On applying the subspace perturbation theory to few-body Hamiltonians
aa r X i v : . [ qu a n t - ph ] N ov Alexander K. Motovilov
On applying the subspace perturbation theoryto few-body Hamiltonians
Abstract
We present a selection of results on variation of the spectral subspace of a Hermitian operator undera Hermitian perturbation and show how these results may work for few-body Hamiltonians.
Keywords
Few-body problem · Binding energy shift · Variation of spectral subspace
The subspace perturbation theory is a branch of the general theory of linear operators (see, e.g. [1; 8]) thatstudies variation of an invariant (in particular, spectral) subspace of an operator under an additive perturbation.In this small survey article we restrict the subject to Hermitian operators and follow the geometric approachthat originates in the works by Davis [4; 5] and Davis and Kahan [6]. Within the Davis-Kahan approach, abound on the variation of a spectral subspace has usually the form of a trigonometric estimate involving justtwo quantities: a norm of the perturbation operator and the distance between the relevant spectral subsets. Wepresent only the estimates that are a priori in their nature and involve the distance between spectral sets of theunperturbed operator (but not of the perturbed one). The results valid for Hermitian operators of any originare collected in Section 2. In Section 3 we give several examples that illustrate the meaning of the abstractresults and show why these results might be useful already in the study of a few-body bound-state problem.Through the whole material we will use only the standard operator norm. We recall that if V is a boundedlinear operator on a Hilbert space H , its norm is given by k V k = sup k f k = (cid:13)(cid:13) V | f i (cid:13)(cid:13) where sup denotes theleast upper bound. Thus, for any | f i ∈ H we have (cid:13)(cid:13) V | f i (cid:13)(cid:13) ≤ k V k k f k . If V is a Hermitian operator withmin (cid:0) spec ( V ) (cid:1) = m V and max (cid:0) spec ( V ) (cid:1) = M V then k V k = max {| m V | , | M V |} . In particular, if V is separableof rank one, that is, if V = l | f ih f | with a normalized | f i ∈ H and l ∈ R , then k V k = | l | . Another exampleconcerns the case where H = L ( R ) and V is a bounded local potential, i.e. h x | V | f i = V ( x ) f ( x ) for any | f i ∈ H , with V ( · ) a bounded function on R . In this case k V k = sup x ∈ R | V ( x ) | . Assume that A is a Hermitian (or, equivalently, self-adjoint) operator on a separable Hilbert space H . If V isa bounded Hermitian perturbation of A then the spectrum, spec ( H ) , of the perturbed operator H = A + V is Based on a talk presented at the 22nd European Conference on Few-Body Problems in Physics (September 9–13, 2013, Cracow,Poland). The paper is to be published in
Few-Body Systems , doi: 10.1007/s00601-013-0752-8. This work was supported by theHeisenberg-Landau Program and by the Russian Foundation for Basic Research.A.K.MotovilovBogoliubov Laboratory of Theoretical Physics, JINRJoliot-Curie 6, 141980 Dubna, Moscow Region, RussiaTel.: +7-496-216-3355E-mail: [email protected] confined in the closed k V k -neighborhood O k V k (cid:0) spec ( A ) (cid:1) of the spectrum of A (see, e.g., [8]). Hence, if a part s of the spectrum of A is isolated from its complement S = spec ( A ) \ s , that is, if d : = dist ( s , S ) > , (1)then the spectrum of H is also divided into two disjoint components, w = spec ( H ) ∩ O k V k ( s ) and W = spec ( H ) ∩ O k V k ( S ) , (2)provided that k V k < d . (3)Under condition (3), one interprets the separated spectral components w and W of the perturbed operator H as the results of the perturbation of the corresponding initial disjoint spectral sets s and S .The transformation of the spectral subspace of A associated with the spectral set s into the spectral sub-space of H associated with the spectral set w may be studied in terms of the difference P − Q between thecorresponding spectral projections P = E A ( s ) and Q = E H ( w ) of A and H . Of particular interest is the casewhere k P − Q k <
1. In this case the spectral projections P and Q are unitarily equivalent and the perturbedspectral subspace Q = Ran ( Q ) may be viewed as obtained by the direct rotation of the unperturbed spectralsubspace P = Ran ( P ) (see, e.g. [6, Sections 3 and 4]). Moreover, the quantity q ( P , Q ) = arcsin ( k P − Q k ) is used as a measure of this rotation. It is called the maximal angle between the subspaces P and Q . A shortreview of the concept of the maximal angle can be found, e.g., in [3, Section 2]; see also [6; 9; 14; 15].Among various questions being answered within the subspace perturbation theory, the first and ratherbasic question is on whether the condition (3) is sufficient for the bound q ( P , Q ) < p (4)to hold, or, in order to secure (4), one should impose a stronger requirement k V k < c d with some c < . Moreprecisely, the question is as follows.