Quasi Exactly Solvable Difference Equations
aa r X i v : . [ n li n . S I] O c t Version: 29 August 2007
Yukawa Institute Kyoto
YITP-07-42July 2007
Quasi Exactly Solvable Difference Equations
Ryu Sasaki
Yukawa Institute for Theoretical Physics,Kyoto University, Kyoto 606-8502, Japan
Abstract
Several explicit examples of quasi exactly solvable ‘discrete’ quantum mechanicalHamiltonians are derived by deforming the well-known exactly solvable Hamiltonians ofone degree of freedom. These are difference analogues of the well-known quasi exactlysolvable systems, the harmonic oscillator (with/without the centrifugal potential) de-formed by a sextic potential and the 1 / sin x potential deformed by a cos 2 x potential.They have a finite number of exactly calculable eigenvalues and eigenfunctions. Exactly solvable and Quasi Exactly Solvable (QES) quantum mechanical systems have playeda very important role in modern physics. The former, the exactly solvable systems, are quitewell-known. In the Schr¨odinger picture, if all the eigenvalues and corresponding eigenfunc-tions are known, the system is exactly solvable. Plenty of such systems are known, for exam-ple, the P¨oschl-Teller and the Morse potential on top of the best-known harmonic oscillatorand the coulomb potential [1] for degree one cases and the Calogero-Sutherland systems[2, 3, 4] for many degrees of freedom cases. Recently the exact Heisenberg operator solutionsand the corresponding annihilation-creation operators are constructed for most of the degreeone exactly solvable quantum mechanics [5] and for the multi-particle Calogero systems [6].The notion of exact solvability was generalised to the so-called ‘discrete’ quantum mechan-ics, in which the Schr¨odinger equation is a difference equation in stead of differential. Theifference analogues of the Calogero-Sutherland systems were constructed by Ruijsenaars-Schneider-van Diejen [7, 8]. The difference equation analogues of the equations determiningthe Hermite, Laguerre and Jacobi polynomials were derived within the Askey-scheme of hy-pergeometric orthogonal polynomials [9, 10]. Later they were reformulated as Hamiltoniandynamics with shape-invariance [11] by Odake-Sasaki [12].In contrast, the latter, Quasi Exactly Solvable (QES) systems have a short history andless known. If a finite number of exact eigenvalues and eigenfunctions are known, the systemis QES [13]. Since the number of exactly solvable states can be chosen as large as wanted, aQES system could be used as a good alternative to an exactly solvable system. Several onedegree of freedom QES systems are listed in [13, 14] and multi-particle QES systems werefirst constructed by Sasaki-Takasaki [15] as deformation of Inozemtsev- Calogero-Sutherlandsystems, which was followed by [16].In the present paper we derive several explicit examples of QES difference equationsas deformation of exactly solvable ‘discrete’ quantum mechanics [12]. They are differ-ence analogues of the well-known quasi exactly solvable systems, the harmonic oscillator(with/without the centrifugal potential) deformed by a sextic potential and the 1 / sin x potential deformed by a cos 2 x potential.This paper is organised as follows. In the next section, the deformation method toobtain a QES from an exactly solvable system is explained in some detail by taking two well-known examples of the ordinary quantum mechanical QES systems. Then