Let\(\Phi \) be a Hermitian\(N\times N\) matrix,E be a positive diagonal\(N\times N\) matrix\(E := diag (E_1, E_2 , \ldots ,E_N )\) without degenerate eigenvalues, and\(\eta \) be a positive real number as a coupling constant. We deal in this paper with the following one Hermitian matrix model defined using thisE:

This matrix model is obtained by changing the potential of the Kontsevich model [15] from\(\Phi ^3\) to\(\Phi ^4\). It was introduced while studying a scalar field defined on a deformed four-dimensional space-time and studied over years [7,8]. An additional oscillator term was added in order to resolve the IR-UV mixing problem. This term leads to an external matrixE with equally spaced eigenvalues. Recent developments are summarized in Branahl et al. [2].

The main theorem of this paper is expressed as follows.

Theorem 1.1

Let\(Z(E, \eta )\) be the partition function defined by

Let\(\Delta (E)\) be the Vandermonde determinant\(\Delta (E) := \prod _{k<l} (E_l -E_k)\). Then the function

is a zero-energy solution of a Schrödinger-type differential equation being 2-nd order in each of its variables,

where\(\mathcal {H}_{HO}\) is the Hamiltonian of theN-body harmonic oscillator without interaction:

In this sense, this matrix model is a solvable system.

Let\(\Phi \) be a Hermitian\(N\times N\) matrix. LetH be a positive Hermitian\(N\times N\) matrix with nondegenerate eigenvalues\(\{E_1, E_2 , \ldots ,E_N ~ | ~ E_i \ne E_j ~\text{ for }~ i \ne j \}\).\(\eta \) is a real positive number. We consider the following action

The partition function is defined by

and we denote the expectation value with this actionS by\(\displaystyle \langle O \rangle := \int \nolimits _{h_N} d \Phi ~ O e^{-S} \). Note that we do not normalize it here, i.e.,\(\langle 1 \rangle = Z(E, \eta ) \ne 1\). Here, the integral measure is the ordinary Haar measure. Using the real variables defined by\(\Phi _{ij}= \Phi _{ij}^{Re} + i \Phi _{ij}^{Im}\), the measure is given as\(\displaystyle \int \nolimits _{h_N} d \Phi := \prod \nolimits _{i}^N \int _{-\infty }^{\infty } d\Phi _{ii} \prod \nolimits _{k<l} \int _{-\infty }^{\infty } d\Phi _{kl}^{Re} \int _{-\infty }^{\infty } d\Phi _{kl}^{Im} \). Note that the partition function\(Z(E, \eta ) \) depends only on the eigenvalues ofH because the integral measure isU(N) invariant. Indeed\(Z(E, \eta ) \) is equal to the partition function obtained from the action defined by\(S'\) in (1.1).

In the following, we use the notation:

For the diagonal elements\(\Phi _{ii} (i= 1,2, \ldots ,N)\), the corresponding partial derivatives are the usual ones. The Schwinger–Dyson equation is derived from

which is expressed as

Taking sum over the indicesi, j and using

a partial differential equation is obtained:

Here,\(\mathcal {L}_{SD}^H \) is a second-order differential operator defined by

Next we rewrite this Schwinger–Dyson equation by using eigenvalues ofH, i.e.,\(E_n (n= 1,2, \ldots , N)\). References [13,14] are helpful in the following calculations. LetP(x) be the characteristic polynomial:

Using thisP(x),

is obtained. Here,\( | M |_{kj} \) denote the minors of the matrixM defined by the determinant of the smaller matrix obtained by removing thek-th row andj-th column fromM. Using the formula (2.9),

The first term in the last line is equal to\(\displaystyle -\sum _{k,i} \frac{P(E_k)}{P'(E_k)} \frac{\partial Z(E, \eta )}{\partial E_{k}} = 0\), becauseP(x) is the characteristic polynomial and\(E_k\) is one of eigenvalues ofH.\(\sum _{i,j} \delta _{ij} (-1)^{i+j} | E_k~ Id_N - H |_{ij} \) in the second term is\(P'(E_k)\). Then we find

Next step, we rewrite the Laplacian\(\displaystyle \sum \nolimits _{i,k} \left( \frac{\partial }{\partial H_{ki}}\frac{\partial }{\partial H_{ik}} \right) Z(E, \eta )\) by\(E_k\). It is a well-known fact that by using the Vandermonde determinant\(\Delta (E) := \prod _{k<l} (E_l -E_k)\) the Jacobian for the change of variables is obtained as follows:

Then, the Laplacian is rewritten as

Here,\(\displaystyle \sum \nolimits _{i \ne j}\) means\(\displaystyle \sum \nolimits _{i,j=1 , i \ne j}^N\). From (2.7), (2.11), and (2.12), we obtain the following.

Theorem 2.1

The partition function defined by (2.2) satisfies

where

In AppendixA, this Schwinger–Dyson equation is checked by using perturbative calculations, as a cross-check.

