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[EQUATION] The reduced eigenvalue problem has two eigenvalues with the expansions [EQUATION] and [EQUATION] By the perturbation theory, for every [MATH] sufficiently small, [MATH] has one simple small negative eigenvalue and one simple small positive eigenvalue. Other eigenvalues are bounded away from zero for small [M... |
Zero eigenvalue of [MATH] is triple and is associated with the subspace spanned by [MATH] We proceed again with full perturbative expansions. First, we express [MATH] |
for the subspace spanned by [MATH] in terms of [MATH] for the subspace spanned by [MATH] [MATH] , and [MATH] . We assume that [MATH] for the small eigenvalues and neglect terms of the order [MATH] and higher. This expansion is given by |
[EQUATION] Next, we substitute these expansions to the first, third and fourth equations of the eigenvalue problem for [MATH] and again neglect terms of the order [MATH] and higher: |
[EQUATION] The double zero eigenvalue persists due to two gauge symmetries, whereas one eigenvalue is expanded by [EQUATION] By the perturbation theory, for every [MATH] sufficiently small, [MATH] has one simple (small) negative eigenvalue and the double zero eigenvalue. Other eigenvalues are bounded away from zero for... |
In the remainder of this section, we consider the two constraints related to the fixed values of [MATH] and [MATH] . The constraints may change the number of negative eigenvalues of the linearization operator [MATH] constrained by the following two orthogonality conditions |
[EQUATION] where [MATH] denotes a real-valued solution of system ( 2.10 ) and [MATH] The constrained space [MATH] is a symplectically orthogonal subspace of [MATH] |
to [MATH] , the two-dimensional subspace associated with the double zero eigenvalue of [MATH] related to the phase rotation symmetries ( 2.2 ) and ( 2.3 ). Alternatively, the constrained space arises when the perturbation [MATH] does not change at the linear approximation the conserved quantities [MATH] and [MATH] defi... |
Let [MATH] denote the stationary state of the stationary equation ( 2.10 ) continued with respect to two parameters [MATH] . Let [MATH] and [MATH] denote the number of negative and zero eigenvalues of |
[MATH] in [MATH] counted with their multiplicities, where [MATH] is the linearized operator at [MATH] Let [MATH] and [MATH] denote the number of negative and zero eigenvalues of [MATH] constrained in [MATH] Assume non-degeneracy of the stationary state [MATH] in the sense that [MATH] By Theorem 4.1 in |
, we have [EQUATION] where [MATH] and [MATH] are the number of positive and zero eigenvalues of the [MATH] matrix [EQUATION] with [MATH] and [MATH] evaluated at the stationary solution [MATH] as a function of the two parameters [MATH] |
Remark 9 For the pair of stationary states ( 1.4 ), it was computed in that [EQUATION] where [MATH] and [MATH] are related to the parameters [MATH] and [MATH] in ( 1.5 ). Substituting ( 4.18 ) into ( 4.17 ) yields |
[EQUATION] hence, [MATH] has one positive and one negative eigenvalue. If [MATH] holds, then [MATH] and [MATH] by ( 4.16 ). We will show in Lemma below that the same count is true for all three branches of Theorem bifurcating from [MATH] |
By the scaling transformation ( 2.1 ), if the stationary state is given by ( 2.9 ) with real [MATH] then the stationary state is continued with respect to parameter [MATH] as |
[EQUATION] hence [MATH] [MATH] , and [MATH] Substituting these relations into [MATH] and [MATH] for [MATH] yields [EQUATION] and |
[EQUATION] where [MATH] and [MATH] Substituting these representations into ( 4.17 ), evaluating derivatives, and setting [MATH] yield the computational formula |
[EQUATION] which can be used to compute [MATH] for the normalized stationary state [MATH] with [MATH] The following lemma gives the variational characterization of the three branches in Theorem bifurcating from [MATH] |
as critical points of [MATH] subject to fixed [MATH] and [MATH] Lemma 5 Consider the three bifurcating branches in Theorem for [MATH] . For every [MATH] with [MATH] sufficiently small, the following is true: |
(i) The branch is a saddle point of [MATH] subject to fixed [MATH] and [MATH] with [MATH] and [MATH] (ii) The branch is a saddle point of [MATH] subject to fixed [MATH] and [MATH] |
with [MATH] and [MATH] (iii) The branch is a minimizer of [MATH] subject to fixed [MATH] and [MATH] with [MATH] and [MATH] The critical points are degenerate only with respect to the two phase rotations 2.2 ) and ( 2.3 ) resulting in [MATH] and [MATH] |