two correspondingdifference equation analogues are derived. Section 3 provides three more explicit examples.The final section is for a summary and comments. There are many different ways of deriving QES Hamiltonians for ordinary quantum me-chanics [13, 14]. However, to the best of our knowledge, a very limited number of explicitexamples of QES difference equations are known in connection with U q ( sl (2)) [17]. In theseexamples, quantum wavefunctions are related to those defined on discrete lattice points only.In the present paper we present several explicit examples of QES ‘discrete’ quantum mechan-ical Hamiltonians of one degree of freedom, whose wavefunctions are continuous functions of x as in the ordinary quantum mechanics. They are obtained by deforming exactly solvable ‘discrete’ quantum mechanical Hamiltonians [12], which have the Askey-scheme of hyperge-2metric orthogonal polynomials [9, 10] as part of the eigenfunctions; the Meixner-Pollaczek,continuous Hahn, continuous dual Hahn, Wilson and Askey-Wilson polynomials. This de-formation method was first applied by Sasaki and Takasaki [15] to derive multi-particle QESbased on the Inozemtsev models.For illustrative purposes, we will explain the deformation method for the two well-knownexamples of degree one QES systems in ordinary quantum mechanics in the next subsection.These examples are the sextic ( x ) potential added to the harmonic oscillator ( x ) potential,and another a cos 2 x potential added to the exactly solvable 1 / sin x potential. The samemethod is applied in subsection 2.2 to derive the first two examples of QES Hamiltonian in‘discrete’ quantum mechanics corresponding to the the sextic potential deformation. Therest of the examples are given in section 3. A best-known example of QES Hamiltonian, the harmonic oscillator plus a sextic ( x ) po-tential is given succinctly by H def = − d dx + (cid:18) dWdx (cid:19) + d Wdx + α M ( x ) , α M ( x ) def = − a M x , M ∈ N . (2.1)Here we tentatively call the last term of the above Hamiltonian, α M ( x ), the compensationterm. Throughout this paper we adopt the unit system 2 m = ~ = 1. The real prepotential W is a deformation of that for the harmonic oscillator W = − bx /
2, with a being thedeformation parameter: W = W ( x ) def = − a x + W = − a x − b x , a, b > . (2.2)By the similarity transformation in terms of the pseudo ground state wavefunction φ ( x ) def = e W ( x ) , we obtain ˜ H def = φ − ◦ H ◦ φ = − d dx − dW ( x ) dx ddx − a M x , (2.3)= − d dx + (cid:0) ax + 2 bx (cid:1) ddx − a M x . (2.4)In the absence of the compensation term α M ( x ), φ ( x ) is actually a ground state wave-function, therefore it has no node and it is square integrable. Another characterisation of3he pseudo ground state wavefunction φ is that it is annihilated by the operator A whichfactorises the main part of the Hamiltonian (2.1): Aφ ( x ) = 0 , A def = − ddx + dW ( x ) dx , A † def = ddx + dW ( x ) dx , (2.5) H = A † A − a M x . (2.6)The action of the Hamiltonian ˜ H (2.3) on monomials of x reads˜ H x n = ( − n ( n − x n − + 2 nbx n + 2 a ( n − M ) x n +2 , n ≤ M − , −M ( M − x M− + 2 M bx M , n = M . (2.7)Since the parity is conserved, it is now obvious that ˜ H keeps the polynomial space V M invariant, ˜ H V M ⊆ V M , (2.8) V M def = ( Span (cid:2) , x , . . . , x k , . . . , x M (cid:3) , M : even , Span (cid:2) x, x , . . . , x k +1 , . . . , x M (cid:3) , M : odd , (2.9)and that ˜ H is a tri-diagonal matrix. Thus we can obtain a finite number of exact eigenvaluesand eigenfunctions of the sextic potential Hamiltonian (2.1) in the form: H φ = E φ, φ ( x ) = φ ( x ) P M ( x ) , P M ∈ V M , ⇐⇒ ˜ H P M = E P M , (2.10)by the diagonalisation of a finite dimensional Hamiltonian matrix ˜ H (2.3) withdim V M = (cid:26) M / , M : even , ( M + 1) / , M : odd . (2.11)Since H is obviously hermitian (or self-adjoint), all the eigenvalues are real and eigenfunctionsbelonging to different eigenvalues are orthogonal with each other. In other words, twopolynomial solutions P M and P ′M are orthogonal with respect to the weight function φ ( x ).The square integrability of all the eigenfunctions of the above form (2.10) R ∞−∞ φ ( x ) dx < ∞ isobvious. The true ground state wave function has the form (2.10) with the lowest eigenvalue,say E and it has no node due to the oscillation theorem. / sin x Potential Deformed by cos 2 x Potential
Another well-known example of quasi exactly solvable system is the exactly solvable 1 / sin x potential ( W = g log sin x ) deformed by a cos 2 x potential. The Hamiltonian has the same4orm as (2.1) with only the prepotential W ( x ) and the compensation term α M ( x ) different: H def = − d dx + (cid:18) dWdx (cid:19) + d Wdx + α M ( x ) , α M ( x ) def = 4 a M sin x, M ∈ N , (2.12) W def = a x + W = a x + g log sin x, g > , < x < π. (2.13)Again by the similarity transformation in terms of the pseudo ground state wavefunction Aφ = 0, φ ( x ) def = e W ( x ) , we obtain˜ H def = φ − ◦ H ◦ φ = − d dx − dW ( x ) dx ddx + 4 a M sin x, (2.14)= − d dx + (2 a sin 2 x − g cot x ) ddx + 4 a M sin x. (2.15)Needless to say φ has no node or singularity and it is square integrable R π φ ( x ) dx < ∞ .The action of the Hamiltonian ˜ H (2.14) on monomials of sin x reads˜ H sin n x (2.16)= ( − n ( n − g ) sin n − x + n ( n + 2 g + 4 a ) sin n x + 4 a ( M − n ) sin n +2 x, n ≤ M − , −M ( M − g ) sin M− x + M ( M + 2 g + 4 a ) sin M x, n = M . Since the parity is conserved, it is now obvious that ˜ H keeps the polynomial space V M invariant, ˜ H V M ⊆ V M , (2.17) V M def = ( Span (cid:2) , sin x, . . . , sin k x, . . . , sin M x (cid:3) , M : even , Span (cid:2) sin x, sin x, . . . , sin k +1 x, . . . , sin M x (cid:3) , M : odd , (2.18)dim V M = (cid:26) M / , M : even , ( M + 1) / , M : odd , (2.19)and that ˜ H is again a tri-diagonal matrix. Thus we can obtain a finite number (dim V M ) ofexact eigenvalues and eigenfunctions in the same way as in the sextic potential Hamiltonian(2.1) case. A difference analogue of the sextic potential Hamiltonian (2.1) is H def = p V ( x ) e − i∂ x p V ( x ) ∗ + p V ( x ) ∗ e i∂ x p V ( x ) − ( V ( x ) + V ( x ) ∗ )+ α M ( x ) , (2.20)5 A † A + α M ( x ) , α M ( x ) def = 2 M x , (2.21) A † def = p V ( x ) e − i ∂ x − p V ( x ) ∗ e i ∂ x , A def = e − i ∂ x p V ( x ) ∗ − e i ∂ x p V ( x ) , (2.22) V ( x ) def = ( a + ix )( b + ix ) V ( x ) , V ( x ) def = c + ix, a, b, c ∈ R + . (2.23)Here as usual V ( x ) ∗ is the complex conjugate of V ( x ). If V is replaced by V and the lastterm in (2.20), α M ( x ), is removed, H becomes the exactly solvable Hamiltonian of a differ-ence analogue of the harmonic oscillator, or the deformed harmonic oscillator in ‘discrete’quantum mechanics [12]. Its eigenfunctions consist of the Meixner-Pollaczek polynomial,which