(i) What is the best possible constant c ∗ in the inequality k V k < c ∗ d ensuring the spectral subspace variationbound (4)?Another, practically important question concerns the maximal possible size of the subspace variation:(ii) What function M : [ , c ∗ ) (cid:2) , p (cid:1) is best possible in the bound q ( P , Q ) ≤ M (cid:16) k V k d (cid:17) for k V k < c ∗ d ? (5)Both the constant c ∗ and the estimating function M are required to be universal in the sense that they shouldnot depend on the Hermitian operators A and V involved.By now, the problems (i) and (ii) have been completely solved only for the particular spectral dispositionswhere one of the sets s and S lies in a finite or infinite gap of the other set, say, s lies in a gap of S . In thiscase c ∗ = and M ( x ) = arcsin ( x ) . (6)This is the essence of the Davis-Kahan sin 2 q theorem in [6].In the generic case (where no assumptions are done on the mutual position of the sets s and S ), thestrongest known answers to the questions (i) and (ii) are the recent ones given by Seelmann [16], within theapproach developed in [3]; see also the earlier works [10] and [13]. In particular, [16, Theorem 1] implies thatthe generic optimal constant c ∗ satisfies inequality c ∗ > . . For the best upper estimate on the function M in the bound (5) we also refer to [16]. Here we only notice that for sure q ( P , Q ) ≤ arcsin p k V k d whenever k V k ≤ p d (see [3, Remark 4.4]; cf. [15, Corollary 2]).Now recall that, under the assumption (1), a bounded operator V is said to be off-diagonal with respect tothe partition spec ( A ) = s ∪ S if it anticommutes with the difference J = P − P ⊥ of the spectral projections P = E A ( s ) and P ⊥ = E A ( S ) , that is, if V J = − JV . The problems like (i) and (ii) have also been addressedin the case of off-diagonal Hermitian perturbations. When considering such a perturbation, one should first take into account that the requirements ensuring the disjointness of the corresponding perturbed spectralcomponents w and W originating in s and S are much weaker than condition (3). In particular, if the sets s and S are subordinated, say max ( s ) < min ( S ) , then for any k V k no spectrum of H = A + V enters theopen interval between max ( s ) and min ( S ) (see, e.g., [17, Remark 2.5.19]). In such a case the maximalangle between the unperturbed and perturbed spectral subspaces admits a sharp bound of the form (5) with M ( x ) = arctan ( x ) , x ∈ [ , ¥ ) . This is the consequence of the celebrated Davis-Kahan tan 2 q theorem [6](also, cf. the extensions of the tan 2 q theorem in [7; 9; 14]). Furthermore, if it is known that the set s lies ina finite gap of the set S then the disjointness of the perturbed spectral components w and W is ensured by the(sharp) condition k V k < √ d . The same condition is optimal for the bound (4) to hold. Both these resultshave been established in [11]. The explicit expression for the best possible function M in the correspondingestimate (5), M ( x ) = arctan x , x ∈ [ , √ ) , was found in [2; 14]. As for the generic spectral disposition withno restrictions on the mutual position of s and S , the condition k V k < √ d has been proven to be optimalin order to guarantee that the gaps between s and S do not close under an off-diagonal V and, thus, thatdist ( w , W ) >
0. The proof was first given in [12, Theorem 1] for bounded A and then in [17, Proposition2.5.22] for unbounded A . The best published lower bound c ∗ > . c ∗ in the off-diagonal case has been established in [13]. Paper [13] also contains the strongest known upperestimate for the optimal function M in the corresponding bound (5) (see [13, Theorem 6.2 and Remark 6.3]). Throughout this section we suppose that the “unperturbed” Hamiltonian A has the form A = H + V where H stands for the kinetic energy operator of an N -particle system in the c.m. frame and the potential V includes only a part of all interactions that are present in the system (say, only two-body forces if N = V is assumed to describe the remaining part of the interactions (say, three-body forces for N =
3; instead, if all the interparticle interactions are already included in V , it may only describe the effectof external fields). We consider the case where V is a bounded operator. Surely, both A and V are assumed tobe Hermitian. In order to apply to H = A + V the abstract results mentioned in the previous section, one onlyneeds to know the norm of the perturbation V and a few very basic things on the spectrum of the operator A .We start our discussion with the simplest example illustrating the sin 2 q and tan 2 q theorems from [6]. Example 3.1
Suppose that E is the ground-state (g.s.) energy of the Hamiltonian A . Assume, in addition,that the eigenvalue E is simple (this is typical for a ground state) and let | y i be the normalized g.s. wavefunction, i.e. A | y i = E | y i , k y k =