In this section, we prove the main theorem (Theorem1.1).

As the first step, we prove the following proposition.

Proposition 3.1

The differential operator\(\mathcal {L}_{SD} \) defined in (2.14) is transformed into the Hamiltonian of theN-body harmonic oscillator as

We denote this Hamiltonian by\(\mathcal {H}_{HO}\):

Proof

The proof is done by direct calculations.

We calculate\(\Delta ^{-1}(E) e^{\frac{N}{2\eta } \sum _i E_i^2} \frac{\eta }{N} \sum _{i=1}^N \left( \frac{\partial }{\partial E_i} \right) ^2 e^{-\frac{N}{2\eta } \sum _i E_i^2} \Delta (E)\) at first.

(3.3) is equal to 0 since

(3.4) is written as follows.

Then, we obtain

\(\square \)

We introduce a transformed partition function\(\Psi (E, \eta )\) by

Note that this transformation is invertible. Then, the following theorem follows from Proposition3.1 immediately.

Theorem 3.2

The transformed partition function\(\Psi (E, \eta ) \) is a zero-energy solution of the Schrödinger-type differential equation:

Here,\(\mathcal {H}_{HO}\) is theN-body harmonic oscillator Hamiltonian (3.2).

ThisN-body harmonic oscillator system has no interaction terms between the oscillators, so it is a trivial quantum integrable system.

Theorem1.1 is proved as above. In the next section, we calculate the solution\(\Psi (E, \eta ) \) more concretely and give another proof using it.

A new expression of the zero-energy solution of theN-body harmonic oscillator system is constructed by a direct calculation of the partition function.

Let us carry out the integration of the off-diagonal components of\(\Phi \) in the definition of the partition function after the change of variables to\(U(N)\times {\mathbb R}^N\). We denote the eigenvalues of\(\Phi \) by\(x_1 , x_2 , \ldots , x_N\). By using a unitary matrixU,\(\Phi \) is diagonalized as\(X=U \Phi U^\dagger \), where\(X= diag(x_1, x_2 , \ldots , x_N )\). Then,

where\(V(x):= \frac{\eta }{4}x^4\).

Let us use the Harish–Chandra–Itzykson–Zuber integral [12,19] for the unitary groupU(N) :

Here,A andB are Hermitian matrices whose eigenvalues are denoted by\(\lambda _{i}(A)\) and\(\lambda _{i}(B)\)\((i=1,\ldots ,N)\), respectively.t is a nonzero complex parameter,\(\displaystyle \Delta (\lambda (A)):=\prod \nolimits _{1\le i<j\le N}(\lambda _{j}(A)-\lambda _{i}(A))\) is the Vandermonde determinant, and\(\displaystyle \tilde{c}_{N}:=\left( \prod \nolimits _{i=1}^{N-1}i!\right) \times \pi ^{\frac{N(N-1)}{2}}\).\(\left( \exp \left( t\lambda _{i}(A)\lambda _{j}(B)\right) \right) \) is the\(N\times N\) matrix with thei-th row and thej-th column being\(\exp \left( t\lambda _{i}(A)\lambda _{j}(B)\right) \). After adapting this formula, the partition function is described by

where\(c_N = \tilde{c}_N (-1/ N)^{\frac{N^2-N}{2}}\) and\(S_N\) denotes the symmetric group. This integral representation (4.3) should be regarded as a Cauchy principal value. Consider the change of variables\(x_i \mapsto x_{\sigma ^{-1}(i)}\). Note that the sign of\(\prod _{l < k}(x_k -x_l)\) changes as\((-1)^\sigma \prod _{l < k}(x_k -x_l)\) , and the following formula is obtained by this change of variables.

Then, the zero-energy solution of (3.9) is obtained by (1.2).

Theorem 4.1

The function

satisfies the Schrödinger-type differential equation (3.9).

Since this fact follows from Theorem3.2, there is no need to prove it, but it would be worthwhile to show the differential equation (3.9) directly from expression (4.5) as a confirmation.

At first, we prove the following Lemma:

Lemma 4.2

Proof

For simplicity,\(\displaystyle \sum \nolimits _i^N N \left( \frac{\eta }{4} x_i^4 + E_i x_i^2 + \frac{1}{2\eta } E_i^2 \right) \) will be abbreviated asf(x, E). From the following identity:

we obtain the formal expression:

Then, we get

From the identity

(4.6) is obtained.\(\square \)

Using Lemma4.2, let us prove Theorem4.1 by direct calculations.

Proof

From (4.5),

For the last equality, we used Lemma4.2.\(\square \)

\(\Psi (E, \eta )\) can also be expressed using Pfaffian. It is described in AppendixB.

As described above, we have also directly proved that the function\(\Psi (E, \eta )\) obtained from the partition function of the matrix model satisfies the Schrödinger-type differential equation for theN-body harmonic oscillator system without interactions.