is a deformation of the Hermite polynomial [12, 18]. The quadratic polynomial factor( a + ix )( b + ix ) can be considered as multiplicative deformation, although the parameters a , b and c are on the equal footing. On the other hand one can consider it as a multiplicativedeformation by a linear polynomial in x : V ( x ) = ( a + ix ) V ( x ) , V ( x ) def = ( b + ix )( c + ix ) , with V describing another difference version of an exactly solvable analogue of the harmonicoscillator [12]. Its eigenfunctions consist of the continuous Hahn polynomial.Next let us introduce the similarity transformation in terms of the pseudo ground state wavefunction φ ( x ): φ ( x ) def = p Γ( a + ix )Γ( a − ix )Γ( b + ix )Γ( b − ix )Γ( c + ix )Γ( c − ix ) , (2.24)˜ H def = φ − ◦ H ◦ φ = V ( x ) (cid:0) e − i∂ x − (cid:1) + V ( x ) ∗ (cid:0) e i∂ x − (cid:1) + 2 M x . (2.25)It is obvious that φ has no node and that it is square integrable. As in the ordinary quantummechanics, φ ( x ) is annihilated by the A operator (2.22)0 = A φ ( x ) = (cid:16) e − i ∂ x p V ( x ) ∗ − e i ∂ x p V ( x ) (cid:17) φ ( x ) . (2.26)It is rather trivial to verify the action of the Hamiltonian ˜ H (2.25) on monomials of x :˜ H x n = ( P [ n/ j =0 a n, j x n +2 − j , n ≤ M − , a n, j ∈ R , P [ M / j =0 a ′ n, j x M− j , n = M , a ′ n, j ∈ R . (2.27)Here [ m ] is the standard Gauss’ symbol denoting the greatest integer not exceeding or equalto m . Since the parity is conserved, ˜ H keeps the polynomial space V M invariant,˜ H V M ⊆ V M , (2.28)6 M def = ( Span (cid:2) , x , . . . , x k , . . . , x M (cid:3) , M : even , Span (cid:2) x, x , . . . , x k +1 , . . . , x M (cid:3) , M : odd , (2.29)dim V M = (cid:26) M / , M : even , ( M + 1) / , M : odd , (2.30)but ˜ H is no longer a tri-diagonal matrix, ( ˜ H ) j k = 0, j ≥ k −
1. The Hamiltonian H (2.21)is obviously hermitian (self-adjoint) and all the eigenvalues are real and eigenfunctions canbe chosen real. We can obtain a finite number of exact eigenvalues and eigenfunctions bysweeping in a similar way as in the sextic potential case (2.10), (2.11). The oscillation theo-rem linking the number of eigenvalues (from the ground state) to the zeros of eigenfunctionsdoes not hold in the difference equations. The square integrability of all the eigenfunctions R ∞−∞ φ ( x ) dx < ∞ is obvious. Another difference analogue of the sextic potential Hamiltonian (2.1) has the same form as(2.20), (2.21) and (2.22), with only the potential function V ( x ) and the compensation term α M ( x ) are different: V ( x ) def = ( a + ix )( b + ix ) V ( x ) , V ( x ) def = ( c + ix )( d + ix ) , a, b, c, d ∈ R + , (2.31) α M ( x ) def = M ( M − a + b + c + d )) x . (2.32)This Hamiltonian can be considered as a deformation by a quadratic polynomial factor ( a + ix )( b + ix ) of the exactly solvable ‘discrete’ quantum mechanics having the continuous Hahnpolynomials as eigenfunctions [12], another difference analogue of the harmonic oscillator.See the comments in section 5 of [21].The pseudo ground state wavefunction φ ( x ) annihilated by the A operator Aφ = 0reads φ ( x ) (2.33) def = p Γ( a + ix )Γ( a − ix )Γ( b + ix )Γ( b − ix )Γ( c + ix )Γ( c − ix )Γ( d + ix )Γ( d − ix ) . Again it has no node and it is square integrable. The similarity transformed Hamiltonianacting on the polynomial space is˜ H def = φ − ◦ H ◦ φ = V ( x ) (cid:0) e − i∂ x − (cid:1) + V ( x ) ∗ (cid:0) e i∂ x − (cid:1) + M ( M − a + b + c + d )) x . (2.34)7t is straightforward to verify the relationship (2.27) and to establish the existence of theinvariant polynomial subspaces of given parity (2.28), (2.29) and (2.30). The hermiticity ofthe Hamiltonian and the square integrability of the eigenfunctions also hold. Thus anotherexample of quasi exactly solvable difference equation is established. The other two examples are the difference equation analogues of the harmonic oscillatorwith the centrifugal potential deformed by the sextic potential. There are two types corre-sponding to the linear and quadratic polynomial deformations. The corresponding exactlysolvable difference equation has the Wilson polynomial [12, 9, 10] as the eigenfunctions. Thelast example is the difference analogue of the model discussed in subsection 2.1.2, 1 / sin x potential deformed by a cos 2 x potential. In this case the corresponding exactly solvable dif-ference equation has the Askey-Wilson polynomials [12, 9, 10] as eigenfunctions. The basicidea for showing quasi exact solvability is almost the same as shown above. The Hamiltonians have the same form as (2.20), (2.21) and (2.22), with only the potentialfunction V ( x ) and the compensation term α M ( x ) are different:Type I : V ( x ) def = ( b + ix ) V ( x ) , α M ( x ) def = M x , (3.1)Type II : V ( x ) def = ( a + ix )( b + ix ) V ( x ) , (3.2) α M ( x ) def = M ( M − a + b + c + d + e + f )) x , (3.3)with a common V ( x ) V ( x ) def = ( c + ix )( d + ix )( e + ix )( f + ix )2 ix (2 ix + 1) , a, b, c, d, e, f ∈ R + − { / } . (3.4)None of the parameters a , b , c , d , e or f should take the value 1 /
2, since it would cancel thedenominator. Because of the centrifugal barrier, the dynamics is constrained to a half line;0 < x < ∞ . The type I case can also be considered as a quadratic polynomial deformationof the exactly solvable dynamics with V ( x ):Type I : V ( x ) def = ( b + ix )( c + ix ) V ( x ) , V ( x ) def = ( d + ix )( e + ix )( f + ix )2 ix (2 ix + 1) , (3.5)8hich has the continuous dual Hahn polynomials [12, 9, 10] as eigenfunctions. This re-interpretation does not change the dynamics, since the Hamiltonian and A and A † operatorsdepend on V ( x ).The pseudo ground state wavefunction φ ( x ) is determined as the zero mode of the A operator Aφ = 0: Type I : φ ( x ) def = qQ j =2 Γ( a j + ix )Γ( a j − ix ) p Γ(2 ix )Γ( − ix ) , (3.6)Type II : φ ( x ) def = qQ j =1 Γ( a j + ix )Γ( a j − ix ) p Γ(2 ix )Γ( − ix ) , (3.7)in which the numbering of the parameters a = a, a = b, a = c, a = d, a = e, a = f, (3.8)is used. It is obvious that both φ have no node in the half line 0 < x < ∞ .The similarity transformed Hamiltonian acting on the polynomial space has the sameform as before (2.34)˜ H def = φ − ◦ H ◦ φ = V ( x ) (cid:0) e − i∂ x − (cid:1) + V ( x ) ∗ (cid:0) e i∂ x − (cid:1) + α M ( x ) . (3.9)Although the potential V ( x ) has the harmful looking denominator 1 / { ix (2 ix + 1) } , it isstraightforward to verify that ˜ H maps a polynomial in x into another:˜ H x n = ( P n +1 j =0 a n, j x n +2 − j , n ≤ M − , a n, j ∈ R , P M j =0 a ′ n, j x M− j , n = M , a ′ n, j ∈ R . (3.10)This is because V , which has the above denominator, keeps the polynomial subspace ofany even degree invariant, reflecting the exact solvability. This establishes that ˜ H keeps thepolynomial space V M invariant,˜ H V M ⊆ V M , (3.11) V