Proof. Case (i): [MATH] [MATH] We compute [MATH] and [MATH] as powers of [MATH] with the following relation between [MATH] and [MATH] |
[EQUATION] Then it follows that [EQUATION] so that [EQUATION] has one positive and one negative eigenvalue. Since [MATH] and [MATH] by Lemma , we have [MATH] |
by ( 4.16 ). At the same time, [MATH] and [MATH] by Lemma Case (ii): [MATH] [MATH] We compute [MATH] and [MATH] as powers of [MATH] with the following relation between [MATH] and [MATH] |
[EQUATION] Then it follows that [EQUATION] so that [EQUATION] has one positive and one negative eigenvalue. Since [MATH] and [MATH] by Lemma ), we have |
[MATH] by ( 4.16 ). At the same time, [MATH] and [MATH] by Lemma Case (iii): [MATH] [MATH] We compute [MATH] and [MATH] as powers of [MATH] with the following relation between [MATH] and [MATH] |
[EQUATION] Then it follows that [EQUATION] The only difference in these expansions compared to the case (ii) is the remainder term as large as [MATH] |
compared to [MATH] . This changes the remainder terms in [MATH] to [MATH] compared to [MATH] but does not affect the conclusion on [MATH] . Moreover, we can see that the computation of [MATH] agrees with the exact expression ( 4.19 ) in Remark . The count [MATH] |
and [MATH] follows by Lemma Remark 10 The presence of conserved quantity [MATH] in ( 2.6 ) does not modify the variational characterization of the stationary states with [MATH] because [MATH] if [MATH] is the stationary state ( 2.9 ) with [MATH] . This follows from the fact that [MATH] |
is independent of [MATH] , which is impossible if [MATH] and [MATH] Hence, any stationary state ( 2.9 ) must satisfy the constraint: |
[EQUATION] which can be verified for all states in Theorem 3.8 and bifurcating from [MATH] for [MATH] 5. Bifurcation from the second eigenmode |
Here we study bifurcations of stationary states in the system of algebraic equations ( 3.1 from the second eigenmode given by ( 2.11 ) with [MATH] Without loss of generality, the scaling transformation ( 2.1 ) yields |
[MATH] and [MATH] By setting [MATH] with real-valued perturbation [MATH] , we rewrite the system ( 3.1 ) with [MATH] in the perturbative form ( 3.2 ), where [MATH] is a block-diagonal operator with the diagonal entries |
[EQUATION] and the only nonzero off-diagonal entries [MATH] , whereas the nonlinear terms are given by [EQUATION] We have the following result on the nonlinear terms. |
Lemma 6 Fix an integer [MATH] . If [MATH] and [EQUATION] then [MATH] and [EQUATION] Proof. The argument repeats the proof of Lemma . Under the conditions ( 5.2 ), every term in [MATH] for [MATH] with [MATH] , and |
[MATH] is inspected and shown to be zero. Bifurcations from the second eigenmode are identified by zero eigenvalues of the diagonal operator [MATH] |
Lemma 7 There exists a sequence of bifurcations at [MATH] with [MATH] [MATH] , and [EQUATION] All bifurcation points are simple except for the three double points |
[MATH] [MATH] , and [MATH] Proof. The diagonal terms [MATH] for [MATH] vanish at the sequence ( 5.4 ). In addition, the double block for [MATH] and [MATH] has zero eigenvalues if and only if [MATH] |
or [MATH] . Therefore, [MATH] is a double bifurcation point. To study other double bifurcation points, we consider solutions of [MATH] for [MATH] and [MATH] for [MATH] Equation [MATH] is equivalent to [MATH] , which has no integer solutions. Equation [MATH] for [MATH] |
is equivalent to [MATH] , which can be solved for [MATH] in terms of [MATH] [EQUATION] Since the right-hand side is monotonically decreasing in [MATH] and [MATH] we can find all integer solutions for [MATH] in the range from [MATH] to [MATH] There exists only two pairs of integer solutions in this range, which give two... |
Simple bifurcation points can be investigated similarly to the proof of Theorem 3.8 This yields the following theorem. Theorem 3 |
Fix [MATH] . There exists a unique branch of solutions [MATH] to system ( 3.1 ) with [MATH] , which can be parameterized by small [MATH] such that [MATH] is smooth in [MATH] and |
[EQUATION] Fix an integer [MATH] with [MATH] and [MATH] . There exists a unique branch of solutions [MATH] to system ( 3.1 ) with [MATH] , which can be parameterized by small [MATH] such that [MATH] is smooth in [MATH] and |
