Perturbation of eigenvalues of the Klein Gordon operators II Kre simir Veseli c

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Perturbation of eigenvalues of the Klein Gordon

operators II

Kreˇsimir Veseli´c∗

Abstract

We give estimates for the changes of the eigenvalues of the Klein Gordon

operator under the change of the potential. In some relevant situations we

improve the existing estimates. We test our results on some exactly solvable

models (Coulomb potential, Klein-Gordon oscillator).

1 Introduction and preliminaries

The abstract time independent Klein-Gordon equation reads formally

(U2−(λ−V)2)ψ= 0 (1.1)

where U, (U2usually meaning the kinetic plus mass energy) is selfadjoint and pos-

itive deﬁnite operator in a Hilbert space Xand V(the potential) symmetric and in

some sense dominated by Uand λis the eigenvalue parameter. The most interesting

application is the standard Klein-Gordon equation with X=L2(R3) and

U2=c2p2+mc2, p = (−ih∇ − e

cA(x))2, V =V(x), x ∈R3(1.2)

where h, c are the common physical constants whereas the mass term mand the

magnetic potential Amay be position-dependent. Or else, U2may be some other

elliptic diﬀerential operator.

Typically there is a spectral gap around zero, then isolated eigenvalues of ﬁnite

multiplicity appear which may either be bounded by the spectral continuum reach-

ing to ±∞ or else the whole spectrum is discrete again approaching ±∞. Of some

interest could be also the case where U, V are ﬁnite matrices. The most interesting

eigenvalues are those around zero.

The aim of this paper is to prove sharp perturbation estimates for discrete eigen-

values λunder the change of Vand U. The main technical tool is the monotonic

dependence of the eigenvalues as functions of the potential. In fact, once one has

∗Fernuniversit¨at Hagen, Fakult¨at f¨ur Mathematik und Informatik, Postfach 940, D-58084 Ha-

gen, Germany, e-mail: Kresimir.Veselic@FernUni-Hagen.de.

arXiv:2210.11623v1 [math-ph] 20 Oct 2022

1 INTRODUCTION AND PRELIMINARIES 2

monotonicity, then for the perturbed potential ˜

V=V+δV (with δV bounded) we

would have

V−inf σ(δV )≤˜

V≤V−sup σ(δV ) (1.3)

which immediately implies the perturbation bound

λ−inf σ(δV )≤˜

λ≤λ+ sup σ(δV ) (1.4)

where λ, ˜

λis the corresponding eigenvalue, respectively. In particular,

λ− kδV k ≤ ˜

λ≤λ+kδV k

While such monotonicities are plausible for the Schroedinger and Dirac operators,

the Klein-Gordon case is less simple, because the potential Venters (1.1) as a

quadratic polynomial. To explain this we give a simple heuristic derivation. Let

V=Vtbe real analytic and produce t-dependent eigenvalue λtand eigenvector ψt

then by diﬀerentiating (1.1) we obtain

λ0

t=(V2

t0ψt, ψt)−2λt(V0

tψt, ψt)

2((Vt−λt)ψt, ψt)(1.5)

(here 0stays for the derivative). Supposing λtto be, say, positive (1.5) will yield

monotonicity, if Vtis negative semideﬁnite — which is a notable restriction. Note

also that growing Vtwill produce falling V2

tonly if Vtall commute (which is secured

with multiplicative potentials, but not generally).

A derivation to this eﬀect was produced in [5].1Here we will make this result

rigorous and more precise. We will also enlarge its validity and then apply it to

obtain eigenvalue perturbation bounds. More speciﬁcally we will

(i) prove that analytic dependence of Von a parameter implies the same for the

eigenvalues and eigenvectors, not just locally, but such as to produce eigenvalue

bounds. In fact, analysing quadratic eigenvalue equation (1.1) involves un-

bounded non-selfadjoint phase space Hamiltonians which are symmetric with

respect to an indeﬁnite scalar product so some extra care has to be taken,

(ii) make sure that the desired monotonicity, as well as the bounds of the type

(1.4) hold for increasingly ordered eigenvalues with their multiplicities, just

as is obtained by standard minimax arguments, (this is not quite trivial even

with common selfadjoint operators if one considers the discrete eigenvalues in

gaps of the essential spectrum, see [20])

(iii) weaken the condition of positivity of the considered eigenvalues into the more

natural, so-called plus property (which will, roughly speaking, cover all eigen-

values coming from the upper continuum as long as they do not clash with

those coming from below). Unfortunately we were able to do this only par-

tially, as yet, for instance for potentials of the form tV .

