Linear operator theory of phase mixing

We begin with the Hamiltonian density for a harmonic oscillator with frequency $\kappa$,

Now we add an anharmonic correction of the form $\varphi\bar{q}^{4}$, representative of the next term in the even Taylor series. Our total Hamiltonian density in scaled coordinates is

Any configuration of $f$ is an element of a function space defined for the independent variables $q$ and $p$. Denote by $\mathscr{H}$ the space of functions of $q$ and $p$ on a domain $\mathcal{D}$. We treat $\mathscr{H}$ as a Hilbert space equipped with an inner product, which we denote $\langle g_{1},g_{2}\rangle$ for any $g_{1},g_{2}\in\mathscr{H}$ (see equation 59). We further assume that there exists a set of linearly independent functions $\{e_{j,k}\}$ that form a basis in $\mathscr{H}$. Any function $g\in\mathscr{H}$ can be written in terms of the basis functions by projecting onto them (equation 60).

Given the function space $\mathscr{H}$, one may consider another, distinct vector space housing its functionals. Such a space is called dual to $\mathscr{H}$, and is denoted $\mathscr{H}^{*}$ (also a Hilbert space). For the purposes of this work, we refer to a functional of $f$ as any linear mapping

The dual space $\mathscr{H}^{*}$ has an inner product and basis that can be understood in terms of the corresponding constructions in $\mathscr{H}$. Given a basis $\{e_{j,k}\}$ of $\mathscr{H}$, there exists a basis for $\mathscr{H}^{*}$, which we denote $\{E_{j,k}\}$. The two are related by the biorthonormal condition, $E_{j,k}[e_{j^{\prime},k^{\prime}}]=\delta_{j}^{j^{\prime}}\delta_{k}^{k^{\prime}}$. The inner product of any two $G_{1},G_{2}\in\mathscr{H}^{*}$ is denoted $\langle G_{1},G_{2}\rangle_{*}$, and defined in equation 61. The subscript asterisk signifies that the operation is in $\mathscr{H}^{*}$.

Finally we highlight some important quantities that exist in $\mathscr{H}^{*}$. The total energy, which is the expectation value of the Hamiltonian with respect to $f$, is the functional $H[f]=\langle f,h\rangle$. Additionally, if we project $f$ onto a set of basis functions in $\mathscr{H}$ as in equation 60, the coefficients are the functionals $E_{j,k}^{\dagger}$. The ^† operation denotes complex conjugation on scalars, and the conjugate transpose on vectors and matrices. If we compute the dynamics of $E_{j,k}$ according to $H$ in $\mathscr{H}^{*}$, we obtain the time-evolution of the basis function expansion coefficients for $f$ in $\mathscr{H}$.

We choose as a basis for $\mathscr{H}$ products of univariate Gaussian-Hermite functions. That is,

Knowing the flow map $\hat{S}^{t}$ is tantamount to solving equation 2. Here, we leverage Koopman (1931) to effectively apply the flow map, without dealing explicitly with the CBE. Discussion of the Koopman formalism in the context of astrophysics can be found in Darling & Widrow (2019b) and Darling & Widrow (2021), but for a more rigorous description of the functional realization we will use here, see Nakao & Mezić (2020). Briefly, the idea is that rather than consider the distribution function directly, one can instead study the dynamics of “observables”. In the case of a partial differential equation like the CBE, the observables take the form of functionals, $G[f]$. Let us define a linear time evolution operator that acts on $\mathscr{H}^{*}$, which we denote $\hat{U}^{t}$. This is called the Koopman operator, and its action on any $G\in\mathscr{H}^{*}$ is,

Treating $\hat{U}^{t}G$ as a function of $t$, we may write its infinitesimal change with respect to $t$ as

Equation 13 is a Schrödinger-type equation. It follows that the operator $\hat{U}^{t}$ satisfies an ordinary differential equation, and has the general solution

Denote the $j$th eigenvalue and eigenfunctional of $\hat{A}$ by $\lambda_{j}$ and $\Psi_{j}[f]$ respectively. The eigenvalue problem is then, $\hat{A}\Psi_{j}=\lambda_{j}\Psi_{j}$. Let $\psi_{j}(q,p)=\frac{\delta\Psi_{j}}{\delta f}$. For convenience, we call this quantity an eigenfunction of $\hat{A}$, but really it is an eigenfunction of the Liouville operator, $[h,\cdot]$. Suppose that $\{\psi_{j}\}$ forms a basis of $\mathscr{H}$, and $\{\Psi_{j}\}$ of $\mathscr{H}^{*}$.