[EQUATION] Proof. The proof of the second assertion repeats the proof of Theorem 3.8 verbatim. The proof of the first assertion is based on the block-diagonalization of the singular matrix for [MATH] with [MATH] |
[EQUATION] The null space is spanned by the vector [MATH] and the vector [MATH] in the decomposition ( 3.9 ) must satisfy the constraint [MATH] The rest of the proof repeats the proof of Theorem 3.8 after a simple observation that [MATH] as [MATH] |
Remark 11 The unique branch bifurcating from [MATH] coincides with the exact solution ( 2.20 ) and 2.21 ) for the lower sign. Indeed, by normalizing [MATH] and taking the limit [MATH] as in ( 2.23 ), we obtain [MATH] [MATH] and [MATH] . The small parameter [MATH] is defined in terms of the small parameter [MATH] |
by [MATH] The three double bifurcation points in Lemma have to be checked separately. In order to characterize branches bifurcating from the double point [MATH] we note the following symmetry. If [MATH] is a generating function for the conformal flow ( 1.1 given by the power series ( 2.24 ), so is [MATH] . If |
[MATH] is a stationary state in the form ( 2.26 with [MATH] satisfying the system ( 3.1 with parameters [MATH] , then the transformed state |
[EQUATION] also satisfies the system ( 3.1 ) with parameters [EQUATION] By Theorem , three branches of solutions bifurcate from the lowest eigenmode at the double bifurcation point [MATH] . Applying the transformation ( 5.7 ) yields three branches bifurcating from the second eigenmode at the double bifurcation point [M... |
Theorem 4 Fix [MATH] There exist exactly three branches of solutions [MATH] to system ( 3.1 ) with [MATH] , which can be parameterized by small [MATH] such that [MATH] is smooth in [MATH] and |
Branches bifurcating from the double point [MATH] can be investigated by computing the normal form. The following theorem represents the main result. |
Theorem 5 Fix [MATH] There exist exactly two branches of solutions [MATH] to system ( 3.1 ) with [MATH] , which can be parameterized by small [MATH] such that [MATH] is smooth in [MATH] and |
[EQUATION] The two branches are characterized by the following: (i) [MATH] [MATH] (ii) [MATH] [MATH] Proof. The proof follows the computations of normal form in Theorem For the double bifurcation point [MATH] we write the decomposition |
[EQUATION] where [MATH] are arbitrary and [MATH] are set from the orthogonality condition [MATH] By performing routine computations and expanding the bifurcation equations at [MATH] and [MATH] |
up to and including the cubic order, we obtain [EQUATION] We are looking for solutions to the system ( 5.12 ) with [MATH] . There exist two nontrivial solutions to the system ( 5.12 ): |
(I) [MATH] [MATH] , and [MATH] (II) [MATH] [MATH] , and [MATH] whereas no solution exists with both [MATH] and [MATH] The Jacobian matrix of the system ( 5.12 ) is given by |
[EQUATION] For both branches (I) and (II), the Jacobian is invertible for small [MATH] , hence the two solutions are continued uniquely with respect to parameters [MATH] and yield branches (i) and (ii). By Lemma with [MATH] , branch (i) corresponds to the reduction ( 5.2 with [MATH] , hence [MATH] persists beyond all o... |
Remark 12 Branch (ii) of Theorem can be obtained by the symmetry transformation ( 5.7 ) from the branch bifurcating from lowest eigenstate at [MATH] |
For the remaining double point [MATH] , bifurcation of stationary state is more complicated. If we compute the normal form up to and including the cubic order, we obtain a trivial normal form, which is satisfied identically if [MATH] . This outcome of the normal form computations suggests that there exists a two-parame... |
[EQUATION] subject to the orthogonality conditions [MATH] and [MATH] . By computing power expansions for small [MATH] with MAPLE, we can extend it to any polynomial order with the first terms given by |
[EQUATION] The three explicit solutions ( 2.17 ), ( 2.29 ), and ( 2.31 are particular solutions of the two-parameter family of stationary states for small [MATH] . Indeed, the twisted state ( 2.17 ) corresponds to |
[MATH] and [MATH] the Blaschke state corresponds to [MATH] and [MATH] and the additional state ( 2.31 ) corresponds to [MATH] and [MATH] |
Remark 13 We have shown by using the general transformation law derived in that the twisted state ( 2.17 ) can be obtained from the [MATH] single-mode state ( 2.11 by applying [MATH] with [MATH] in the symmetry transformation ( 2.7 with [MATH] . In the present time, we do not know how to obtain the two-parameter branch... |
