Wavefunctions from Energies: Applications in simple potentials.

Recently DentonRosetta:19 , precise perturbative calculations of neutrino oscillation probabilities yield a surprising result: given the eigenvalues of the Hamiltonian and its minor sub–matrices, it is possible to write down all the probabilities by a quite simple expression. In a following two–pages paper the authors Denton:19 described and proved this eigenvector–eigenvalue identity (1), which is the subject of the present work. Despite its simplicity and the possible implications, the identity was not broadly well known. Lately, it achieves a blaze of publicity, after a popular article wolcholver:19 reported the unexpected relationship between the eigenvalues and eigenvectors in the neutrino oscillation calculations. The authors of Denton:19 rewrote the paper, adding thirty pages with several proofs of the identity, together with a discussion about its complicated history in the literature, and the reasons for its unawareness. They also showed that the equation appeared in different references, being independently rediscovered in diverse fields. In most cases, the papers were weakly connected, lacking in significant propagation of citations.

The eigenvector–eigenvalue identity Denton:19 relates the eigenvectors $v_{i}$ of an $n\times n$ Hermitian matrix $H$, with its eigenvalues $\lambda_{1},\lambda_{2},\ldots,\lambda_{n}$. In this work, it is slightly modified as follows:

where $v_{ji}$ is the $j^{\mathrm{t}h}$ component of the normalized eigenvector $v_{i}$ associated to the eigenvalue $\lambda_{i}$. The eigenvalues $\lambda_{1}^{(j)},\lambda_{2}^{(j)},\cdots,\lambda_{n-1}^{(j)}$ correspond to the minor $M_{j}$ of $H$, formed by removing the $j^{\mathrm{t}h}$ row and column.

Identity (1) has many remarkable aspects to discuss. We can mention many positive and very promising issues. First, notice that at no point one ever actually needs to know any of the entries of the matrix to calculate its eigenvectors. From the physical point of view, this can be very useful in systems in which the eigenvalues are got by measurements. The most direct application could be the recovering of the electronic densities from the experimental spectra, which is, perhaps, the holy grail of the physical chemistry. Second, it can lead to the development of new numerical methods to diagonalize large matrices, faster and more efficiently. The standard computer routines use much more memory and resources if the eigenvectors are needed, in place of solely the eigenvalues.

However, these two advantages have some weaknesses. Indeed, the values of the original matrix $H$ are not needed, but since the expression (1) involves the minors $M_{j}$, this statement is not completely straight. It will be undisputed, only if the minor’s eigenvalues $\lambda_{i}^{(j)}$ are available, without the explicit knowledge of the matrix elements. From the numerical point of view, the new algorithms do not seem to be competitive, unless for very large matrices. The reason is that for every eigenvector $v_{i}$, all the eigenvalues $\lambda_{k}^{(j)}$ of each minor $M_{j}$, are needed. Thus, the algorithm requires to obtain all the eigenvalues of $n-1$ different matrices, which will result in a much more costly way to solve an eigenproblem.

In the present work, all these aspects are examined. First, we prove the eigenvector–eigenvalue identity for the solution of the corresponding Schrödinger equation for simple one–dimensional potentials. The wavefunctions obtained by direct diagonalization, and by using the expression (1) are compared for harmonic oscillators, Coulombic potentials, and the infinite potential well. Then, the spectra of the minor matrices are analyzed. We found that these eigenvalues are not randomly scattered, else they behave systematically, forming a pattern that could be described analytically. Finally, a deep analysis of the minor eigenvalues $\lambda^{(j)}$ for the simplest potential (the infinite potential well), lead us to understand the origin of this singular behavior, which could be useful for the calculation of eigenvectors of other potentials.

