text: Diﬀuse Galactic Light 3 aﬀecting dust scattering rather than to accurately represent 1.5×1010 a = 0.7; g = 0.0 a = 0.7; g = 0.3 the distribution of interstellar dust. a = 0.7; g = 0.6 A full radiative transfer model is complex because it a = 0.4; g = 0.0 a = 0.4; g = 0.3 is non-linear, non-local and multiwavelength in its formu- a = 0.4; g = 0.6 a = 0.1; g = 0.0 lation (reviewed by Steinacker et al. 2013). However, I am 1.0×1010 a = 0.1; g =0.3 only concerned with the scattering of photons at single wave- a = 0.1; g =0.6 lengths which is a much simpler problem (Steinacker et al. 2013, Section 5.1). I have written a set of routines in ANSI C which are available from the ASCL (Murthy 2015). The 5.0×109 program is intended for use in the UV where the source dis- tribution is well characterized and the volume of space is limited because of the high optical depth to UV radiation. However, the code is modular and documented and may be freely modiﬁed for other purposes. 0.0×100 0 20 40 60 80 100 My code is very similar to other Monte Carlo programs Number of Scatters to predict the scattered light (eg. Wood & Reynolds (1999); 300 a = 0.1; g = 0.0 Gordon et al. (2001)). I will illustrate the program ﬂow in a = 0.1; g = 0.3 a = 0.1; g = 0.6 my implementation by following a single photon through its a = 0.4; g = 0.1 multiple scatterings. I generate a new random number from a = 0.4; g = 0.3 250 a = 0.4; g = 0.6 a uniform distribution in each step below except in Steps a = 0.7; g = 0.1 2 and 5 where two numbers are needed to determine the a = 0.7; g = 0.3 a = 0.7; g = 0.6 direction of the photon. I have used the genunf (generate a random number from a uniform distribution) function from 200 † the randlib library to generate the random numbers and, if necessary, weighted the distribution to match the desired probabilities. 150 (i) A photon is emitted from a random star, where the probability of selecting a given star is weighted by the rel- ative number of photons from that star, in a random direc- tion. Each photon begins with a unit weight which will be 100 0 20 40 60 80 100 reduced at each scattering. Number of Scatters (ii) Two random numbers are generated to calculate the direction of motion, one for theta (in the range from 0 — − − − − π, measured from the z axis) and one for phi (in the range Figure 3. The total ﬂux (ph cm 2 s 1 sr 1 ˚A 1) in the simulation from 0 — 2π). The position of the star is known (from the is shown on the top and the amount of time per 10 million in- Hipparcos catalog) in Cartesian coordinates (x, y, z) and the put photons (on a 3.7 GHz Quad-Core Intel Xeon E5 MacPro) is shown on the bottom. In both, the maximum number of scatter- angles are converted into a direction vector assuming a step ings per input photon before terminating the run is plotted on the size of 1 bin. x axis. Each point represents a run of 10 million input photons (iii) Another random number is generated to determine with diﬀerent value of a and g. the optical depth the photon travels. I have divided the Galaxy into 1000 x 1000 x 1000 cells with a side of 1 pc and ﬁlled each cell with dust as per the individual model. with the change of reference back to the original Cartesian The cross-section of the dust was taken from Draine (2003), coordinates using a rotation angle matrix. which is a parametrization of the canonical extinction curve − with R (= AV/E(B V)) = 3.1, the so-called Milky Way dust. (vi) The procedure is repeated from step 3 until the eﬀec- tive weight of the photon drops below a predetermined factor I then follow the photon along until the cumulative optical or the number of scatterings exceeds a speciﬁed limit (nscat- depth along the path exceeds the predetermined value. This ter). Note that single scattering corresponds to nscatter = 0. yields the Cartesian coordinates x, y, and z of the scattering In that case, the photon stops at the ﬁrst interaction but the location. eﬀective peeling method results in a ﬂux at the detector. (iv) I use the “eﬀective peeling” technique (Yusef-Zadeh et al. 1984) to send a fraction of the photon back to the detector and subtract this (small) amount of energy from the eﬀective weight of the photon. The detector in this case is assumed to observe the entire sky with an angular resolution ◦ 4 RESULTS of 0.1 per square bin. (v) The eﬀective weight of the photon is reduced by the I have plotted the total ﬂux in the Galaxy at 1500 ˚A as albedo and a new scattering direction is determined as in a function of the number of scatterings for diﬀerent values step 2. The z direction is now the original direction of motion of the optical constants along with the time taken for each run in Fig. 3. The ﬂux saturates at about 5 scatterings per photon and I have therefore capped the number of scatters † http://hpux.connect.org.uk/hppd/hpux/Maths/Misc/randlib- at that level leading to a signiﬁcant savings in execution 1.3/readme.html time without aﬀecting the total ﬂux. These numbers are MNRAS 000, 1–12 (2015) )stinu notohP( sexulF )s( emiT pro Fit TRIAL version pro Fit TRIAL version ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Diﬀuse

