Towards verifications of Krylov complexity

Krylov complexity is considered to provide a measure of the growth of operators evolving under Hamiltonian dynamics. The main strategy is the analysis of the structure of Krylov subspace $\mathcal{K}_M(\mathcal{H},\eta)$ spanned by the multiple applications of the Liouville operator $\mathcal{L}$ defined by the commutator in terms of a Hamiltonian $\mathcal{H}$, $\mathcal{L}:=[\mathcal{H},\cdot]$ acting on an operator $\eta$, $\mathcal{K}_M(\mathcal{H},\eta)=\text{span}\{\eta,\mathcal{L}\eta,\ldots,\mathcal{L}^{M-1}\eta\}$. For a given inner product $(\cdot,\cdot)$ of the operators, the orthonormal basis $\{\mathcal{O}_n\}$ is constructed from $\mathcal{O}_0=\eta/\sqrt{(\eta,\eta)}$ by Lanczos algorithm. The moments $\mu_m=(\mathcal{O}_0,\mathcal{L}^m\mathcal{O}_0)$ are closely related to the important data $\{b_n\}$ called Lanczos coefficients. I present the exact and explicit expressions of the moments $\{\mu_m\}$ for 16 quantum mechanical systems which are {\em exactly solvable both in the Schr\"odinger and Heisenberg pictures}. The operator $\eta$ is the variable of the eigenpolynomials. Among them six systems show a clear sign of `non-complexity' as vanishing higher Lanczos coefficients $b_m=0$, $m\ge3$.


Introduction
In the study of quantum chaos, Krylov complexity is proposed as a measure of the growth of operators in the Heisenberg picture [1,2,3,4,5,6,7].The definition of Krylov complexity is universal, that is, applicable to a very complicated system as well as an extremely ordered system, for example, an exactly solvable system in the Heisenberg picture.As materials to support the validity of the concept of Krylov complexity by contrast [8,9,10], I present one of the most basic ingredients of the theory, the moments {µ m } (2.20) of a special operator η called sinusoidal coordinate of many exactly solvable quantum systems in the Heisenberg picture.They are ten discrete quantum mechanical systems of finite dimensions and two infinite ones plus four ordinary one dimensional quantum mechanics.In all these systems, the eigenvalues of the Hamiltonian are simple explicit functions of the system parameters and the sinusoidal coordinate is the variable in the eigenpolynomials {P n (η)} of the Hamiltonian H.
This paper is organised as follows.In section 2, the basic concepts of Krylov complexity are briefly recapitulated [1] through the orthonormalisation of operators in a Krylov subspace [14] by Lanczos algorithm [15].In section 3 the outline of exactly solvable discrete quantum systems in the Schrödinger picture is briefly reviewed [16].Corresponding solutions of Heisenberg equation of motion [17,18], which are the principal tool of the present paper, are discussed in some detail in section 4. The main results of the paper, the exact and explicit expressions of the moments {µ m } are derived for ten exactly solvable discrete quantum mechanics in section 5 and four ordinary one-dimensional quantum mechanics in section 6.The inner product including the Boltzmann factor is introduced in section 6 to deal with unbounded Hamiltonians.The exact expressions of the moments for the bounded and unbounded Hamiltonians look very similar as shown in two main Theorems 5.2 and 6.1.Among these 16 exactly solvable systems, six systems related to the Krawtchouk and dual Hahn §5.1, Meixner and Charlier §6.1, Hermite and Laguerre §6.2 polynomials show a very clear sign of 'non-complexity' as vanishing higher Lanczos coefficients b m = 0, m ≥ 3.
These six systems all share a linear spectrum E(n) ∝ n.The final section is for a summary and some comments.