To corroborate the expression (1) in physical problems, we will validate it first, in simple potential problems, solving the one–dimensional Schrödinger equation

where

We choose the most commonly studied simple potentials, say, the harmonic oscillator

the Coulomb potential

and the infinite potential well

For each potential, the corresponding Schrödinger equation was approximated in first–order finite differences, resulting in tridiagonal matrices. First, the full Hamiltonians have been diagonalized, by using standard computational packages lapack , getting the eigenvectors $v$ and eigenvalues $\lambda$. Next, we successively construct the minor matrices $M_{j}$, deleting from the full Hamiltonian matrix the $j^{\mathrm{th}}$ row and column. In that way, the corresponding $\lambda^{(j)}$ eigenvalues have been obtained. Finally, Eq. (1) is used to get the “reconstructed” eigenvalues. The operations involved in this rebuilding require some care. The order of the operations is important, and also, it is better to add logarithms than to multiply terms having different orders of magnitudes. One of the principal numerical advantages of the proposed approach resides in the fact that the matrices are not needed, else, only their eigenvalues. Therefore, we also take care to exploit that, performing the calculation in a particular sequence, in which only one minor eigenvalues are stored in memory, at any step.

For all these potentials, and for many different Hamiltonian sizes, the reconstructed results agree with the original eigenvectors within machine precision. Thus, the first conclusion reached in the present work is that for any number $n$ of points used in the numerical grid, the eigenvalue–eigenvector identity can be considered numerically exact.

Of course, there is a drawback in the expression (1) and it is the requirement of the calculation of the eigenvalues for $n$ different matrices, which, in general, takes more computational efforts than the direct diagonalization of an $n\times n$ matrix. The algorithm can be improved significantly if one can found some relationship between the original eigenvalues $\lambda_{i}$ and the corresponding minor eigenvalues $\lambda_{i}^{(j)}$. To this end, we first plot these eigenvalues, to see if they follow some identifiable behavior. We show, in Figure 1, the $\lambda_{i}^{(j)}$ values corresponding to an infinite potential well $V_{\mathrm{W}}(x)$ (Eq. (6)) of size $L=20$ a.u. and amplitude $V_{0}=0$, calculated with a numerical grid having $n=33$ points. From this figure, we can not get any useful information, because of the similarities among the $\lambda_{i}^{(j)}$ eigenvalues, for any minor matrix $j$.

For a better representation, we define and plot, in Figure 2 the differences $D(i,j)\equiv\lambda_{i}-\lambda_{i}^{(j)}$, which could be helpful to understand the meaning of the minor’s eigenvalues.

These differences have been calculated for three different potentials, an infinite potential well, a Coulomb potential, and a harmonic oscillator potential. An unexpected regular pattern appears in the three cases, encouraging the search for analytical expressions capable to represent these values.

Although the curves shown in Fig. 2 suggest the possibility to find analytical functions to represent the minor’s eigenvalues, this task is not straightforward. Let us analyze the eigenvalues of the minor matrices for the infinite potential well $V_{\mathrm{W}}(x)$, which seems to be the potential having the most regular behavior. It is important to stress that we are dealing with the numerical solutions of a first order finite–differences approximation, which are different than the exact solutions. We will first attempt to understand the state of affairs in the numerical solutions, and will treat the exact solutions in further works (see a brief discussion in Appendix B).

For the first minor matrix $M_{1}$, the differences $D(i,1)$ are arranged in a smooth curve which is easily approximated, as is shown in Figure 3, by the expression

where $a_{1}$ is very close to $n$, the number of points in the numerical grid.

The analytic approximation is more difficult for the next minor matrix $M_{2}$. Here, there are two different noticeable regions $d$, and each one should be approximated separately, i.e.,

For $j=2$ the first coefficient $a_{2}^{1}=a_{1}\approx n$, but, for the second half of the eigenvalues ($d=2$), the appropriate value is $a_{2}^{2}\approx\frac{n}{2}$ (formal expressions for $a_{j}^{d}$ are given in Appendix A).

The complications related to pursuing an understanding of the behavior of the minor’s eigenvalues do not end here. We found that the size of the Hamiltonian also affects how the $D(i,2)$ are approximated. Let see these values for a case in which the number of points in the numerical grid is even. Figure 5 shows the differences between the eigenvalues of a Hamiltonian matrix with $n=34$ points (in place of the previous values showed in Fig. 4, where $n=33$).