text: Galactic

text: Light

text: aﬀecting

text: dust

text: scattering

text: rather

text: than

text: to

text: accurately

text: represent

text: a

text: g

text: a

text: g

text: the

text: distribution

text: of

text: interstellar

text: dust.

text: a

text: g

text: A

text: full

text: radiative

text: transfer

text: model

text: is

text: complex

text: because

text: it

text: a

text: g

text: a

text: g

text: is

text: non-linear,

text: non-local

text: and

text: multiwavelength

text: in

text: its

text: formu-

text: a

text: g

text: a

text: g

text: lation

text: (reviewed

text: by

text: Steinacker

text: et

text: al.

text: However,

text: I

text: am

text: a

text: g

text: only

text: concerned

text: with

text: the

text: scattering

text: of

text: photons

text: at

text: single

text: wave-

text: a

text: g

text: lengths

text: which

text: is

text: a

text: much

text: simpler

text: problem

text: (Steinacker

text: et

text: al.

text: Section

text: I

text: have

text: written

text: a

text: set

text: of

text: routines

text: in

text: ANSI

text: C

text: which

text: are

text: available

text: from

text: the

text: ASCL

text: (Murthy

text: The

text: program

text: is

text: intended

text: for

text: use

text: in

text: the

text: UV

text: where

text: the

text: source

text: dis-

text: tribution

text: is

text: well

text: characterized

text: and

text: the

text: volume

text: of

text: space

text: is

text: limited

text: because

text: of

text: the

text: high

text: optical

text: depth

text: to

text: UV

text: radiation.

text: However,

text: the

text: code

text: is

text: modular

text: and

text: documented

text: and

text: may

text: be

text: freely

text: modiﬁed

text: for

text: other

text: purposes.

text: My

text: code

text: is

text: very

text: similar

text: to

text: other

text: Monte

text: Carlo

text: programs

text: Number

text: of

text: Scatters

text: to

text: predict

text: the

text: scattered

text: light

text: (eg.

text: Wood

text: Reynolds

text: a

text: g

text: Gordon

text: et

text: al.

text: I

text: will

text: illustrate

text: the

text: program

text: ﬂow

text: in

text: a

text: g

text: a

text: g

text: my

text: implementation

text: by

text: following

text: a

text: single

text: photon

text: through

text: its

text: a

text: g

text: multiple

text: scatterings.

text: I

text: generate

text: a

text: new

text: random

text: number

text: from

text: a

text: g

text: a

text: g

text: a

text: uniform

text: distribution

text: in

text: each

text: step

text: below

text: except

text: in

text: Steps

text: a

text: g

text: and

text: where

text: two

text: numbers

text: are

text: needed

text: to

text: determine

text: the

text: a

text: g

text: a

text: g

text: direction

text: of

text: the

text: photon.

text: I

text: have

text: used

text: the

text: genunf

text: (generate

text: a

text: random

text: number

text: from

text: a

text: uniform

text: distribution)

text: function

text: from

text: the

text: randlib

text: library

text: to

text: generate

text: the

text: random

text: numbers

text: and,

text: if

text: necessary,

text: weighted

text: the

text: distribution

text: to

text: match

text: the

text: desired

text: probabilities.

