certainly have , thus

By splitting the integral into the ranges and we see that the right-hand side is , and Theorem 1 follows.

Exercise 7 Show that Theorem 1 conversely implies Theorem 6.

The alternate form (8) of the Euler product identity connects the primes (represented here via proxy by the von Mangoldt function) with the logarithmic derivative of the zeta function, and can be used as a starting point for describing further relationships between and the primes. Most famously, we shall see later in these notes that it leads to the remarkably precise Riemann-von Mangoldt explicit formula:

Theorem 8 (Riemann-von Mangoldt explicit formula) For any non-integer , we have

where ranges over the non-trivial zeroes of with imaginary part in . Furthermore, the convergence of the limit is locally uniform in .

Actually, it turns out that this formula is in some sense too precise; in applications it is often more convenient to work with smoothed variants of this formula in which the sum on the left-hand side is smoothed out, but the contribution of zeroes with large imaginary part is damped; see Exercise 22. Nevertheless, this formula clearly illustrates how the non-trivial zeroes of the zeta function influence the primes. Indeed, if one formally differentiates the above formula in , one is led to the (quite nonrigorous) approximation

or (writing )

Thus we see that each zero induces an oscillation in the von Mangoldt function, with controlling the frequency of the oscillation and the rate to which the oscillation dies out as . This relationship is sometimes known informally as “the music of the primes”.

Comparing Theorem 8 with Theorem 6, it is natural to suspect that the key step in the proof of the latter is to establish the following slight but important extension of Theorem 3(ii), which can be viewed as a very small step towards the Riemann hypothesis:

Theorem 9 (Slight enlargement of zero-free region) There are no zeroes of on the line .

It is not quite immediate to see how Theorem 6 follows from Theorem 8 and Theorem 9, but we will demonstrate it below the fold.

Although Theorem 9 only seems like a slight improvement of Theorem 3(ii), proving it is surprisingly non-trivial. The basic idea is the following: if there was a zero at , then there would also be a different zero at (note cannot vanish due to the pole at ), and then the approximation (9) becomes

But the expression can be negative for large regions of the variable , whereas is always non-negative. This conflict eventually leads to a contradiction, but it is not immediately obvious how to make this argument rigorous. We will present here the classical approach to doing so using a trigonometric identity of Mertens.

In fact, Theorem 9 is basically equivalent to the prime number theorem:

Exercise 10 For the purposes of this exercise, assume Theorem 6, but do not assume Theorem 9. For any non-zero real , show that

as , where denotes a quantity that goes to zero as after being multiplied by . Use this to derive Theorem 9.

This equivalence can help explain why the prime number theorem is remarkably non-trivial to prove, and why the Riemann zeta function has to be either explicitly or implicitly involved in the proof.

This post is only intended as the briefest of introduction to complex-analytic methods in analytic number theory; also, we have not chosen the shortest route to the prime number theorem, electing instead to travel in directions that particularly showcase the complex-analytic results introduced in this course. For some further discussion see this previous set of lecture notes, particularly Notes 2 and Supplement 3 (with much of the material in this post drawn from the latter).

A statement from mathematics department faculty at Stanford and MIT

[The following statement is signed by several mathematicians at Stanford and MIT in support of one of their recently admitted graduate students, and I am happy to post it here on my blog. -T]

We were saddened and horrified to learn that Ilya Dumanski, a brilliant young mathematician who has been admitted to our graduate programs at Stanford and MIT, has been imprisoned in Russia, along with several other mathematicians, for participation in a peaceful demonstration. Our thoughts are with them. We urge their rapid release, and failing that, that they be kept in humane conditions. A petition in their support has been started at

https://www.ipetitions.com/petition/a-call-for-immediate-release-of-arrested-students/

