Marcel Filoche, Svitlana Mayboroda and I have just finished our preprint “The effective potential of a {M}-Matrix”. This article examines the analogue of the effective potential of Schrödinger operators {-  Delta + V} provided by the “Landscape function” {u}when working with a certain type of self-adjunct matrix called {M} Matrix instead of a Schrödinger operator.

Suppose you have an eigenfunction

 displaystyle (-  Delta + V)  phi = E  phi

of a Schrödinger operator {-  Delta + V}, Where {Delta} is the Laplace on {{ bf R} ^ d}, {V: { bf R} ^ d  rightarrow { bf R}} is a potential, and {E} is an energy. Where would one expect the eigenfunction { phi} to be focused? If the potential {V} is smooth and varies slowly, the principle of correspondence suggests that the eigenfunction { phi} should mainly focus on the potential energy sources { {x: V (x)  leq E }}with an exponentially decreasing amount of tunnels between boreholes. One way to rigorously establish such an exponential decay is an argument by Agmon, which we will outline later in this post, which results in an exponentially decreasing upper limit (in a {L ^ 2} Sense) of eigenfunctions { phi} in terms of distance to the fountain { {V  leq E }} in relation to a particular “Agmon metric” on {{ bf R} ^ d} determined by the potential {V} and energy level {E} (or any upper limit { overline {E}} on this energy). Similar exponential decay results can also be obtained for discrete Schrödinger matrix models in which the domain {{ bf R} ^ d} is replaced by a discrete set like the grid {{ bf Z} ^ d}and the Laplace {Delta} is replaced by a discrete analog such as a Laplace graph.

If the potential {V} is very “rough”such as the random potentials resulting from the theory of Anderson localization, the agmon boundaries, although still true, become very weak due to the depressions { {V  leq E }} are fairly densely distributed across the domain {{ bf R} ^ d}and the eigenfunction can tunnel between different wells with relative ease. As my two co-authors first discovered in 2012, you can replace the gross potential in these situations {V} from a smooth effective potential{1 / u}where the eigenfunctions are typically localized to a single connected component of the effective wells { {1 / u  leq E }}. Indeed, a good choice of effective potential comes from locating the Landscape function{u}, that is the solution to the equation {(-  Delta + V) u = 1} with reasonable behavior in infinity, and that is not negative from the maximum principle, and then the opposite {1 / u} this landscape function serves as an effective potential.

There are now several explanations as to why this particular choice {1 / u} is a good effective potential. Perhaps the simplest (such as in this recent article by Arnold, David, Jerison, and my two co-authors) is the following observation: if { phi} is an eigenvector for {-  Delta + V} with energy {E}, then { phi / u} is an eigenvector for {-  frac {1} {u ^ 2}  mathrm {div} (u ^ 2  nabla  cdot) +  frac {1} {u}} with the same energy {E}, so the original Schrödinger operator {-  Delta + V} is conjugated with a (variable coefficient, but still in divergence form) Schrödinger operator with potential {1 / u} Instead of {V}. Closely related to this, we have the integration according to part identity

 displaystyle  int _ {{ bf R} ^ d} |   nabla f |  ^ 2 + V |  f |  ^ 2  dx =  int _ {{ bf R} ^ d} u ^ 2 |   nabla (f / u) |  ^ 2 +  frac {1} {u} |  f |  ^ 2  dx      (1)

for any reasonable function {f}This again underlines the emergence of the effective potential {1 / u} .

These particular explanations seem to be quite specific to the Schrödinger equation (continuous or discrete); For example, we couldn’t find any similar identities to explain an effective potential for the Bi-Schrödinger operator { Delta ^ 2 + V} .

In this article we show the (perhaps surprising) fact that there is still effective potential for operators who have very little resemblance to Schrödinger operators. Our chosen model is the one {M}-Matrix: self-adjunct positive definitive matrices {A} whose off-diagonal entries are negative. This model includes discrete Schrödinger operators (with non-negative potentials), but can enable significantly more non-local interactions. The analogue of the landscape function would then be the vector {u: = A ^ {- 1} 1}, Where {1} denotes the vector with all entries {1}. Our main result is, roughly speaking, that it is an eigenvector {A  phi = E  phi} of {A} is then exponentially in the “potential wells” localized. {K: =  {j:  frac {1} {u_j}  leq E }}, Where {u_j} denotes the coordinates of the landscape function {u}. In particular, we note the inequality

 displaystyle  sum_k  phi_k ^ 2 e ^ {2  rho (k, K) /  sqrt {W}} ( frac {1} {u_k} - E) _ +  leq W  max_ {i, j} |  a_ {ij} |