text: (i)

text: A

text: photon

text: is

text: emitted

text: from

text: a

text: random

text: star,

text: where

text: the

text: probability

text: of

text: selecting

text: a

text: given

text: star

text: is

text: weighted

text: by

text: the

text: rel-

text: ative

text: number

text: of

text: photons

text: from

text: that

text: star,

text: in

text: a

text: random

text: direc-

text: tion.

text: Each

text: photon

text: begins

text: with

text: a

text: unit

text: weight

text: which

text: will

text: be

text: reduced

text: at

text: each

text: scattering.

text: Number

text: of

text: Scatters

text: (ii)

text: Two

text: random

text: numbers

text: are

text: generated

text: to

text: calculate

text: the

text: direction

text: of

text: motion,

text: one

text: for

text: theta

text: (in

text: the

text: range

text: from

text: π,

text: measured

text: from

text: the

text: z

text: axis)

text: and

text: one

text: for

text: phi

text: (in

text: the

text: range

text: Figure

text: The

text: total

text: ﬂux

text: (ph

text: cm

text: s

text: sr

text: ˚A

text: in

text: the

text: simulation

text: from

text: The

text: position

text: of

text: the

text: star

text: is

text: known

text: (from

text: the

text: is

text: shown

text: on

text: the

text: top

text: and

text: the

text: amount

text: of

text: time

text: per

text: million

text: in-

text: Hipparcos

text: catalog)

text: in

text: Cartesian

text: coordinates

text: (x,

text: y,

text: z)

text: and

text: the

text: put

text: photons

text: (on

text: a

text: GHz

text: Quad-Core

text: Intel

text: Xeon

text: MacPro)

text: is

text: shown

text: on

text: the

text: bottom.

text: In

text: both,

text: the

text: maximum

text: number

text: of

text: scatter-

text: angles

text: are

text: converted

text: into

text: a

text: direction

text: vector

text: assuming

text: a

text: step

text: ings

text: per

text: input

text: photon

text: before

text: terminating

text: the

text: run

text: is

text: plotted

text: on

text: the

text: size

text: of

text: bin.

text: x

text: axis.

text: Each

text: point

text: represents

text: a

text: run

text: of

text: million

text: input

text: photons

text: (iii)

text: Another

text: random

text: number

text: is

text: generated

text: to

text: determine

text: with

text: diﬀerent

text: value

text: of

text: a

text: and

text: g.

text: the

text: optical

text: depth

text: the

text: photon

text: travels.

text: I

text: have

text: divided

text: the

text: Galaxy

text: into

text: x

text: x

text: cells

text: with

text: a

text: side

text: of

text: pc

text: and

text: ﬁlled

text: each

text: cell

text: with

text: dust

text: as

text: per

text: the

text: individual

text: model.

text: with

text: the

text: change

text: of

text: reference

text: back

text: to

text: the

text: original

text: Cartesian

text: The

text: cross-section

text: of

text: the

text: dust

text: was

text: taken

text: from

text: Draine

text: coordinates

text: using

text: a

text: rotation

text: angle

text: matrix.

text: which

text: is

text: a

text: parametrization

text: of

text: the

text: canonical

text: extinction

text: curve

text: with

text: R

text: AV/E(B

text: V))

text: the

text: so-called

text: Milky

text: Way

text: dust.

text: (vi)

text: The

text: procedure

text: is

text: repeated

text: from

text: step

text: until

text: the

text: eﬀec-

text: tive

text: weight

text: of

text: the

text: photon

text: drops

text: below

text: a

text: predetermined

text: factor

text: I

text: then

text: follow

text: the

text: photon

text: along

text: until

text: the

text: cumulative

text: optical

text: or

text: the

text: number

text: of

text: scatterings

text: exceeds

text: a

text: speciﬁed

text: limit

text: (nscat-

text: depth

text: along

text: the

text: path

text: exceeds

text: the

text: predetermined

text: value.

text: This

text: ter).

text: Note

text: that

text: single

text: scattering

text: corresponds

text: to

text: nscatter

text: yields

text: the

text: Cartesian

text: coordinates

text: x,

text: y,

text: and

text: z

text: of

text: the

text: scattering

text: In

text: that

text: case,

text: the

text: photon

text: stops

text: at

text: the

text: ﬁrst

text: interaction

text: but

text: the

text: location.