The matrix model studied in this paper is related to a renormalizable scalar\(\Phi ^4\) theory on Moyal space [8] in the largeN limit. There are mainly two approaches to study the question of integrability of this matrix model: One relies on the model, where one replaces the\(\Phi ^4\) interaction by a constant times\(\Phi ^3\). This gives the Kontsevich model, for which it is known, that the logarithm of the partition function is the\(\tau \) function for the KdV hierarchy and fulfills a Hirota bilinear equation [10,13,15,21]. Another approach follows topological recursion. While the Kontsevich model follows topological recursion, it turned out that the\(\Phi ^4\) model follows the more sophisticated blobbed topological recursion, (proven for genus one and two) [2,3,11]. Due to these complications, it was unexpected, to obtain such a simple answer. ThisN-body harmonic oscillator system is known as an integrable system and this system has been studied for a long time. See, for example, [17,18,20] and references therein. Note that the solution required by the Schrödinger-type equation (3.9) is a zero-energy solution, which is different from the well-known harmonic oscillator solutions by using Hermite polynomials for nonzero-energy solutions. In particular, the case\(N=1\) corresponds to what is called the Weber equation. In the following, we will consider the case\(N=1\) as a particularly simplest case and see how\(\Psi (E, \eta )\) corresponds to a solution to the Weber equation.

Introducing new variables\(\displaystyle u_i := \sqrt{\frac{N}{\eta }}E_i\), the Schrödinger-type equation (3.9) is deformed into

So the\(N=1\) case, this is a kind of the Weber equation [6]:

The series solution of this Weber equation is given as follows. For\(y(u)= \sum _{n=0}^{\infty } a_n u^n\), (5.2) requires

So the boundary conditions are given by\(y(0)=a_0\) and\(y'(0) =a_1\). For\(N=1\) case, the partition function is

and using this\(Z(u,\eta ),\) (3.8) implies that the solution of (5.2) is given as

Indeed, we can prove that\(\Psi (u)\) satisfies (5.2) as follows. As similar to the proof for Lemma4.2,

where\(-f(u)= -\sqrt{\eta } u x^2 - \frac{\eta }{4} x^4 -\frac{u^2}{2}\). Using this formula, (5.2) is derived:

Furthermore, for\(u>0\) , by using modified Bessel function of the second kind\(K_{\frac{1}{4}}(u)\),\(\Psi (u) \) can also be written as

We find that the boundary condition for this solution is required as

Thus, in the case of\(N=1\), the results are derived using known special functions.

Additional comments

During the peer-review process of this paper, a follow-up paper [9] was submitted by the authors, Kanomata and Wulkenhaar, which was published first. Paper [9] corresponds to the application of the technology of this paper. For the benefit of the readers of this journal, we comment on it below.

In this paper, we have seen that the Schwinger–Dyson equation satisfied by the partition function of the Hermitian matrix model (1.1) derives the Schrödinger equation for the Hamiltonian ofN-body harmonic oscillator system. TheN-body harmonic oscillator system can be extended to the integrable Calogero–Moser model [5,16]. It is thus natural to think that there should be matrix models whose partition functions satisfy the Schrödinger equation for the Calogero–Moser model. The calculations with Hermitian matrices in this paper are performed by replacing them with real symmetric matrices and showing that the Schwinger–Dyson equation satisfied by the partition function corresponds to the Schrödinger equation of the Calogero–Moser model in Grosse et al. [9]. Furthermore, since the Calogero–Moser model admits a Virasoro algebra representation, it gives rise to a family of differential equations satisfied by the partition function\({Z}(E,\eta )\) in Grosse et al. [9]. It is also possible to construct Virasoro algebra representations forN-body harmonic oscillator systems. Therefore, a similar family of differential equations satisfied by the partition function\({Z}(E,\eta )\) can also be constructed for the Hermitian matrix model of this paper.

Next, we also comment on the generalization of this model. As mentioned above, the matrix model of the Hermitian matrix and the matrix model of the real symmetric matrix correspond to the harmonic oscillator system and the Calogero–Moser model, respectively. As a similar extension, it is natural to extend the matrix\(\Phi \) from a Hermitian to a quaternion self-dual matrix without changing the form of the action (1.1). In this case, it is still expected to correspond to the Schrödinger equation in the Calogero–Moser model.

A more non-trivial generalization is to consider higher-order potentials like\(\Phi ^6 , \Phi ^8 ,\) etc. In the case of this paper, the\(\Phi ^4\) interaction term leads to the Laplacian in the Schrödinger equation, as seen from its derivation process. Equations in (2.6) and the subsequent calculations show this. By similar considerations, it can be expected that higher-order differential terms will appear, as the order of interactions increases. In such cases, it remains a challenging problem for the future what kind of theory can be developed. We intend to return to such models and to treat also different observables beyond partition functions.