Signed,

Roman Bezrukavnikov (MIT)
Alexei Borodin (MIT)
Daniel Bump (Stanford)
Sourav Chatterjee (Stanford)
Otis Chodosh (Stanford)
Ralph Cohen (Stanford)
Henry Cohn (MIT)
Brian Conrad (Stanford)
Joern Dunkel (MIT)
Pavel Etingof (MIT)
Jacob Fox (Stanford)
Michel Goemans (MIT)
Eleny Ionel (Stanford)
Steven Kerckhoff (Stanford)
Jonathan Luk (Stanford)
Eugenia Malinnikova (Stanford)
Davesh Maulik (MIT)
Rafe Mazzeo (Stanford)
Haynes Miller (MIT)
Ankur Moitra (MIT)
Elchanan Mossel (MIT)
Tomasz Mrowka (MIT)
Bjorn Poonen (MIT)
Alex Postnikov (MIT)
Lenya Ryzhik (Stanford)
Paul Seidel (MIT)
Mike Sipser (MIT)
Kannan Soundararajan (Stanford)
Gigliola Staffilani (MIT)
Nike Sun (MIT)
Richard Taylor (Stanford)
Ravi Vakil (Stanford)
Andras Vasy (Stanford)
Jan Vondrak (Stanford)
Brian White (Stanford)
Zhiwei Yun (MIT)

246B, Notes 3: Elliptic functions and modular forms

Previous set of notes: Notes 2. Next set of notes: Notes 4.

On the real line, the quintessential examples of a periodic function are the (normalised) sine and cosine functions , , which are -periodic in the sense that

By taking various polynomial combinations of and we obtain more general trigonometric polynomials that are -periodic; and the theory of Fourier series tells us that all other -periodic functions (with reasonable integrability conditions) can be approximated in various senses by such polynomial combinations. Using Euler’s identity, one can use and in place of and as the basic generating functions here, provided of course one is willing to use complex coefficients instead of real ones. Of course, by rescaling one can also make similar statements for other periods than . -periodic functions can also be identified (by abuse of notation) with functions on the quotient space (known as the additive -torus or additive unit circle), or with functions on the fundamental domain (up to boundary) of that quotient space with the periodic boundary condition . The map also identifies the additive unit circle with the geometric unit circle , thanks in large part to the fundamental trigonometric identity ; this can also be identified with the multiplicative unit circle . (Usually by abuse of notation we refer to all of these three sets simultaneously as the “unit circle”.) Trigonometric polynomials on the additive unit circle then correspond to ordinary polynomials of the real coefficients of the geometric unit circle, or Laurent polynomials of the complex variable .

What about periodic functions on the complex plane? We can start with singly periodic functions which obey a periodicity relationship for all in the domain and some period ; such functions can also be viewed as functions on the “additive cylinder” (or equivalently ). We can rescale as before. For holomorphic functions, we have the following characterisations:

Proposition 1 (Description of singly periodic holomorphic functions)

• (i) Every -periodic entire function has an absolutely convergent expansion

  

  where is the nome , and the are complex coefficients such that

  

  Conversely, every doubly infinite sequence of coefficients obeying (2) gives rise to a -periodic entire function via the formula (1).
• (ii) Every bounded -periodic holomorphic function on the upper half-plane has an expansion

  

  where the are complex coefficients such that

  

  Conversely, every infinite sequence obeying (4) gives rise to a -periodic holomorphic function which is bounded away from the real axis (i.e., bounded on for every ).

In both cases, the coefficients can be recovered from by the Fourier inversion formula

for any in (in case (i)) or (in case (ii)).

Proof: If is -periodic, then it can be expressed as for some function on the “multiplicative cylinder” , since the fibres of the map are cosets of the integers , on which is constant by hypothesis. As the map is a covering map from to , we see that will be holomorphic if and only if is. Thus must have a Laurent series expansion with coefficients obeying (2), which gives (1), and the inversion formula (5) follows from the usual contour integration formula for Laurent series coefficients. The converse direction to (i) also follows by reversing the above arguments.