1. Set s = { E } , S = spec ( A ) \{ E } and d = dist ( s , S ) = min ( S ) − E (we remark that the set S is definitely not empty since it should contain at least the essential spectrum of A ). Ifthe norm of V is such that the condition (3) holds, then the g.s. energy E ′ of the total Hamiltonian H = A + V is also a simple eigenvalue with a g.s. vector | y ′ i , k y ′ k =
1. The g.s. energy E ′ lies in the closed k V k -neighborhood of the g.s. energy E , that is, | E − E ′ | ≤ k V k . The corresponding spectral projections P = E A ( s ) and Q = E H ( w ) of A and H associated with the one-point spectral sets s = { E } and w = { E ′ } read as P = | y ih y | and Q = | y ′ ih y ′ | . One verifies by inspection that arcsin (cid:0) k P − Q k (cid:1) = arccos |h y | y ′ i| , whichmeans that the maximal angle q ( P , Q ) between the one-dimensional spectral subspaces P = Ran ( P ) = span ( | y i ) and Q = Ran ( Q ) = span ( | y ′ i ) is, of course, nothing but the angle between the g.s. vectors | y i and | y ′ i . Then the Davis-Kahan sin 2 q theorem (see (5) and (6)) implies that arccos |h y | y ′ i| ≤ arcsin k V k d . This bound on the rotation of the ground state under the perturbation V is sharp. In particular, it implies thatunder condition (3) the angle between | y i and | y ′ i can never exceed 45 ◦ .If, in addition, it is known that the perturbation V is off-diagonal with respect to the partition spec ( A ) = s ∪ S then for any (arbitrarily large) k V k no spectrum of H enters the gap between the g.s. energy E and theremaining spectrum S of A . Moreover, there are the following sharp universal bounds for the perturbed g.s.energy E ′ : E − e V ≤ E ′ ≤ E , where e V = k V k tan (cid:16) arctan k V k d (cid:17) < k V k (7)(see [12, Lemma 1.1] and [17, Proposition 2.5.21]). At the same time, the Davis-Kahan tan 2 q theorem [6]implies that arccos |h y | y ′ i| ≤ arctan k V k d < p . With a minimal change, the above consideration is extended to the case where the initial spectral set s consists of the n + E ≤ E ≤ . . . ≤ E n , n ≥
1, of A . We only want to underline thatif V is off-diagonal than for any k V k the whole perturbed spectral set w of H = A + V originating from s willnecessarily lie in the interval [ E − r V , E n ] where the shift r V is given by (7), while the interval (cid:0) E n , min ( S ) (cid:1) will contain no spectrum of H . Furthermore, the tan 2 q -theorem-like estimates for the maximal angle betweenthe spectral subspaces P = Ran (cid:0) E A ( s ) (cid:1) and Q = Ran (cid:0) E H ( w ) (cid:1) may be done even for some unbounded V (but, instead of d and k V k , in terms of quadratic forms involving A and V ), see [7; 14].Along with the sin 2 q theorem, our second example also illustrates the tan q bound proven in [2; 14]. Example 3.2
Assume that s = { E n + , E n + , . . . , E n + k } , n ≥ , k ≥ , is the set of consecutive bindingenergies of A and S = spec ( A ) \ s = S − ∪ S + where S − is the increasing sequence of the energy lev-els E , E , . . . , E n of A that are lower than min ( s ) ; S + denotes the remaining part of the spectrum of A ,i.e. S + = spec ( A ) \ ( s ∪ S − ) . Together with (1), this assumption means that the set s lies in the finite gap (cid:0) max ( S − ) , min ( S + ) (cid:1) of the set S . Under the single condition (3), not much can be said about the location ofthe perturbed spectral sets w and W , except for (2), but the Davis-Kahan sin 2 q theorem [6] still well appliesto this case and, thus, we again have the bound q ( P , Q ) ≤ arcsin k V k d < p .Much stronger conclusions can be done if the operator V is off-diagonal with respect to the partitionspec ( A ) = s ∪ S . As it was already mentioned in Section 2, for off-diagonal V the gap-non-closing conditionreads as k V k < √ d (and even a weaker but more detail condition k V k < √ dD with D = min ( S + ) − max ( S − ) is allowed, see [11; 14]). In this case the lower bound for the spectrum of H = A + V is E − e V where themaximal possible energy shift e V , e V < d , is given again by (7). Furthermore, w ⊂ [ E n + − e V , E n + k + e V ] andthe open intervals ( E n , E n + − e V ) and (cid:0) E n + k + e V , min ( S + ) (cid:1) contain no spectrum of H . For tighter enclosuresfor w and W involving, say, the the gap length D we refer to [11; 12; 17]. In the case under consideration, thesharp bound for the rotation of the spectral subspace P = Ran (cid:0) E A ( s ) (cid:1) is given by q ( P , Q ) ≤ arctan k V k d < arctan √ D is known and k V k < √ dD ,then a more detail and stronger but still optimal estimate for q ( P , Q ) involving D is available (see [2, Theorem4.1]).Both Examples 3.1 and 3.2 show how one may obtain a bound on the variation of the spectral subspaceprior to any calculations with the total Hamiltonian H . To perform this, only the knowledge of the values of d and k V k is needed. Furthermore, if V is off-diagonal, with just these two values one can also provide thestronger estimates (via e V ) for the binding energy shifts. References
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