text: eﬀective

text: peeling

text: method

text: results

text: in

text: a

text: ﬂux

text: at

text: the

text: detector.

text: (iv)

text: I

text: use

text: the

text: “eﬀective

text: peeling”

text: technique

text: (Yusef-Zadeh

text: et

text: al.

text: to

text: send

text: a

text: fraction

text: of

text: the

text: photon

text: back

text: to

text: the

text: detector

text: and

text: subtract

text: this

text: (small)

text: amount

text: of

text: energy

text: from

text: the

text: eﬀective

text: weight

text: of

text: the

text: photon.

text: The

text: detector

text: in

text: this

text: case

text: is

text: assumed

text: to

text: observe

text: the

text: entire

text: sky

text: with

text: an

text: angular

text: resolution

text: RESULTS

text: of

text: per

text: square

text: bin.

text: (v)

text: The

text: eﬀective

text: weight

text: of

text: the

text: photon

text: is

text: reduced

text: by

text: the

text: I

text: have

text: plotted

text: the

text: total

text: ﬂux

text: in

text: the

text: Galaxy

text: at

text: ˚A

text: as

text: albedo

text: and

text: a

text: new

text: scattering

text: direction

text: is

text: determined

text: as

text: in

text: a

text: function

text: of

text: the

text: number

text: of

text: scatterings

text: for

text: diﬀerent

text: values

text: step

text: The

text: z

text: direction

text: is

text: now

text: the

text: original

text: direction

text: of

text: motion

text: of

text: the

text: optical

text: constants

text: along

text: with

text: the

text: time

text: taken

text: for

text: each

text: run

text: in

text: Fig.

text: The

text: ﬂux

text: saturates

text: at

text: about

text: scatterings

text: per

text: photon

text: and

text: I

text: have

text: therefore

text: capped

text: the

text: number

text: of

text: scatters

text: at

text: that

text: level

text: leading

text: to

text: a

text: signiﬁcant

text: savings

text: in

text: execution

text: time

text: without

text: aﬀecting

text: the

text: total

text: ﬂux.