For part (ii), we observe that the map is also a covering map from to the punctured disk , so we can argue as before except that now is a bounded holomorphic function on the punctured disk. By the Riemann singularity removal theorem (Exercise 35 of 246A Notes 3) extends to be holomorphic on all of , and thus has a Taylor expansion for some coefficients obeying (4). The argument now proceeds as with part (i).

The additive cylinder and the multiplicative cylinder can both be identified (on the level of smooth manifolds, at least) with the geometric cylinder , but we will not use this identification here.

Now let us turn attention to doubly periodic functions of a complex variable , that is to say functions that obey two periodicity relations

for all and some periods , which to avoid degeneracies we will assume to be linearly independent over the reals (thus are non-zero and the ratio is not real). One can rescale by a common scaling factor to normalise either or , but one of course cannot simultaneously normalise both parameters in this fashion. As in the singly periodic case, such functions can also be identified with functions on the additive -torus , where is the lattice , or with functions on the solid parallelogram bounded by the contour (a fundamental domain up to boundary for that torus), obeying the boundary periodicity conditions

for in the edge , and

for in the edge .

Within the world of holomorphic functions, the collection of doubly periodic functions is boring:

Proposition 2 Let be an entire doubly periodic function (with periods linearly independent over ). Then is constant.

In the language of Riemann surfaces, this proposition asserts that the torus is a non-hyperbolic Riemann surface; it cannot be holomorphically mapped non-trivially into a bounded subset of the complex plane.

Proof: The fundamental domain (up to boundary) enclosed by is compact, hence is bounded on this domain, hence bounded on all of by double periodicity. The claim now follows from Liouville’s theorem. (One could alternatively have argued here using the compactness of the torus .

To obtain more interesting examples of doubly periodic functions, one must therefore turn to the world of meromorphic functions – or equivalently, holomorphic functions into the Riemann sphere . As it turns out, a particularly fundamental example of such a function is the Weierstrass elliptic function

which plays a role in doubly periodic functions analogous to the role of for -periodic real functions. This function will have a double pole at the origin , and more generally at all other points on the lattice , but no other poles. The derivative

of the Weierstrass function is another doubly periodic meromorphic function, now with a triple pole at every point of , and plays a role analogous to . Remarkably, all the other doubly periodic meromorphic functions with these periods will turn out to be rational combinations of and ; furthermore, in analogy with the identity , one has an identity of the form

for all (avoiding poles) and some complex numbers that depend on the lattice . Indeed, much as the map creates a diffeomorphism between the additive unit circle to the geometric unit circle , the map turns out to be a complex diffeomorphism between the torus and the elliptic curve

with the convention that maps the origin of the torus to the point at infinity. (Indeed, one can view elliptic curves as “multiplicative tori”, and both the additive and multiplicative tori can be identified as smooth manifolds with the more familiar geometric torus, but we will not use such an identification here.) This fundamental identification with elliptic curves and tori motivates many of the further remarkable properties of elliptic curves; for instance, the fact that tori are obviously an abelian group gives rise to an abelian group law on elliptic curves (and this law can be interpreted as an analogue of the trigonometric sum identities for ). The description of the various meromorphic functions on the torus also helps motivate the more general Riemann-Roch theorem that is a fundamental law governing meromorphic functions on other compact Riemann surfaces (and is discussed further in these 246C notes). So far we have focused on studying a single torus . However, another important mathematical object of study is the space of all such tori, modulo isomorphism; this is a basic example of a moduli space, known as the (classical, level one) modular curve . This curve can be described in a number of ways. On the one hand, it can be viewed as the upper half-plane quotiented out by the discrete group ; on the other hand, by using the -invariant, it can be identified with the complex plane ; alternatively, one can compactify the modular curve and identify this compactification with the Riemann sphere . (This identification, by the way, produces a very short proof of the little and great Picard theorems, which we proved in 246A Notes 4.) Functions on the modular curve (such as the -invariant) can be viewed as -invariant functions on , and include the important class of modular functions; they naturally generalise to the larger class of (weakly) modular forms, which are functions on which transform in a very specific way under -action, and which are ubiquitous throughout mathematics, and particularly in number theory. Basic examples of modular forms include the Eisenstein series, which are also the Laurent coefficients of the Weierstrass elliptic functions . More number theoretic examples of modular forms include (suitable powers of) theta functions , and the modular discriminant . Modular forms are -periodic functions on the half-plane, and hence by Proposition 1 come with Fourier coefficients ; these coefficients often turn out to encode a surprising amount of number-theoretic information; a dramatic example of this is the famous modularity theorem, (a special case of which was) used amongst other things to establish Fermat’s last theorem. Modular forms can be generalised to other discrete groups than (such as congruence groups) and to other domains than the half-plane , leading to the important larger class of automorphic forms, which are of major importance in number theory and representation theory, but which are well outside the scope of this course to discuss.