1This kind of monotonicity seems to be observed ﬁrst in [12]. The result in [12] was stated

under similar restrictive conditions, but it seems to us that the proof oﬀered there has some gaps

we were unable to ﬁll.

2 THE HAMILTONIAN FORMULATION AND ANALYTICITY 3

(iv) also weaken the non-positivity of the potential Vbecause even for non-positive

potentials the two-sided inclusion (1.3) requires monotonicity for slightly in-

deﬁnite potentials.

(v) give a collection of mostly exactly solvable examples illustrating the estimates.

A particularly interesting case will be that of perturbed Coulomb potential,

where the local deformation δV with

|δV (x)| ≤ β1

|x|, β < 1

leads to particularly tiny bound for the perturbed eigenvalue λ0

λ(1 + β)≤λ0≤λ(1 + β) (1.6)

where λ(t) is the corresponding eigenvalue corresponding to the potential

−t/|x|and is given by an explicit formula. We will also make comparison

with the bounds obtained recently in [14] and show that our bounds comple-

ment the ones from [14].

Another example will be that of the Klein-Gordon oscillator in L2(Rn) with

U2=−∆ + x2and V= 0 and then the bounds (1.4) will actually hold for both

sides of the spectrum.

The plan of the paper is as follows. In Section 1. we give deﬁnitions and fun-

damental properties of the Hamiltonians considered, in Section 2. we derive the

mononotonicity and in Section 3. give the resulting sharp eigenvalue bounds of the

type (1.4), (4.22) together with illustrating examples.

2 The Hamiltonian formulation and analyticity

The eigenvalue analysis necessitates rewriting a quadratic eigenvalue problem as a

linear one with ’doubled dimension’. By setting

ψ1=ψ, ψ1= (λI −V)ψ

we arrive at the eigenvalue equation Kψ =µψ with

K=V I

U2V.(2.7)

that is, K=JL where L

L=JK =U2V

V I (2.8)

is again formally Hermitian. So, Kis J-Hermitian. Our general assumption is

k(V−µI)U−1k<1,(2.9)

2 THE HAMILTONIAN FORMULATION AND ANALYTICITY 4

for some real µ. This commonly used condition insures reasonable spectral properties

of the operator K, see e.g. [14] and the literature cited there.2The set Iof all such µ

is obviously an open interval and we shall call it the deﬁniteness interval. Under this

condition Lis rigorously deﬁned by means of quadratic forms in the factorisation

L−µJ =U0

0I I U−1(V−µI)

(V−µI)U−1I U0

0I(2.10)

which is selfadjoint positive deﬁnite (being a symmetric product of three such fac-

tors). Thus we obtain a J-selfadjoint operator

K=JL, J =0I

I0(2.11)

that is, a selfadjoint operator with respect to the indeﬁnite scalar product

[ψ, φ] = (Jψ, φ).

These operators have real spectrum and a rich spectral calculus, see [8]. By our

condition (2.9) we have

σ(K) = σ−(K)∪σ+(K), σ−(K)<I< σ+(K).

Moreover, as it is readily seen the eigenvalues on the right/left from µhave [·,·]-

positive/negative eigenvactors, and will be called plus/minus-eigenvalues, respec-

tively. Also obvious is the fact that all eigenvalues are semisimple.

The spectra of the operators H, K are connected with those of the quadratic

families like (1.1).

The following Facts were shown in [19], Thms 2.4 and 4.2.

1. The sesquilinear form

qµ(ψ, φ) = (Uψ, Uφ)−((V−µI)ψ, (V−µI)φ) (2.12)

is closed and sectorial for every µ∈Cas deﬁned on D(U) and it generates a

closed sectorial operator Qµwhose domain D(Qµ) is independent of µ.

Qµ=Qλ+ 2(µ−λ)V+ (µ2−λ2)I(2.13)

for any µ, λ.

3. Denoting by ρthe set of µ-s for which Q−1

µis everywhere deﬁned and bounded

and by σits complement we have

ρ=ρ(K), σ =σ(K).

2The Hamiltonian considered there is not the one from the present paper but the eigenvalues

and their multiplicities are the same.

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PerturbationofeigenvaluesoftheKleinGordonoperatorsIIKresimirVeselic*AbstractWegiveestimatesforthechangesoftheeigenvaluesoftheKleinGordonoperatorunderthechangeofthepotential.Insomerelevantsituationsweimprovetheexistingestimates.Wetestourresultsonsomeexactlysolvablemodels(Coulombpotential,Klein-Gord...

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