text: These

text: numbers

text: are

text: MNRAS

text: )stinu

text: notohP(

text: sexulF

text: )s(

text: emiT

text: pro

text: Fit

text: TRIAL

text: version

text: pro

text: Fit

text: TRIAL

text: version

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Diﬀuse Galactic Light 9 800 North discussion and references). (Murthy 2014a) derived airglow South contributions of 220 ph cm−2 s−1 sr−1 ˚A−1 in the FUV and − − − − 350 ph cm 2 s 1 sr 1 ˚A 1 in the NUV using a diﬀerent (em- 600 pirical) method of analysis of the GALEX data, which would − − − − leave a residual of about 100 and 250 ph cm 2 s 1 sr 1 ˚A 1 400 in the FUV and NUV, respectively. It is diﬃcult to separate airglow from the other contributors without spectral diag- nostics and I will defer a self-consistent determination of the 200 oﬀsets for a future work. 6 SUMMARY 0 I have presented a Monte Carlo model for calculating 0.00 0.01 0.02 0.03 0.04 0.05 E(B – V) the dust scattered starlight over the entire Galaxy. The main 1000 North conclusions are as follows: South (i) A multiple scattering model increases the scattered 800 ﬂux by about 30% over the single scattering approximation regardless of the optical depth. 600 (ii) The total scattered ﬂux is proportional to the square of the albedo and is greatest for isotropically scattering grains. 400 (iii) 90% of the diﬀuse ﬂux originates from less than 1000 stars and 25% from only 10 stars. 200 (iv) About half of the diﬀuse radiation seen at the Earth is scattered within 200 pc of the Sun with no radiation arising 00.00 0.01 0.02 0.03 0.04 0.05 from further than 600 pc away. Most photons travel less than E(B – V) 200 pc before another interaction. (v) The all-sky diﬀuse radiation is ﬁt well with 0.3<a<0.5 Figure 18. Observed ﬂux for FUV (top) and NUV (bottom) at high latitudes plotted versus the reddening. The red points and g < 0.6. The albedo is constrained by the total ﬂux while are Southern hemisphere and the blue points are Northern hemi- g is constrained by the amount of scattering far from bright sphere. stars. (vi) The model predictions are close to the observed val- 800 ues at low and mid-latitudes for low optical depths. The ﬁt is North South poorer at larger optical depths where the geometry is more complex. 600 (vii) There is reasonable agreement between the model (a = 0.36;g = 0.5) and the data at high latitudes but with considerable scatter. − − − 400 (viii) There is an oﬀset of 300 — 700 ph cm 2 s 1 sr 1 − ˚A 1 in both bands and at all latitudes which cannot be due to dust scattered radiation. Murthy (2014a) estimated that 200 the residual airglow was 220 ph cm−2 s−1 sr−1 ˚A−1 in the − − − − FUV and 350 ph cm 2 s 1 sr 1 ˚A 1 in the NUV implying − − that the oﬀset at high latitudes is 100 and 250 ph cm 2 s 1 0 − − 0 50 Model FUV 1(0p0hoton units) 150 200 sr 1 ˚A 1 in the FUV and NUV bands, respectively. This is 1000.0 the component that is usually identiﬁed with extragalactic North South light (Bowyer 1991; Henry 1991). The oﬀsets are larger at 800.0 low latitudes and may be due to unaccounted sources such as molecular hydrogen emission or as yet undetermined sources (Henry et al. 2015). 600.0 (ix) I have tabulated results from the literature in Table 4. In most cases, I have taken the results as speciﬁed by the 400.0 authors which are diﬃcult to translate into the 1σ results more often seen. In all cases, there is enough uncertainty in 200.0 the data and the modelling that the formal limits are sug- gestive rather than deﬁnitive. There are a range of preferred 0.0 values for the optical constants and the oﬀset but I hope 0.0 50.0 100.0 150.0 200.0 Model NUV (photon units) that the volume of data and improved modelling will yield tighter constraints on the dust properties. Figure 19. Observed ﬂux at high latitudes plotted versus pre- (x) I have released the software under a non-restrictive dicted ﬂux. Colours are the same as Fig. 17. license (Murthy 2015) and have uploaded the model ﬁles to a public archive (Murthy 2016). These ﬁles are the runs for Model 2 for a range of the optical constants and may MNRAS 000, 1–12 (2015) )stinu notohp( VUF devresbO )stinu notohp( VUN devresbO )stinu notohp( VUF devresbO )stinu notohp( VUN devresbO ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Diﬀuse

label: paragraph

text: Galactic

text: Light

font: default

text: North

text: discussion

text: and

text: references).

text: (Murthy

text: derived

text: airglow

text: South

text: contributions

text: of

text: ph

text: ˚A−1

text: in

text: the

text: FUV

text: and

font: TRUSMP+txsy

text: ph

text: cm

text: s

text: sr

text: ˚A

text: in

text: the

text: NUV

text: using

text: a

text: diﬀerent

text: (em-

text: pirical)

text: method

text: of

text: analysis

text: of

text: the

text: GALEX

text: data,

text: which

text: would

text: leave

text: a

text: residual

text: of

text: about

text: and

text: ph

text: cm

text: s

text: sr

text: ˚A

text: in

text: the

text: FUV

text: and

text: NUV,

text: respectively.

text: It

text: is

text: diﬃcult

text: to

text: separate

text: airglow

text: from

text: the

text: other

text: contributors

text: without

text: spectral

text: diag-

text: nostics

text: and

text: I

text: will

text: defer

text: a

text: self-consistent

text: determination

text: of

text: the

text: oﬀsets

text: for

text: a

text: future

text: work.

text: SUMMARY

text: I

text: have

text: presented

text: a

text: Monte

text: Carlo

text: model

text: for

text: calculating

text: E(B

text: V)

text: the

text: dust

text: scattered

text: starlight

text: over

text: the

text: entire

text: Galaxy.