In this paper, we focus on the case where $\frac{\partial\hat{A}}{\partial t}=0$, which corresponds to a time-independent potential. This holds for our target Hamiltonian in equation 7, but is not true in general for a self-consistent distribution function satisfying both the CBE and Poisson equation. To study the response to both an external potential and self-consistent density perturbations, one must consider $\frac{\partial\hat{A}}{\partial t}\neq 0$. We will discuss this briefly in Sections 6 and 7, but treatment of that problem is not in the scope of this work.

In the time-independent generator case, equation 15 simplifies to $\hat{U}^{t}=\mathrm{e}^{\hat{A}t}$, and we have

Now that we have established the formalism in which we will carry out our calculations, it is advantageous to restate the original problem. Recall that the target Hamiltonian is stated in equation 7. In terms of total energy functionals, this corresponds to $H=H_{0}+\varphi H_{4}$, where $H_{0}=\langle f,h_{0}\rangle$ and $H_{4}=\langle f,\frac{4}{\kappa^{4}\beta^{2}}q^{4}\rangle$. It follows that the generators are $\hat{A}_{0}=[\cdot,H_{0}]_{f}$, $\hat{A}_{4}=[\cdot,H_{4}]_{f}$, and $\hat{A}=\hat{A}_{0}+\varphi\hat{A}_{4}$. Our goal then is to compute the spectrum of $\hat{A}$, so that we can use equation 18.

In order to compute the spectrum of $\hat{A}_{0}$, we construct a $M\times M$ matrix, which we denote $\bm{\mathsf{A}}_{\mathsf{0}}$. Its elements are obtained by computing the inner products,

With a matrix representation of $\hat{A}_{0}$, we can compute a subset of its spectrum. Let $\bm{\mathsf{A}}_{\mathsf{0}}\bm{\Psi}^{(0)}=\bm{\Psi}^{(0)}\bm{\Lambda}^{(0)}$ denote the eigendecomposition of $\bm{\mathsf{A}}_{\mathsf{0}}$, where

In general, there can be a combination of real and imaginary eigenvalues. Complex eigenvalues come in conjugate pairs, as do their associated eigenvectors. All $\bm{\mathsf{A}}_{\mathsf{0}}$ eigenvalues $\lambda_{j}^{(0)}$ are determined by the characteristic polynomial,

where $\bm{\mathsf{I}}$ denotes the $M\times M$ identity matrix. We compute the $\lambda_{j}^{(0)}$ by finding the roots of equation 27. It is possible that some of these roots are repeated, and it is such repeated roots that we refer to as degenerate eigenvalues of $\bm{\mathsf{A}}_{\mathsf{0}}$. We will refer to the number of times a given eigenvalue is repeated as its multiplicity, denoted $d_{j}$.

The eigenvectors $\bm{\psi}_{j}^{(0)}$ associated with $\lambda_{j}^{(0)}$ are computed by solving,

We show in Fig. 1 and 2 the sets of linearly independent eigenfunctions associated with $\lambda_{j}^{(0)}=-i\kappa$ and $\lambda_{j}^{(0)}=-i2\kappa$ respectively. The former corresponds to the standard bending mode structure, and the latter the breathing mode. In each case, the presented function is the real part of one member of a conjugate pair. When the full complex functions are taken together with their conjugates, multiplication with the relevant complex exponentials produces real valued clockwise rotating bending and breathing patterns.

The dynamics of an arbitrary initial condition according to $h_{0}$ are determined by the spectrum of $\hat{A}_{0}$. Explicitly, $f$ is mapped to a new function $\hat{S}_{0}^{t}f$ at time $t$ by the flow of equation 2, for $h=h_{0}$. We have from equation 18,

For the harmonic potential, the dynamics of $f$ are a rigid rotation of the initial condition about the origin. It is not immediately apparent from equation 29, but we can verify that no deformation to the distribution occurs with the conserved quantities in the spectrum of $\hat{A}_{0}$. There are several eigenvectors of $\bm{\mathsf{A}}_{\mathsf{0}}$ associated with a zero eigenvalue. Each of these corresponds to a conserved quantity. By solving equation 28 for $\lambda_{j}^{(0)}=0$, we find the sequence:

The equivalent expression to equation 21 for the $\hat{A}_{4}$ matrix elements is obtained by making the substitutions $G\rightarrow E_{j,k}$, and $H\rightarrow H_{4}$ in equation 14. We have,

Application of equation 33, with the coefficients from the $n=3$ column of Table 1 yields

To apply the ordinary differential equation solution for $\hat{U}^{t}$, and compute $\hat{S}^{t}f$ from equation 18, we need the spectrum of $\hat{A}$. The matrix