Here, the differences for the first 16 eigenvalues $D^{(1)}(i,2)$ are approximated with (8) using a set of parameters $A_{j}^{d}$ and $a_{j}^{d}$, and the last 16 $D^{(2)}(i,2)$ by using another set, like in the previous case. But, the difference $D(17,2)=\lambda_{17}-\lambda_{17}^{(2)}$ does not belong to either of both curves. Surprisingly, we found that this value agrees exactly with the difference got for the first minor, which is $D(17,1)=\lambda_{17}-\lambda_{17}^{(1)}$.

More intricate is the approximation for the $M_{3}$ minor’s eigenvalues ($j=3$). We start the analysis with a Hamiltonian matrix having $n=35$ points. As is shown in Figure 6, we need to establish three different regions $d$, where the size of each one is pretty near $1/3$ (see Appendix A). Since 34 is not divisible by 3, we do not know beforehand how many points $i$ belong to each region $d$.

We found that the approximation (8) works well for the three ranges if considering 11 points belonging to the first and last regions and 12 points to the middle. The corresponding parameters $a$ are roughly $a_{3}^{1}\approx n$, $a_{3}^{2}\approx\frac{2n}{3}$, $a_{3}^{3}\approx\frac{n}{3}$. As pointed out before, this distribution is particular for a given Hamiltonian size. The $n=33$–case is illustrated in Figure 7. Here, 10 eigenvectors can be fitted very well with (8), for each region $d$. We run into troubles at two points corresponding to $i=\frac{n}{3}$ and $i=\frac{2n}{3}$. As shown in the figure, the same unexpected property found before holds for this case: for these points, the eigenvalue differences for $M_{3}$ coincide exactly with the corresponding values of the first minor $M_{1}$.

Nevertheless, the picture is different for $n=34$, as is shown in Figure 8. Here, the first and last region holds 10 points, the central region 11 points, and, again, two values lie outside these curves. Now, these values agree exactly with those corresponding to $M_{2}$.

In conclusion, the eigenvalue differences $D^{(d)}(i,j)$ follow the general behavior given in Eq.(8), in which the width coefficients

But, it seems very difficult to assign a simple formula for each minor index $j$, for each region $d$, and for any number of points $n$.

We emphasize that the expression (9) is approximate. An accurate fitting for the $a$ values can be done, requiring a lot of work since adjustments must be done to generalize it for every number of points $n$. These corrections are small and will not be significant for large matrices. However, establishing the number of points assigned to each sector $d$ is not straightforward since it depends on the number of points $n$ and varies for each minor index $j$. Assigning a wrong number of points to a particular region accumulates enormous errors in the reconstruction formula (1), turning any approximation of $D$ useless. Furthermore, there are many problematic points $k$, that do not belong to any of the regions $d$. The $D(k,j)$ differences for these points agree with the values calculated for other minors $D(k,p)$, where the minor index $p$ also depends on $j$ and the number of points $n$. We found the analytical expression that relates these indexes:

Taking all these aspects into consideration, we can reproduce all the $D$ values, and thus, all the eigenvalues $\lambda_{i}^{(j)}$ belonging to all the minor matrices $M_{j}$, for any Hamiltonian size $n$. As an example, we illustrate, in Figure 9, the eigenvalue differences $D(i,16)=\lambda_{i}-\lambda_{i}^{(16)}$ for the minor matrix $M_{16}$, for a Hamiltonian matrix with $n=250$ points. All the points $k$ that are not fitted with the curves represented in solid lines have exactly the values $D(k,11)$ corresponding to the minor matrix $M_{11}$ (dashed curves), in agreement with our findings, since $p=\mathrm{mod}(235,16)=11$.