text: The

text: main

text: North

text: conclusions

text: are

text: as

text: follows:

text: South

text: (i)

text: A

text: multiple

text: scattering

text: model

text: increases

text: the

text: scattered

text: ﬂux

text: by

text: about

text: over

text: the

text: single

text: scattering

text: approximation

text: regardless

text: of

text: the

text: optical

text: depth.

text: (ii)

text: The

text: total

text: scattered

text: ﬂux

text: is

text: proportional

text: to

text: the

text: square

text: of

text: the

text: albedo

text: and

text: is

text: greatest

text: for

text: isotropically

text: scattering

text: grains.

text: (iii)

text: of

text: the

text: diﬀuse

text: ﬂux

text: originates

text: from

text: less

text: than

text: stars

text: and

text: from

text: only

text: stars.

text: (iv)

text: About

text: half

text: of

text: the

text: diﬀuse

text: radiation

text: seen

text: at

text: the

text: Earth

text: is

text: scattered

text: within

text: pc

text: of

text: the

text: Sun

text: with

text: no

text: radiation

text: arising

text: from

text: further

text: than

text: pc

text: away.

text: Most

text: photons

text: travel

text: less

text: than

text: E(B

text: V)

text: pc

text: before

text: another

text: interaction.

text: (v)

text: The

text: all-sky

text: diﬀuse

text: radiation

text: is

text: ﬁt

text: well

text: with

text: Figure

text: Observed

text: ﬂux

text: for

text: FUV

text: (top)

text: and

text: NUV

text: (bottom)

text: at

text: high

text: latitudes

text: plotted

text: versus

text: the

text: reddening.

text: The

text: red

text: points

text: and

text: g

text: The

text: albedo

text: is

text: constrained

text: by

text: the

text: total

text: ﬂux

text: while

text: are

text: Southern

text: hemisphere

text: and

text: the

text: blue

text: points

text: are

text: Northern

text: hemi-

text: g

text: is

text: constrained

text: by

text: the

text: amount

text: of

text: scattering

text: far

text: from

text: bright

text: sphere.

text: stars.

text: (vi)

text: The

text: model

text: predictions

text: are

text: close

text: to

text: the

text: observed

text: val-

text: ues

text: at

text: low

text: and

text: mid-latitudes

text: for

text: low

text: optical

text: depths.

text: The

text: ﬁt

text: is

text: North

text: South

text: poorer

text: at

text: larger

text: optical

text: depths

text: where

text: the

text: geometry

text: is

text: more

text: complex.

text: (vii)

text: There

text: is

text: reasonable

text: agreement

text: between

text: the

text: model

text: (a

text: and

text: the

text: data

text: at

text: high

text: latitudes

text: but

text: with

text: considerable

text: scatter.

text: (viii)

text: There

text: is

text: an

text: oﬀset

text: of

text: ph

text: cm

text: s

text: sr

text: ˚A

text: in

text: both

text: bands

text: and

text: at

text: all

text: latitudes

text: which

text: cannot

text: be

text: due

text: to

text: dust

text: scattered

text: radiation.

text: Murthy

text: estimated

text: that

text: the

text: residual

text: airglow

text: was

text: ph

text: ˚A−1

text: in

text: the

text: FUV

text: and

text: ph

text: cm

text: s

text: sr

text: ˚A

text: in

text: the

text: NUV

text: implying

text: that

text: the

text: oﬀset

text: at

text: high

text: latitudes

text: is

text: and

text: ph

text: cm

text: s

text: Model

text: FUV

text: units)

text: sr

text: ˚A

text: in

text: the

text: FUV

text: and

text: NUV

text: bands,

text: respectively.

text: This

text: is

text: the

text: component

text: that

text: is

text: usually

text: identiﬁed

text: with

text: extragalactic

text: North

text: South

text: light

text: (Bowyer

text: Henry

text: The

text: oﬀsets

text: are

text: larger

text: at

text: low

text: latitudes

text: and

text: may

text: be

text: due

text: to

text: unaccounted

text: sources

text: such

text: as

text: molecular

text: hydrogen

text: emission

text: or

text: as

text: yet

text: undetermined

text: sources

text: (Henry

text: et

text: al.