6. Variational characterization of the bifurcating states We shall now give variational characterization of the bifurcating states from the second eigenmode. The second variation of the action functional [MATH] in ( 4.1 ) is given by the quadratic forms in ( 4.2 ) with the self-adjoint operators |
[MATH] given by ( 4.3 ). The following lemma gives variational characterization of the second eigenmode at the bifurcation points [MATH] in Lemma |
Lemma 8 The following is true: For [MATH] , the [MATH] single-mode state ( 2.11 ) is a degenerate saddle point of [MATH] with three positive eigenvalues, zero eigenvalue of multiplicity three, and infinitely many negative eigenvalues bounded away from zero. |
For [MATH] and [MATH] , the [MATH] single-mode state ( 2.11 ) is a degenerate minimizer of [MATH] with zero eigenvalue of multiplicity three and infinitely many positive eigenvalues bounded away from zero. |
For [MATH] with [MATH] and [MATH] , the [MATH] single-mode state ( 2.11 ) is a degenerate saddle point of [MATH] with an even number of negative eigenvalues, zero eigenvalue of odd multiplicity, and infinitely many positive eigenvalues bounded away from zero. |
Proof. By the scaling transformation ( 2.1 ), we take [MATH] and [MATH] for which the explicit form ( 4.3 ) yields [EQUATION] We note that [MATH] given by ( 5.1 ) and |
[MATH] is only different from [MATH] at the diagonal entry at [MATH] (which is [MATH] instead of [MATH] ) and for the off-diagonal entries at [MATH] and [MATH] |
(which are [MATH] instead of [MATH] ). For [MATH] , the [MATH] block at [MATH] and [MATH] [EQUATION] is positive definite for all [MATH] with a simple zero eigenvalue only arising for [MATH] and [MATH] . The diagonal entries of [MATH] |
for [MATH] are given by [EQUATION] These entries are negative if [MATH] with a simple zero eigenvalue and strictly positive if [MATH] . For [MATH] with [MATH] the diagonal entries are simplified in the form: |
[EQUATION] For [MATH] and [MATH] , two eigenvalues are zero, six are negative, and all others are positive. For [MATH] and [MATH] , two eigenvalues are zero, one is negative, and all others are positive. For [MATH] , one eigenvalue is zero and all others are positive. For [MATH] and [MATH] one eigenvalue is zero, finit... |
Multiplying these counts for [MATH] by a factor of [MATH] due to the matrix operator [MATH] and adding an additional zero entry of [MATH] at [MATH] yields the assertion of the lemma. |
Among all stationary states bifurcating from the second eigenmode, we shall only consider the potential minimizers of energy [MATH] subject to fixed [MATH] and [MATH] . By Lemma , this includes only two branches bifurcating from [MATH] and [MATH] . In both cases, we are able to compute the number of negative eigenvalue... |
Lemma 9 Consider the two bifurcating branches in Theorem 5.6 for [MATH] and [MATH] . For every small [MATH] we have [MATH] and [MATH] . For each branch, [MATH] has a double zero eigenvalue, |
[MATH] has no zero eigenvalue, and the rest of the spectrum of [MATH] and [MATH] is strictly positive and is bounded away from zero. |
Proof. By the second item of Lemma , the corresponding operators [MATH] and [MATH] at the bifurcation point [MATH] have respectively the simple and double zero eigenvalue, whereas the rest of their spectra is strictly positive and is bounded away from zero. By the two symmetries ( 2.2 ) and ( 2.3 ), the double zero eig... |
and the assertion of the lemma for [MATH] follows by the perturbation theory. On the other hand, the simple zero eigenvalue of [MATH] is not preserved for [MATH] |
and will generally shift to either negative or positive values. We will show that it shifts to the negative values for [MATH] , hence [MATH] in both cases and the assertion of the lemma for [MATH] follows by the perturbation theory. |
For [MATH] , the Lyapunov–Schmidt decomposition of Theorem 5.6 yields power expansion [EQUATION] and [MATH] [MATH] [MATH] [MATH] and [MATH] for [MATH] . Similarly to the proof of Lemma we compute the [MATH] -by- [MATH] block of the operator [MATH] at [MATH] and truncate it up to and including [MATH] terms. The correspo... |
[EQUATION] We are looking for a small eigenvalue of [MATH] , where [MATH] Assuming [MATH] and expressing [MATH] in terms of [MATH] yield |
[EQUATION] Substituting these expressions into the eigenvalue problem [MATH] and truncating it up to and including the order of [MATH] , we obtain |