M def = Span (cid:2) , x , . . . , x k , . . . , x M (cid:3) , dim V M = M + 1 . (3.12)The hermiticity of the Hamiltonians is obvious and the square-integrability of the eigenfunc-tions R ∞ φ ( x ) dx < ∞ holds true. This establishes the quasi exact solvability.9he corresponding quantum mechanical system has the prepotential W ( x ) and the com-pensation term α M ( x ) as W ( x ) = − a x − b x + g log x, a, b, g > , < x < ∞ , (3.13) α M ( x ) = − a M x . (3.14)This and the above two difference analogue systems share the same invariant polynomialsubspace (3.11), (3.12). The undeformed exactly solvable system, i.e. (3.13) with a = 0, hasthe Laguerre polynomials as eigenfunctions. The corresponding undeformed exactly solvabledifference equations determined by V (3.4), V (3.5) have the Wilson and the continuousdual Hahn polynomials as eigenfunctions. These are three and two parameter deformationof the Laguerre polynomial [12, 9, 10]. / sin x Potential Deformedby cos 2 x Potential
This system is a quasi exactly solvable deformation of the exactly solvable dynamics whichhas the Askey-Wilson polynomials [12, 9, 10] as eigenfunctions. Let us first introduce thevariables and notation appropriate for the Askey-Wilson polynomials. We use variables θ , x and z , which are related as 0 < θ < π, x = cos θ, z = e iθ . (3.15)The dynamical variable is θ and the inner product is h f | g i = R π dθf ( θ ) ∗ g ( θ ). We denote D def = z ddz . Then q D is a q -shift operator, q D f ( z ) = f ( qz ), with 0 < q <
1. The Hamiltonianis obtained by deforming the potential function V ( z ) by a linear polynomial in z : H def = p V ( z ) q D p V ( z ) ∗ + p V ( z ) ∗ q − D p V ( z ) − ( V ( z ) + V ( z ) ∗ ) + α M ( z ) , (3.16)= A † A + α M ( z ) , α M ( z ) def = − abcdeq − (1 − q M )( z + 1 z ) , (3.17) A † def = − i (cid:16)p V ( z ) q D − p V ( z ) ∗ q − D (cid:17) , A def = i (cid:16) q D p V ( z ) ∗ − q − D p V ( z ) (cid:17) , (3.18) V ( z ) def = (1 − az ) V ( z ) , V ( z ) def = (1 − bz )(1 − cz )(1 − dz )(1 − ez )(1 − z )(1 − qz ) , (3.19) − < a, b, c, d, e < . (3.20)The pseudo ground state wavefunction φ ( z ) is determined as the zero mode of the A Aφ = 0: φ ( z ) def = s ( z , z − ; q ) ∞ ( az, az − , bz, bz − , cz, cz − , dz, dz − , ez, ez − ; q ) ∞ , (3.21)where ( a , · · · , a m ; q ) ∞ = Q mj =1 Q ∞ n =0 (1 − a j q n ). Obviously φ has no node or singularity in0 < θ < π . We look for exact eigenvalues and eigenfunctions of the Hamiltonian (3.16) inthe form: H φ = E φ, φ ( z ) = φ ( z ) P M ( x ) , (3.22)in which P M ( x ) is a degree M polynomial in x or in z + 1 /z = 2 cos θ = 2 x . The similaritytransformed Hamiltonian acting on the polynomial space has the form˜ H def = φ − ◦ H ◦ φ = V ( z ) (cid:0) q D − (cid:1) + V ( z ) ∗ (cid:0) q − D − (cid:1) − abcdeq − (1 − q M )( z + 1 z ) . (3.23)Without the deformation factor 1 − az and the compensation term, the above Hamiltonian˜ H keeps the polynomial subspace in z + 1 /z of any degree invariant. It is straightforward toshow ˜ H V M ⊆ V M , (3.24) V M def = Span " , z + 1 z , . . . , (cid:18) z + 1 z (cid:19) k , . . . , (cid:18) z + 1 z (cid:19) M , dim V M = M + 1 . (3.25)The hermiticity of the Hamiltonian is obvious and the square-integrability of the eigenfunc-tions holds also true. This establishes the quasi exact solvability. First let us summarise: Five explicit examples of quasi exactly solvable