Having decoded the startling pattern conformed by the minor’s eigenvalues, many questions remain open. The behavior discovered is particular to the infinite potential well. From Figure 2 it is reasonable to conjecture about the existence of similar relationships on the other potentials. All the eigenvalues have been calculated numerically. A question arises is whether the same pattern holds also for the exact analytical results. The most important issue to solve is the possibility to recognize a physical meaning in our findings. Perhaps, understanding this matter could help to answer the other questions.

The first physical realization of the eigenvector–eigenvalue identity has been done by Voss and Balloon Voss:20 . They showed that one–dimensional arrays of coupled resonators, described by square matrices with real eigenvalues, provide simple physical systems where this formula can be applied in practice. The subsystems consist of arrays with the $j^{\mathrm{th}}$ resonator removed. Thus, from their spectra alone, the oscillation modes of the full system can be obtained.

Concerning the infinite potential well, the physical meaning of the removal of the first row and column is very simple: it conforms another infinite potential well whose width decreases from $L$ to $L-\Delta x$, where $\Delta x$ is the numerical mesh size. The same potential may be thought of as an infinite well of size $L$, with an additional infinite wall located at $\Delta x$. Accordingly, the removal of the $j^{{\mathrm{t}h}}$ row and column means an infinite potential well with an infinite wall at $x=j\Delta x$. As an example, we plot the first eigenvectors of the submatrix $M_{20}$, for an infinite potential well represented by $n=200$ points, in Figure 10 (above). In the relative coordinates it means that an infinite wall is present at $x/L=0.1$. As is shown in the figure the first eigenvectors correspond to the eigenvectors of a $L-L/10$ infinite potential well. We also have to consider the other potential well of size $L/10$, which, for this particular case have eigenvalues coincident with the full Hamiltonian spectra for every $i=10$ functions. The lower part of Figure 10 shows the eigenvalues $v_{10}^{(20)}$, $v_{20}^{(20)}$, and $v_{30}^{(20)}$, having the same energy as $v_{10}$, $v_{20}$, and $v_{30}$ from the full Hamiltonian.

The eigenvalue–eigenvector identity expresses a curious and surprising relationship between the eigenvector of a matrix and the eigenvalues of its minors. It can be very useful from the numerical point of view, encouraging the study of new techniques to solve large eigenproblems. In general, it is much simple and fast to calculate the eigenvalues than the eigenvectors. Moreover, for very large matrices, the memory storage requirement for standard eigenvectors solver can lead to serious problems, which could be overcome by this approach. A critical element to strengthening the use of the identity is the ability to avoid the calculation of the eigenvalues of all the minor matrices.

In this work, we first tested the eigenvalue–eigenvector identity on different systems. The formula has been used to reconstruct, numerically, the eigenvectors of simple one–dimensional Hamiltonians. Using only the eigenvalues of the minor matrices, the formula reproduces the eigenvectors within machine precision, even for large cases. In most cases, the eigenvalue $\lambda_{i}$ is close to the $\lambda_{i}^{(j)}$ value. However, analyzing their differences $D(i,j)=\lambda_{i}-\lambda_{i}^{(j)}$ , we found systematic patterns, allowing to extract analytically the minor’s eigenvectors without diagonalizing all the submatrices.

To understand the patterns, we focused our study on the infinite potential well. Elucidating the behavior of the differences $D(i,j)$ is not straightforward since they depend on the eigenvalue index $i$, the minor index $j$, and also on the number $n$ of points in the numerical grid representing the Hamiltonian. We found that, indeed, it is possible to find an analytical expression for $D(i,j)$, and therefore, for all the $\lambda_{i}^{(j)}$. The existence of regular patterns suggest that some hidden information about the eigenvectors of the full Hamiltonian is buried under the minor matrices’ eigenvalues structure. A crucial aspect to understand this information is to found the physical reasons that generate those patterns. For the case studied here, the minor matrix represents a clear physical case that consists of the same Hamiltonian but with an infinite wall located at the point corresponding to the index $j$. The generalization to other potentials in 3 dimensions and for real scenarios, can be a matter of further interesting development.

The data that support the findings of this study are available from the corresponding author upon reasonable request.