246B, Notes 2: Some connections with the Fourier transform

Previous set of notes: Notes 1. Next set of notes: Notes 3.

In Exercise 5 (and Lemma 1) of 246A Notes 4 we already observed some links between complex analysis on the disk (or annulus) and Fourier series on the unit circle:

• (i) Functions that are holomorphic on a disk are expressed by a convergent Fourier series (and also Taylor series) for (so in particular ), where

  

  conversely, every infinite sequence of coefficients obeying (1) arises from such a function .
• (ii) Functions that are holomorphic on an annulus are expressed by a convergent Fourier series (and also Laurent series) , where

  

  conversely, every doubly infinite sequence of coefficients obeying (2) arises from such a function .
• (iii) In the situation of (ii), there is a unique decomposition where extends holomorphically to , and extends holomorphically to and goes to zero at infinity, and are given by the formulae

  

  where is any anticlockwise contour in enclosing , and and

  

  where is any anticlockwise contour in enclosing but not .

This connection lets us interpret various facts about Fourier series through the lens of complex analysis, at least for some special classes of Fourier series. For instance, the Fourier inversion formula becomes the Cauchy-type formula for the Laurent or Taylor coefficients of , in the event that the coefficients are doubly infinite and obey (2) for some , or singly infinite and obey (1) for some .

It turns out that there are similar links between complex analysis on a half-plane (or strip) and Fourier integrals on the real line, which we will explore in these notes.

We first fix a normalisation for the Fourier transform. If is an absolutely integrable function on the real line, we define its Fourier transform by the formula

From the dominated convergence theorem will be a bounded continuous function; from the Riemann-Lebesgue lemma it also decays to zero as . My choice to place the in the exponent is a personal preference (it is slightly more convenient for some harmonic analysis formulae such as the identities (4), (5), (6) below), though in the complex analysis and PDE literature there are also some slight advantages in omitting this factor. In any event it is not difficult to adapt the discussion in this notes for other choices of normalisation. It is of interest to extend the Fourier transform beyond the class into other function spaces, such as or the space of tempered distributions, but we will not pursue this direction here; see for instance these lecture notes of mine for a treatment.

Exercise 1 (Fourier transform of Gaussian) If is a coplex number with and is the Gaussian function , show that the Fourier transform is given by the Gaussian , where we use the standard branch for .

The Fourier transform has many remarkable properties. On the one hand, as long as the function is sufficiently “reasonable”, the Fourier transform enjoys a number of very useful identities, such as the Fourier inversion formula

the Plancherel identity

and the Poisson summation formula

On the other hand, the Fourier transform also intertwines various qualitative properties of a function with “dual” qualitative properties of its Fourier transform ; in particular, “decay” properties of tend to be associated with “regularity” properties of , and vice versa. For instance, the Fourier transform of rapidly decreasing functions tend to be smooth. There are complex analysis counterparts of this Fourier dictionary, in which “decay” properties are described in terms of exponentially decaying pointwise bounds, and “regularity” properties are expressed using holomorphicity on various strips, half-planes, or the entire complex plane. The following exercise gives some examples of this:

Exercise 2 (Decay of implies regularity of ) Let be an absolutely integrable function.