difference equationsof one degree of freedom are derived by multiplicatively deforming the the known exactlysolvable difference equations for the Meixner-Pollaczek, continuous Hahn, continuous dualHahn, Wilson and Askey-Wilson polynomials. The finite dimensional Hamiltonian matrix,no longer tri-diagonal, can be solved exactly by sweeping. All the eigenvalues and eigen-functions are real, but the oscillation theorem, connecting the excitation level to the numberof zeros, does not hold. Similarity and contrast with the known QES examples in ordinaryquantum mechanics; the harmonic oscillator (with or without the centrifugal potential) de-formed by a sextic potential and the 1 / sin x potential deformed by a cos 2 x potential, aredemonstrated in some detail. 11 few comments are in order. First let us stress that the mere existence of the finitedimensional invariant polynomial subspace ˜ H V M ⊆ V M is not sufficient for the quasi ex-act solvability. The theory must be endowed with a pseudo ground state wavefunction φ ,which must be nodeless and square integrable. Moreover, the reverse similarity transformedHamiltonian H def = φ ◦ ˜ H ◦ φ − must be hermitian, in order to guarantee the real spectrum.Let us consider an additively deformed potential with a cubic term V ( x ) def = ax + V ( x ) , α M ( x ) def = a M ( M − x, a ∈ R , (4.1)with an exactly solvable V , for example, V ( x ) = b + ix, or ( b + ix )( c + ix ) , b, c ∈ R + , (4.2)corresponding to the Meixner-Pollaczek and the continuous Hahn polynomials. It is rathertrivial to verify the existence of the invariant polynomial subspace:˜ H def = V ( x ) (cid:0) e − i∂ x − (cid:1) + V ( x ) ∗ (cid:0) e i∂ x − (cid:1) + a M ( M − x, (4.3)˜ H V M ⊆ V M , V M def = Span (cid:2) , x, x , . . . , x k , . . . , x M (cid:3) . (4.4)Although we have not been able to derive the explicit form of the pseudo ground statewavefunction φ ( x ) as a solution of Aφ = 0, it seems rather unlikely that the φ satisfiesthe above mentioned criteria. This is because the corresponding quantum mechanical case,the quartic oscillator, W ( x ) = ax − b x , α M ( x ) = 6 a M x, (4.5)also have the invariant polynomial subspace (4.4). The square integrability of φ ( x ) = e W ( x ) does not hold whichever sign a might take.Let us discuss the hermiticity of the Hamiltonians (2.20) and (3.16). The hermiticitymeans h g |H f i = hH g | f i for a given inner product h g | f i for arbitrary elements f and g ina certain dense subspace of the appropriate Hilbert space. The obvious choice for such asubspace is spanned by the ‘pseudo ground state’ wavefunction φ times polynomials. Thetypes of the polynomials are:( a ) : polynomials in x for the Hamiltonians in section 2.2 , (4.6) h g | f i = Z ∞−∞ g ( x ) ∗ f ( x ) dx, f ( x ) = φ ( x ) P ( x ) , g ( x ) = φ ( x ) Q ( x ) , (4.7)( b ) : polynomials in x for the Hamiltonians in section 3.1 , (4.8)12 g | f i = Z ∞ g ( x ) ∗ f ( x ) dx, f ( x ) = φ ( x ) P ( x ) , g ( x ) = φ ( x ) Q ( x ) , (4.9)( c ) : polynomials in x = cos θ for the Hamiltonians in section 3.2 , (4.10) h g | f i = Z π g ( θ ) ∗ f ( θ ) dθ, f ( θ ) = φ ( z ) P (cos θ ) , g ( θ ) = φ ( z ) Q (cos θ ) . (4.11)The Hamiltonians (2.20) and (3.16) consist of three parts: H = H + H + H , H = α M − ( V + V ∗ ) , (4.12)and H = p V ( x ) e − i∂ x p V ( x ) ∗ , H = p V ( x ) ∗ e i∂ x p V ( x ) , for (2.20) , (4.13) H = p V ( z ) q D p V ( z ) ∗ , H = p V ( z ) ∗ q − D p V ( z ) , for (3.16) . (4.14)It is obvious that H is hermitian by itself. When H acts on f , the argument is shifted from x to x − i or from θ to θ − i log q . With the compensating change of integration variablefrom x to x + i or from θ to θ + i log q one can formally show h g |H f i = hH g | f i in astraightforward way. Similarly we have h g |H f i = hH g | f i by another change of integrationvariable. This is the ‘formal hermiticity.’Actually, the shift of integration variable, to be realised by the Cauchy integral, wouldinvolve additional integration contours:( a ) : ( −∞ , ± i − ∞ ) , (+ ∞ , ± i + ∞ ) for the Hamiltonians in section 2.2 , (4.15)( b ) : (0 , ± i ) , (+ ∞ , ± i + ∞ ) for the Hamiltonians in section 3.1 , (4.16)( c ) : (0 , ± i log q ) , ( π, π ± i log q ) for the Hamiltonians in section 3.2 . (4.17)It should be noted that all the singularities arising from V and V ∗ in cases ( b ) and ( c )are cancelled by the zeros coming from the pseudo ground state wavefunctions φ and φ ∗ ,and the Cauchy integration formula applies in all cases. The contribution of the additionalcontour integrals (4.15)–(4.17) cancel with each other and the shifts of integration variablesis justified.To be more specific, the contribution from the contours at infinity in ( a ) and ( b ) vanishidentically due to the strong damping by φ and φ ∗ . The contribution from the two verticalcontours in ( b ) passing the origin cancel with each other due to the evenness of φ , φ ( x ) = φ ( − x ) and the polynomials in x , P (( − x ) ) = P ( x ). Likewise the contribution from thetwo vertical contours in ( c ) passing the origin cancel with each other. Those in ( c ) passing13 also cancel with each other due to the 2 π periodicity of φ , φ ( θ ) = φ (2 π + θ ). Note thatthe hermiticity in ( b ) and ( c ) cases holds only for the sum H + H . This concludes thecomments on hermiticity.It is now evident that the scope of the present method, deformation of W or V for gener-ating a quasi exactly solvable system from an exactly solvable dynamics, is rather limited. Itis highly unlikely to get a quasi exactly solvable system, if W contains a term higher than x or cos 4 x , or if V has a form Q nj =1 ( a j + ix ), n ≥ (cid:16)Q nj =1 ( a j + ix ) (cid:17) / { ix (2 ix + 1) } , n ≥ i.e. quasi-exactly solvable Calogero-Sutherland systems [2, 3] are derived by Sasakiand Takasaki [15].For simplicity of presentation, we have restricted the parameters a , b , c , d , e and f in V ( x )to be real, positive etc. In most cases the ranges of parameters can be relaxed without losingquasi exact solvability. For example, in the first example of subsection 2.2.1, a configuration a > b and c are complex conjugate, b = c ∗ with positive real parts, is also possible.It is a challenge to see if these newly derived quasi exactly solvable difference equations canbe understood by the existing ideas of QES; the sl (2) algebra [14] and/or its deformations,various generalised super symmetry ideas [19]. Can we use the Bethe ansatz method to solvethese problems? Are these QES systems equivalent to some spin systems [20]? Acknowledgements
We thank Choon-Lin Ho for useful comments. This work is supported in part by Grants-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science andTechnology, No.18340061 and No.19540179.
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