[EQUATION] The small eigenvalue of this reduced problem is given by the expansion [EQUATION] Hence [MATH] due to the shift of the zero eigenvalue of [MATH] at [MATH] to [MATH] for [MATH] |
For [MATH] , the Lyapunov–Schmidt decomposition of Theorem 5.6 yields power expansion [EQUATION] with [MATH] [MATH] [MATH] [MATH] for [MATH] , where [MATH] for every [MATH] [MATH] |
by Lemma . Similarly to the proof of Lemma we compute the [MATH] -by- [MATH] block of the operator [MATH] at [MATH] [MATH] , and [MATH] |
and truncate it up to and including [MATH] terms. The corresponding matrix block denoted by [MATH] is given by [EQUATION] Looking at the small eigenvalue of [MATH] , where [MATH] we can normalize [MATH] and obtain |
[EQUATION] and [EQUATION] Hence [MATH] due to the shift of the zero eigenvalue of [MATH] at [MATH] to [MATH] for [MATH] Finally, we consider the two constraints related to the fixed values of [MATH] and [MATH] by using the same computational formulas ( 4.16 ) and ( 4.17 ). Let [MATH] denote the stationary state of the ... |
[EQUATION] which can be used to compute [MATH] for the normalized stationary state [MATH] with [MATH] The following lemma confirms that the two branches in Lemma are indeed local minimizers of [MATH] subject to fixed [MATH] and [MATH] |
Lemma 10 Consider the two bifurcating branches in Theorem 5.6 for [MATH] and [MATH] . For every small [MATH] , the two branches are local minimizers of [MATH] subject to fixed [MATH] and [MATH] |
with [MATH] and [MATH] Proof. For [MATH] , we use power expansions in Lemma and compute [EQUATION] so that [EQUATION] Note that the expression for [MATH] agrees with the exact computations in Remark Therefore, [MATH] has one positive and one negative eigenvalue, so that [MATH] |
by ( 4.16 ). For [MATH] , we use power expansions in Lemma and compute [EQUATION] so that [EQUATION] has again one positive and one negative eigenvalue, hence [MATH] |
by ( 4.16 ). 7. Numerical approximations We confirm numerically that the stationary states ( 1.4 ) with ( 1.5 have the same variational characterization in the entire existence interval for [MATH] in [MATH] Therefore, they remain constrained minimizers of [MATH] for fixed [MATH] and [MATH] |
Figure shows the smallest eigenvalues of [MATH] computed at the upper branch of solution 1.4 ). In agreement with item (iii) in Lemma for small [MATH] we have [MATH] and [MATH] for every [MATH] The small parameter [MATH] is related to the small parameter [MATH] in Lemma |
by [MATH] , see Remark . The dashed line shows the asymptotic dependencies ( 4.12 ), ( 4.13 ), and ( 4.14 for the small eigenvalues of [MATH] and [MATH] |
Figure shows the smallest eigenvalues of [MATH] computed at the lower branch of solution 1.4 ). In agreement with Lemma for small [MATH] we have [MATH] and [MATH] in the entire region of existence of the stationary state. The small parameter [MATH] is related to the small parameter [MATH] in Lemma |
by [MATH] , see Remark 11 The dashed line shows the asymptotic dependence ( 6.2 ) for the small eigenvalue of [MATH] Figure shows the smallest eigenvalues of [MATH] computed at the stationary state bifurcating from the second eigenmode at [MATH] . We use here parameter [MATH] |
for continuation of the stationary state as in Lemma In agreement with Lemma , we have [MATH] and [MATH] for small [MATH] However, this result does not hold for larger values of [MATH] far from the bifurcation point because additional eigenvalues of [MATH] and [MATH] become negative eigenvalue for [MATH] Therefore, the... |
Remark 14 The presence of zero eigenvalue in the spectrum of [MATH] at [MATH] on Figure singles out new bifurcation of the stationary state along the branch. We have checked that the numerical results are stable with respect to truncation. In the present time, it is not clear how to identify new solution branches which... |
# Source: arxiv 1807.00428 # Title: Ground-state energies of the open and closed $p+ip$-pairing models from the Bethe Ansatz # Sections: all # Downloaded: 2026-03-02T08:54:00.706641+00:00 |
Ground-state energies of the open and closed [MATH] -pairing models from the Bethe Ansatz Abstract Using the exact Bethe Ansatz solution, we investigate methods for calculating the ground-state energy for the [MATH] -pairing Hamiltonian. We first consider the Hamiltonian isolated from its environment (closed model) thr... |
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