• (i) If has super-exponential decay in the sense that for all and (that is to say one has for some finite quantity depending only on ), then extends uniquely to an entire function . Furthermore, this function continues to be defined by (3).
• (ii) If is supported on a compact interval then the entire function from (i) obeys the bounds for . In particular, if is supported in then .
• (iii) If obeys the bound for all and some , then extends uniquely to a holomorphic function on the horizontal strip , and obeys the bound in this strip. Furthermore, this function continues to be defined by (3).
• (iv) If is supported on (resp. ), then there is a unique continuous extension of to the lower half-plane (resp. the upper half-plane which is holomorphic in the interior of this half-plane, and such that uniformly as (resp. ). Furthermore, this function continues to be defined by (3).

Hint: to establish holomorphicity in each of these cases, use Morera’s theorem and the Fubini-Tonelli theorem. For uniqueness, use analytic continuation, or (for part (iv)) the Cauchy integral formula.

Later in these notes we will give a partial converse to part (ii) of this exercise, known as the Paley-Wiener theorem; there are also partial converses to the other parts of this exercise.

From (3) we observe the following intertwining property between multiplication by an exponential and complex translation: if is a complex number and is an absolutely integrable function such that the modulated function is also absolutely integrable, then we have the identity

whenever is a complex number such that at least one of the two sides of the equation in (7) is well defined. Thus, multiplication of a function by an exponential weight corresponds (formally, at least) to translation of its Fourier transform. By using contour shifting, we will also obtain a dual relationship: under suitable holomorphicity and decay conditions on , translation by a complex shift will correspond to multiplication of the Fourier transform by an exponential weight. It turns out to be possible to exploit this property to derive many Fourier-analytic identities, such as the inversion formula (4) and the Poisson summation formula (6), which we do later in these notes. (The Plancherel theorem can also be established by complex analytic methods, but this requires a little more effort; see Exercise 8.)

The material in these notes is loosely adapted from Chapter 4 of Stein-Shakarchi’s “Complex Analysis”.

The effective potential of an M-matrix

Marcel Filoche, Svitlana Mayboroda, and I have just uploaded to the arXiv our preprint “The effective potential of an -matrix“. This paper explores the analogue of the effective potential of Schrödinger operators provided by the “landscape function” , when one works with a certain type of self-adjoint matrix known as an -matrix instead of a Schrödinger operator.

Suppose one has an eigenfunction

of a Schrödinger operator , where is the Laplacian on , is a potential, and is an energy. Where would one expect the eigenfunction to be concentrated? If the potential is smooth and slowly varying, the correspondence principle suggests that the eigenfunction should be mostly concentrated in the potential energy wells , with an exponentially decaying amount of tunnelling between the wells. One way to rigorously establish such an exponential decay is through an argument of Agmon, which we will sketch later in this post, which gives an exponentially decaying upper bound (in an sense) of eigenfunctions in terms of the distance to the wells in terms of a certain “Agmon metric” on determined by the potential and energy level (or any upper bound on this energy). Similar exponential decay results can also be obtained for discrete Schrödinger matrix models, in which the domain is replaced with a discrete set such as the lattice , and the Laplacian is replaced by a discrete analogue such as a graph Laplacian.

When the potential is very “rough”, as occurs for instance in the random potentials arising in the theory of Anderson localisation, the Agmon bounds, while still true, become very weak because the wells are dispersed in a fairly dense fashion throughout the domain , and the eigenfunction can tunnel relatively easily between different wells. However, as was first discovered in 2012 by my two coauthors, in these situations one can replace the rough potential by a smoother effective potential , with the eigenfunctions typically localised to a single connected component of the effective wells . In fact, a good choice of effective potential comes from locating the landscape function , which is the solution to the equation with reasonable behavior at infinity, and which is non-negative from the maximum principle, and then the reciprocal of this landscape function serves as an effective potential.