We assume that the basis formed by the eigenvectors of $\bm{\mathsf{A}}_{\mathsf{0}}$ spans the eigenspace of $\bm{\mathsf{A}}$. That is,

We can use the corrections computed here as-is for the non-degenerate subset of the $\bm{\mathsf{A}}_{\mathsf{0}}$ spectrum, where $\lambda_{j}^{(0)}\neq\lambda_{k}^{(0)}$ if $j\neq k$. The situation is more complicated if $\lambda_{j}^{(0)}=\lambda_{k}^{(0)}$. As discussed in Section 4.2, the spectrum of $\bm{\mathsf{A}}_{\mathsf{0}}$ in general contains degenerate subspaces, where there are multiple distinct eigenvectors associated with the same eigenvalue (see for example Fig. 1 and 2). This poses an issue for the correction in equation 48, where we must sum over all of the linearly independent eigenvectors. If $j$ and $k$ are such that both eigenvectors lie within a degenerate subspace, the denominator here is zero. To proceed, we treat the $\bm{\mathsf{A}}_{\mathsf{4}}$ contribution separately for each degenerate subspace.

We begin by adding a second index label to the eigenvalues of $\bm{\mathsf{A}}_{\mathsf{0}}$. That is, let $\lambda_{j,k}^{(0)}$ denote the $k$th repetition of the $j$th unique eigenvalue of $\bm{\mathsf{A}}_{\mathsf{0}}$. The same indexing scheme is applied to the eigenvectors. As stated in Section 4.2, all eigenvectors corresponding to repeated eigenvalues are themselves linearly independent. So although we have $\lambda_{j,k}^{(0)}=\lambda_{j,k^{\prime}}^{(0)}$, for the eigenvectors, $\bm{\psi}_{j,k}^{(0)}\neq\bm{\psi}_{j,k^{\prime}}^{(0)}$.

For any fixed $j$, the $\bm{\psi}_{j,k}^{(0)}$ are linearly independent and span a subspace of the $\bm{\mathsf{A}}_{\mathsf{0}}$ eigenspace corresponding to a single eigenvalue $\lambda_{j}^{(0)}$. Any linear combination of these vectors is itself an eigenvector of $\bm{\mathsf{A}}_{\mathsf{0}}$, with eigenvalue $\lambda_{j}^{(0)}$. We can make use of this property in determining the $\bm{\mathsf{A}}_{\mathsf{4}}$ contribution. We begin by looking at the spectrum of $\bm{\mathsf{A}}_{\mathsf{4}}$ when it is projected onto each degenerate subspace of $\bm{\mathsf{A}}_{\mathsf{0}}$ separately. This process is as follows.

For each eigenvalue $\lambda_{j}^{(0)}$ with multiplicity $d_{j}>1$, we define the subspace basis as

The $\bm{\xi}_{j,k}$ comprise weights for linear combinations of the degenerate subspace basis vectors $\bm{\psi}^{(0)}_{j,k}$. Contracting with the subspace basis $\bm{\Psi}_{j}^{(0)}$ in equation 49, we can get back vectors in $\mathbb{C}^{M}$. Explicitly, we write

We need these new eigenvectors to resolve an ambiguous definition of the unperturbed $\bm{\psi}_{j,k}^{(0)}$. If we did not do this, the transition between $\varphi=0$ and $\varphi>0$ could constitute a discontinuous change in the eigenvectors. We must avoid this to assure that we have a smooth variation of the eigenvectors with respect to change in the perturbation parameter $\varphi$. Otherwise, the original power series assumption (equation 40) would be invalid. From this point on, we replace all of the degenerate subspace eigenvectors, $\bm{\psi}_{j,k}^{(0)}$ with the new vectors we just found, $\tilde{\bm{\psi}}^{(0)}_{j,k}$. For notational simplicity, we drop the tilde. Any time from this point on that we refer to the degenerate subspace vectors, it should be assumed that they are the ones defined by equation 52 that diagonalize $\bm{\mathsf{A}}_{\mathsf{4}}$ in their respective subspaces. The eigenfunctions are obtained from the eigenvectors in the same way as for the harmonic oscillator, using an analog to equation 26.

To compute the $\mathcal{O}(\varphi)$ corrections for the degenerate eigenvalues and eigenvectors, we follow a similar procedure to Section 4.2, carrying out the necessary algebra in Appendix B. To summarize, we again assume a power series in $\varphi$ for both the eigenvalues and eigenvectors of $\bm{\mathsf{A}}$ (equation 68), and then compute the coefficients of the $\mathcal{O}(\varphi)$ terms in these series.