if { phi} is normalized in { ell ^ 2}where the Connectivity{W} is the maximum number of non-zero entries of {A} in every row or column, {a_ {ij}} are the coefficients of {A}, and { rho} is a certain moderately complicated but explicit metric function in the spatial domain. Informally, this inequality claims that the eigenfunction { phi_k} should like to expire {e ^ {-  rho (k, K) /  sqrt {W}}} or faster. In fact, our numbers show a very strong log-linear relationship between { phi_k} and { rho (k, K)}although it seems that our exponent {1 /  sqrt {W}} is not quite optimal. We also provide an associated localization result, which can be technically described, but very roughly asserts that a given eigenvector is actually localized to a single connected component of {K} unless there is a resonance between two depressions (by which we mean that an eigenvalue for a localization of {A} A deepening is extremely close to an eigenvalue for a localization {A} connected to another well); Such a localization is also strongly supported by numerics. (Analogous results for Schrödinger operators were previously obtained from the aforementioned paper by Arnold, David, Jerison, and my two co-authors, and from Quantum Graphs in a recent paper by Harrell and Maltsev.)

Our approach is based on Agmon’s methods, which we interpret as the double commutator method, and in particular on the use of the negative determinateness of certain double commutator operators. In the case of Schrödinger operators {-  Delta + V}This negative certainty is provided by identity

 displaystyle  langle [[-Delta+V,g], g]u, u  rangle = -2  int _ {{ bf R} ^ d} |   nabla g |  ^ 2 |  u |  ^ 2  dx  leq 0      (2)

for sufficiently reasonable functions {u, g: { bf R} ^ d  rightarrow { bf R}}where we see {G} (to like {V}) as a multiplier operator. To take advantage of this, we’ll use commutator identity

 displaystyle  langle g [psi, -Delta+V] u, g  psi u  rangle =  frac {1} {2}  langle [[-Delta+V, g psi], g  psi]u, u  rangle

 displaystyle -  frac {1} {2}  langle [[-Delta+V, g], g] psi u,  psi u  rangle

valid for everyone {g,  psi, u: { bf R} ^ d  rightarrow { bf R}} after a short calculation. The double commutator identity then tells us that

 displaystyle  langle g [psi, -Delta+V] u, g  psi u  rangle  leq  int _ {{ bf R} ^ d} |   nabla g |  ^ 2 |   psi u |  ^ 2  dx.

If we choose {u} be and let be a non-negative weight { psi: =  phi / u} for an eigenfunction { phi}, then we can write

 displaystyle  [psi, -Delta+V] u = [psi, -Delta+V - E] u =  psi (-  Delta + V - E) u

and we conclude from it

 displaystyle  int _ {{ bf R} ^ d}  frac {(-  Delta + VE) u} {u} |  g |  ^ 2 |   phi |  ^ 2  dx  leq  int _ {{ bf R.} ^ d} |   nabla g |  ^ 2 |   phi |  ^ 2  dx.       (3)

We have considerable freedom in this inequality to choose the functions {u, g}. If we choose {u = 1}we get the clean inequality

 displaystyle  int _ {{ bf R} ^ d} (VE) |  g |  ^ 2 |   phi |  ^ 2  dx  leq  int _ {{ bf R} ^ d} |   nabla g |  ^ 2 |   phi |  ^ 2  dx.

If we take {G} be a function that is the same {1} on the fountain { {V  leq E }} but rises exponentially away from these wells, so that

 displaystyle |   nabla g |  ^ 2  leq  frac {1} {2} (VE) |  g |  ^ 2

Outside the well we can get the estimate

 displaystyle  int_ {V> E} (VE) |  g |  ^ 2 |   phi |  ^ 2  dx  leq 2  int_ {V <E} (EV) |  phi | ^ 2  dx,

which then gives an exponential type decay of { phi} away from the fountain. This is basically the classic exponential decay estimate of agmon; you can basically take {G} the distance to the wells { {V  leq E }} in terms of the Euclidean metric weighted by an appropriately normalized version of conformal {VE}. If we choose instead {u} be the landscape function {u = (-  Delta + V) ^ {- 1} 1}, (3) then gives

 displaystyle  int _ {{ bf R} ^ d} ( frac {1} {u} - E) |  g |  ^ 2 |   phi |  ^ 2  dx  leq  int _ {{ bf R} ^ d} |   nabla g |  ^ 2 |   phi |  ^ 2  dx,

and by selection {G} Suitably, an exponential decay estimate outside of the effective wells results { { frac {1} {u}  leq E }}using a weighted metric { frac {1} {u} -E}.

It turns out that this argument can be applied without much difficulty to the {M}-Matrix setting. The analog of the crucial double commutator identity is (2)

 displaystyle  langle [[A,D], D]u, u  rangle =  sum_ {i  neq j} a_ {ij} u_i u_j (d_ {ii} - d_ {jj}) ^ 2  leq 0

for each diagonal matrix {D =  mathrm {diag} (d_ {11},  dots, d_ {NN})}. The rest of the agmon-type arguments are looped through after the natural changes.

We also found numerically that some aspects of landscape theory persist beyond that {M} matrix setting, although the double commutators are no longer negative definitive, so this may not be the end of the story, but it at least shows that the utility of the landscape isn’t just based on identities like (1).

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