There are now several explanations for why this particular choice is a good effective potential. Perhaps the simplest (as found for instance in this recent paper of Arnold, David, Jerison, and my two coauthors) is the following observation: if is an eigenvector for with energy , then is an eigenvector for with the same energy , thus the original Schrödinger operator is conjugate to a (variable coefficient, but still in divergence form) Schrödinger operator with potential instead of . Closely related to this, we have the integration by parts identity

for any reasonable function , thus again highlighting the emergence of the effective potential .

These particular explanations seem rather specific to the Schrödinger equation (continuous or discrete); we have for instance not been able to find similar identities to explain an effective potential for the bi-Schrödinger operator .

In this paper, we demonstrate the (perhaps surprising) fact that effective potentials continue to exist for operators that bear very little resemblance to Schrödinger operators. Our chosen model is that of an -matrix: self-adjoint positive definite matrices whose off-diagonal entries are negative. This model includes discrete Schrödinger operators (with non-negative potentials) but can allow for significantly more non-local interactions. The analogue of the landscape function would then be the vector , where denotes the vector with all entries . Our main result, roughly speaking, asserts that an eigenvector of will then be exponentially localised to the “potential wells” , where denotes the coordinates of the landscape function . In particular, we establish the inequality

if is normalised in , where the connectivity is the maximum number of non-zero entries of in any row or column, are the coefficients of , and is a certain moderately complicated but explicit metric function on the spatial domain. Informally, this inequality asserts that the eigenfunction should decay like or faster. Indeed, our numerics show a very strong log-linear relationship between and , although it appears that our exponent is not quite optimal. We also provide an associated localisation result which is technical to state but very roughly asserts that a given eigenvector will in fact be localised to a single connected component of unless there is a resonance between two wells (by which we mean that an eigenvalue for a localisation of associated to one well is extremely close to an eigenvalue for a localisation of associated to another well); such localisation is also strongly supported by numerics. (Analogous results for Schrödinger operators had been previously obtained by the previously mentioned paper of Arnold, David, Jerison, and my two coauthors, and to quantum graphs in a very recent paper of Harrell and Maltsev.)

Our approach is based on Agmon’s methods, which we interpret as a double commutator method, and in particular relying on exploiting the negative definiteness of certain double commutator operators. In the case of Schrödinger operators , this negative definiteness is provided by the identity

for any sufficiently reasonable functions , where we view (like ) as a multiplier operator. To exploit this, we use the commutator identity

valid for any after a brief calculation. The double commutator identity then tells us that

If we choose to be a non-negative weight and let for an eigenfunction , then we can write

and we conclude that

We have considerable freedom in this inequality to select the functions . If we select , we obtain the clean inequality

If we take to be a function which equals on the wells but increases exponentially away from these wells, in such a way that

outside of the wells, we can obtain the estimate

which then gives an exponential type decay of away from the wells. This is basically the classic exponential decay estimate of Agmon; one can basically take to be the distance to the wells with respect to the Euclidean metric conformally weighted by a suitably normalised version of . If we instead select to be the landscape function , (3) then gives

and by selecting appropriately this gives an exponential decay estimate away from the effective wells , using a metric weighted by .

It turns out that this argument extends without much difficulty to the -matrix setting. The analogue of the crucial double commutator identity (2) is

for any diagonal matrix . The remainder of the Agmon type arguments go through after making the natural modifications.

Numerically we have also found some aspects of the landscape theory to persist beyond the -matrix setting, even though the double commutators cease being negative definite, so this may not yet be the end of the story, but it does at least demonstrate that utility the landscape does not purely rely on identities such as (1).