Orthonormalisation of operators in the Krylov subspace
Let us start with a brief review of the general setting of the Krylov complexity along the line of the seminal work [1] and others [2,3,4,5] in order to introduce proper notions and appropriate notation.The orthonormalisation of operators in the Krylov subspace is the basic ingredient for the evaluation of Krylov complexity.For simplicity of presentation, the discrete quantum mechanical systems are discussed at the beginning.That is, the basic vector space is assumed to be C N +1 , N ∈ N in this section and up to §5.Later in §6, infinite dimensional Hilbert space is introduced for the treatment of the traditional quantum mechanical systems.
Let us begin with the notation.The ordinary Krylov subspace [14] is spanned by a series of vectors generated by multiple applications of a certain operator H on a vector v, K M (H; v) def = span{v, Hv, H 2 v, . . ., H M −1 v}.
For the evaluation of Krylov complexity [1,2,3,4,5], a different type of subspaces is necessary.It is spanned by a series of operators generated by multiple applications of adjoint actions of a Hamiltonian H on an operator, to be named as η throughout this paper, in which the Liouville operator L denotes the commutator by the Hamiltonian H; The Hamiltonian H, η and other operators V, W etc are all (N + 1) × (N + 1) matrices with the complex components, The Hamiltonian H is a positive semi-definite hermitian matrix, in which † denotes the hermitian conjugation and * means the complex conjugation.Throughout this paper the operator η is assumed to be a real diagonal matrix, Two types of bra-ket notation are used.The bra ((x| and ket |x)) correspond to the x-th The orthonormal basis corresponding to the eigenvectors of the Hamiltonian H is denoted by the bra n| and the ket |n , The inner product of operators V and W is defined by the trace, which is real if V and W are both hermitian or anti-hermitian.The norm of an operator V is denoted by, It should be stressed that the Liouville operator L flips from the right to left side and vice versa, under the present definition of the inner product, It is also obvious that (V, LV) vanishes if V is hermitian or anti-hermitian, The orthonormalisation of the Krylov subspace K m (H, η) (2.1), {O n }, n = 0, 1, . . . is much simpler than the ordinary Gram-Schmidt orthonormalisation due to the built-in structure of K M (H, η).The orthonormalisation à la Lanczos [15] starts with ) and it goes on until a zero norm operator is produced.By the above property (2.11), O 1 is automatically orthogonal with the previous O 0 , The next step is the orthogonalisation of LO 1 with O 0 , (2.15) The orthogonality (O 1 , W 1 ) = 0 is obvious by (2.11) and (2.14).By construction, the orthogonality (O 0 , W 1 ) = 0 holds.As (2.16) The flip property (V, LW) = (LV, W) (2.10) plays an important role in the above and further calculations.The process goes on as It is easy to prove that O k+1 is orthogonal to all the previous ones (O j , O k+1 ) = 0, j = 0, 1, . . ., k, by assuming the previous ones are orthonormal and the induction is complete.
in which [m] is the Gauss's symbol meaning the greatest integer not exceeding m.The squares of the Lanczos coefficients up to n, {b 2 1 , . . ., b 2 n }, which are the length squared of the basis before normalisation, are expressed as rational functions of the moments µ 2m , (2.20) For example (cf [19] Table 3 (2.21) Such formulas can be checked by considering the formal scaling properties, Remark 2.3 In some work [1](A.4), a formula involving the determinant of Hankel matrix of moments was erroneously reported.The l.h.s.scales as λ 2n and the diagonal part of the r.h.s.matrix, in which O(t) is the Heisenberg operator solution of η, Reflecting the unit norm of O(t), n ϕ(t) 2 = 1 holds.
In the rest of this paper I present the explicit forms of the Heisenberg operator solution (2.24) and the moments {µ 2m } of more than a dozen quantum mechanical systems.Based on the exact knowledge of the moments, the Lanczos coefficients {b n } and the functions {ϕ n (t)} (2.23) can be evaluated as precisely as wanted.

Exactly solvable discrete quantum mechanics
Here I present ten exactly solvable discrete quantum mechanical systems, see [20] for a review.
The eigenvectors of the Hamiltonians are the hypergeometric orthogonal polynomials of the Askey scheme [11,12,13,16].They are the Krawtchouk (K), Hahn (H), dual Han (dH), Racah (R), quantum q-Krawtchouk (qqK), q-Krawtchouk (qK), affine q-Krawtchouk (aqK), q-Hahn (qH), dual q-Hahn (dqH) and q-Racah (qR) polynomials.The Hamiltonian H of these exactly solvable quantum mechanics is a tridiagonal (N + 1) × (N + 1) real symmetric matrix, in which the functions B(x) and D(x) are positive except for the boundary conditions, The orthonormal eigenvectors of the Hamiltonian H are in which P n (η) is a degree n polynomial in η.The sinusoidal coordinate η is a linear or quadratic function of x or q ±x (0 < q < 1) which vanishes at x = 0, η(0) = 0 [16], Likewise, the eigenvalue E is a linear or quadratic function of n or q ±n which vanishes at In the formulas (3.5), (3.6) the parameter d is specific in each system.As functions of x, {P n η(x) } are terminating (q)-hypergeometric functions [11,12,13] and they are normalised by a uniform condition, As orthogonal polynomials in η with the above normalisation condition, {P n (η)} satisfy three term recurrence relation [16], in which the coefficients A n , C n are negative except for the boundary conditions so that P −1 and P N +1 do not enter into the theory.
For definiteness, I show the data of the simplest example, the Krawtchouk (K) system: As will be shown shortly, most of the data, except for η, E, A n and C n , are not needed for the evaluation of the moments {µ 2m } and the functions {ϕ n (t)}.

Solutions of Heisenberg equation of motion
As shown in (2.24), the explicit form of the Heisenberg operator solution is essential for the determination of the functions {ϕ n (t)}.Although the Heisenberg solution of the harmonic oscillator potential (x 2 ) was known in the early days of quantum mechanics, it was late 1970's that those for four other potentials in one-dimensional quantum mechanics were reported by Nieto and Simmons [21]- [24].They were for the potentials x 2 + 1/x 2 , 1/ sin 2 x, −1/ cosh 2 x and the Morse potential.The term 'sinusoidal coordinate' was also coined by them, meaning that η undergoes sinusoidal motion with frequencies depending on the energy.About a quarter century after Nieto and Simmons, the list of exact Heisenberg operator solutions was enlarged by Odake and myself to include many discrete quantum mechanics [16,17,18,25] and some multi-particle dynamics [26].
The essence is the discovery [18] (this paper will be cited as I) that the Hamiltonian H and the sinusoidal coordinate η of exactly sovable systems in the Schrödinger picture all satisfy a simple commutation relation in which R 0 (H), R 1 (H) and R −1 (H) are polynomials in H of maximal degree 2, 1 and 2, respectively, reflecting the power counting of H on both sides.Those R i (H)'s may contain some system parameters but not dynamical operators other than H.In terms of the Liouville operator L, the above commutation relation reads which is obviously generalised to with the obvious initial conditions By solving the recurrence relations, one arrives at, in which two operators α ± (H) are the roots of the quadratic equation, They also satisfy the relations as shown in [18](I.2.22) and (I.2.23) ) Summing up {L m η} leads to the exact Heisenberg operator solution of η [17,18], as the second factor is time-dependent and non-vanishing.
One only needs the values of (5.3) ) By combining them, one arrives at ) by using (4.11).It is easy to see that ( due to (4.15).These lead to and one arrives at in which the boundary conditions (3.9)A N = 0 ans C 0 = 0 are used.Changing n − 1 → n in the second sum leads to in which (4.13) is used.
Remark 5.1 It should be stressed that the product of the coefficients of the three term recurrence relations (3.8) These results are summarised as the following Theorem 5.2 The moments of the exactly solvable discrete quantum systems have a very simple exact expression It applies to ten systems related to the Krawtchouk (K), Hahn (H), dual Han (dH), Racah (R), quantum q-Krawtchouk (qqK), q-Krawtchouk (qK), affine q-Krawtchouk (aqK), q-Hahn (qH), dual q-Hahn (dqH) and q-Racah (qR) polynomials.
In the rest of this section, the necessary data for the evaluation of the moments µ 2m of the ten discrete quantum systems are provided.They are B(x) and D(x) for the definition of the Hamiltonian H and for specifying the parameter ranges, the sinusoidal coordinates η(x), the energy eigenvalues E(n), R 0 (H) and R 1 (H) for the derivation of α ± E(n) and the coefficients of the three term recurrence relation A n and C n .It should be stressed that for these exactly solvable systems, the inside of the square root ) is always a complete square and α ± E(n) are polynomials in n or q ±n of maximal degree two.
For the full details of these exactly solvable discrete quantum systems, a paper by Odake and myself [16] should be consulted.

Krawtchouk and dual Hahn
These two systems have E(n) = n and R 0 = 1 and R 1 = 0 so that α ± E(n) = ±1 and all the moments are identical K, dH :  The other data are Dual Hahn (a, b > 0) (5.15)

affine q-Krawtchouk and dual q-Hahn
These systems share the same functions R 0 and R 1 For both systems A n and C n are quadratic polynomials in q n so that µ 2m can be exactly calculated in a closed form.

quantum q-Krawtchouk
The system has the functions R 0 and R 1 Other data are (p > q −N ) The A n and C n are quadratic polynomials in q −n so that µ 2m can be exactly calculated in a closed form,

q-Krawtchouk
The system has the functions R 0 and R 1 with Other data are (p > 0) As A n and C n are rational function of q n , exact calculation of µ 2m is rather complicated, )

Hahn and Racah
These two systems have similar structures.
Hahn The system has the functions R 0 and R 1 with Other data are As A n and C n are rational functions of n, exact calculation of µ 2m is rather complicated, (5.21) 6 .
Racah The system has the functions R 0 and R 1 with As A n and C n are rational functions of n, exact calculation of µ 2m is rather complicated, ) 5.6 q-Hahn and q-Racah These two systems have similar structures.
q-Hahn The system has the functions R 0 and R 1 with Other data are (0 < a, b < 1), ) .
As A n and C n are rational function of q n , exact calculation of µ 2m is rather complicated, ) q-Racah The system has the functions R 0 and R 1 with Other data are ( As A n and C n are rational function of q n , exact calculation of µ 2m is rather complicated, ) 6 Evaluation of µ m of other exactly solvable quantum mechanical systems In order to evaluate the moments {µ m } for verifications of Krylov complexity of general quantum mechanical systems through orthonormalisation of a Krylov subspace, an appropriate definition of the inner product of operators is essential.The simplest trace one ) is obviously ill-defined for most operators, which are unbounded.A simple prescription to deal with unbounded operators is to introduce the finite temperature (T) effects through the Boltzmann factor e −βH , β = 1/T .The finite temperature inner product suppresses the contributions of higher energy eigenstates and make the trace summable.
The general forms of the inner product including the Boltzmann factor were discussed in detail in section VIII of [1] and [5].Here I choose, following [1,5], the Wightman inner product With this symmetric inner product the flip property (V, LW) β = (LV, W) β (2.10) holds and Thus the Lanczos orthonormalisation reviewed in section 2 works equally for the Wightman inner product (6.1).
The Hamiltonian H and the sinusoidal coordinate η of the exactly solvable quantum mechanical systems demonstrated in [17,18,16,25] all satisfy the same relationship therefore the essential formulas and as in (5.9).
as in (5.10).These lead to The squared norm of the sinusoidal coordinate η is simplified as before, e −βE(n)/2 e −βE(l)/2 n|η|l l|η|n These results are summarised as the following Theorem 6.1 The moments of the exactly solvable ordinary quantum systems have a very simple exact expression It applies to two non-compact discrete quantum systems with the eigenpolynomials, the Meixner (M) and Charlier (C), as well as the exactly solvable systems related to the Hermite (H), Laguerre (L), Gegenbauer (G), and Jacobi (J) polynomials.
In the rest of this section, the necessary data for the evaluation of the moments µ 2m of the above six exactly solvable quantum mechanical systems are provided.For the ordinary one-dimensional quantum systems, they are the potential U(x) in the Hamiltonian H = p 2 + U(x) in which p is the canonical momentum operator conjugate to x, [p, x] = −i, the sinusoidal coordinates η(x), the energy eigenvalues E(n), R 0 (H) and R 1 (H) for the derivation of α ± E(n) and the coefficients of the three term recurrence relation A n B n and C n .

Meixner and Charlier
These two are non-compact exactly solvable discrete quantum systems.For the summability of the trace, the finite temperature effects (6.1) are needed.These two systems have very simple structure This simply means that all the moments are equal µ 2m = µ 2 , m ∈ N. The Lanczos orthonormalisation stops at O 2 .The other data are As expected, the Boltzmann factor e −βH introduced for the trace definition in 6.1 makes the infinite sum of these two µ 2 's convergent.The summation can be carried out explicitly.

Hermite and Laguerre
These two are the best known examples of exactly solvable one-dimensional quantum mechanical systems, the harmonic oscillator for the Hermite (H) and the harmonic oscillator with a centrifugal potential for the Laguerre (L).Like K and dH cases in §5.1, all the moments form geometrical sequences, µ 2m = λ 2(m−1) µ 2 , λ = 2 for H and λ = 4 for L. The Krylov orthogonalisation stops at O 2 , a clear sign of 'non-complexity' of these solvable systems.The data for each system are: This is a very well-known result.

Gegenbauer and Jacobi
These two systems seem to provide very interesting materials for verifying Krylov complexity.

Summary and comments
For a group of exactly solvable compact discrete quantum systems, the moments of the operators in a Krylov subspace spanned by a Liouville operator L := [H, •] and the sinusoidal coordinate η are evaluated explicitly.They provide the essential tool for measuring the growth of operators evolving under Hamiltonian dynamics [1].Understanding of the complexity of exactly solvable quantum systems would reveal the nature of quantum chaos by contrast.The moments of exactly solvable non-compact discrete dynamics and ordinary one-dimensional quantum systems are also evaluated explicitly by adopting Wightman inner product involving the Boltzmann factor.These exactly solvable discrete quantum mechanical systems can be regarded as a very special type of matrix models.
There are many exactly solvable ordinary quantum mechanical systems having finitely many discrete energy levels.Among them, for example, the Morse potential and the soliton potential (−1/ cosh 2 x) are exactly solvable in the Heisenberg picture, too [18] and the formula L m η = η A m (H) + Lη B m (H) + C m (H) (4.4) is applicable.It is quite natural to expect that their 'complexity' is qualitatively different from that of the systems having infinitely many energy levels only.I cannot apply the Lanczos algorithm to such systems as I do not know how to include the contribution of the continuous energy levels in the operator inner product.
Another type of exactly solvable discrete quantum mechanical systems is also known [18,25] and they are also solvable in the Heisenberg picture.Their eigenvectors contain the hypergeometric orthogonal polynomials of Askey scheme, the Wilson, Askey-Wilson, continuous (dual) (q) Hahn, Meixner-Pollaczek, Al-Salam-Chihara, continuous (big) q-Hermite, continuous q-Jacobi (Laguerre) polynomials.It is expected that the moments of these quantum systems can be evaluated explicitly in a similar manner.
Exact Heisenberg operator solutions are also known for a family of multi-particle dynamics, the Calogero models based on any root systems [26].It is a good challenge to generalise the present method for multi-particle systems.
Four explicit examples of multivariate discrete orthogonal polynomials, the multivariate Krawtchouk, Meixner and two types of Rahman polynomials, are constructed recently by myself [27]- [29].They are eigenvectors of respective Hamiltonians having nearest neighbour and other types of interactions.Investigation of their complexity through Krylov orthonormalisation would expose the contrast between integrable and chaotic multi-particle dynamics.
One of the motivations of the present research is to display the main ingredients of Krylov complexity with the explicit description of the Hamiltonian H and the operator η, which is lacking in some reports.

Remark 2 . 1 2 . 2 . 2
The orthonormalisation is complete when all the Lanczos coefficients {b n } are determined.The orthonormalisation stops at O k when b k+1 vanishes, b k+1 = 0. Two explicit examples of the stopped orthonormalisation will be shown in §5.1.Since the Hilbert space is C N +1 , the totality of the basis {O n } is less than (N + 1) Remark The orthonormal basis O n has the following structure

n|η 2
|n , n|ηLη|n , n|η|n , and they are expressed by the coefficients A n and C n of the three term recurrence relation (3.8).By using the explicit expression ((x|n = φ 0 (x)P n (η)d n and the three term recurrence relation, one obtains

Proposition 5 . 3
shed light on an important property of the moments as stated in the following When all the moments are equal, µ 2m = µ 2 , m ∈ N, the Lanczos orthogonalisation stops at O 2 .Likewise, when all the moments form a geometrical sequence

Z=
def Tr[[e −βH ]], β = 1/T.However, as shown later in the examples §6.1- §6.3, the factor Z cancels out in the calculation of the moments.The norm of an operator is defined by

5
Evaluation of µ 2m of exactly solvable discrete quantum mechanics n|η 2 |n n|A m (H)|n + n|ηLη|n n|B m (H)|n + n|η|n n|C m (H)|n /||η|| 2 , n∈X n|O 0 L m O 0 |n = n∈X n|ηL m η|n /||η|| 2 = n∈X m (H)|n , n|B m (H)|n , n|C m (H)|n are known the Lanczos orthogonalisation stops at O 2 .This case reduces to the constant case by the scaling of the Hamiltonian H → λ −1 H, which is absorbed by the time rescaling t → λt.The very early stopping of the Lanczos orthogonalisation may be considered as a clear sign of 'non-complexity' of integrable systems.