text: (ix)

text: I

text: have

text: tabulated

text: results

text: from

text: the

text: literature

text: in

text: Table

text: In

text: most

text: cases,

text: I

text: have

text: taken

text: the

text: results

text: as

text: speciﬁed

text: by

text: the

text: authors

text: which

text: are

text: diﬃcult

text: to

text: translate

text: into

text: the

text: 1σ

text: results

text: more

text: often

text: seen.

text: In

text: all

text: cases,

text: there

text: is

text: enough

text: uncertainty

text: in

text: the

text: data

text: and

text: the

text: modelling

text: that

text: the

text: formal

text: limits

text: are

text: sug-

text: gestive

text: rather

text: than

text: deﬁnitive.

text: There

text: are

text: a

text: range

text: of

text: preferred

text: values

text: for

text: the

text: optical

text: constants

text: and

text: the

text: oﬀset

text: but

text: I

text: hope

text: Model

text: NUV

text: (photon

text: units)

text: that

text: the

text: volume

text: of

text: data

text: and

text: improved

text: modelling

text: will

text: yield

text: tighter

text: constraints

text: on

text: the

text: dust

text: properties.

text: Figure

text: Observed

text: ﬂux

text: at

text: high

text: latitudes

text: plotted

text: versus

text: pre-

text: (x)

text: I

text: have

text: released

text: the

text: software

text: under

text: a

text: non-restrictive

text: dicted

text: ﬂux.

text: Colours

text: are

text: the

text: same

text: as

text: Fig.

text: license

text: (Murthy

text: and

text: have

text: uploaded

text: the

text: model

text: ﬁles

text: to

text: a

text: public

text: archive

text: (Murthy

text: These

text: ﬁles

text: are

text: the

text: runs

text: for

text: Model

text: for

text: a

text: range

text: of

text: the

text: optical

text: constants

text: and

text: may

text: MNRAS

text: )stinu

text: notohp(

text: VUF

text: devresbO

text: )stinu

text: notohp(

text: VUN

text: devresbO

text: )stinu

text: notohp(

text: VUF

text: devresbO

text: )stinu

text: notohp(

text: VUN

text: devresbO

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: On the other hand, the ﬁrst case, has a more general character: if the perturbation decays3 in time, either the described assumption on the form of f or on the eigenvalues turn out to be unnecessary. Basically, the presence of resonance phenomena is no longer an obstruction for the existence of the normal form, see also [FW15c]. The paper, based on the Lie series formalism developed by A. Giorgilli et al., can be regarded, at the same time, as a non-autonomous version of [Gio]. 1.2 Setting Let us consider the following Hamiltonian (cid:88) n H(x, y, η, t) = h(x, y, η) + f(x, y, t), h(x, y, η) := η + λ x y , (1) l l l l=1 where (x, y, η) ∈ D := [−r, r](cid:80)n×[−r, r]n×R, with n ≥ 1 and r > 0, λ ∈ C and t ∈ R+ := [0, +∞). l The assumptions on f will be discussed below. The system (1) is nothing but the “autonomous H equivalent” of (x, y, t) = n λ x y + f(y, x, t), once η has been deﬁned as the conjugate l=1 l l l variable to t. D The standard use of the analytic tools requires the complexiﬁcation of the domain as follows. ∈ D Q × S Given R (0, 1/2] set := , where R R R Q := {(x, y) ∈ C2n : |x|, |y| ≤ R}, S := {η ∈ C : |(cid:61)η| ≤ R}, R R It will be required that, for all t ∈ R+, f belongs to the space of real-analytic functions on Q◦ Q ∈ D R and continuous on the boundary, which we denote with C( ). In such a way H C( ). ∈ Q (cid:88) R R In particular, the space of all the G C( ) is endowed with the Taylor norm R (cid:107) (cid:107) | | | | G(x, y, t) := g (t) R α+β , (2) (cid:80) R (cid:80)α,β ∈N α,β n | | wwhheicrhe,Gif(Gx,∈y,Ct)(Q=: ) foαr,βa∈lNl nt g∈α,Rβ(+t,)xthαeynβ |agnd4(t)α| ≤:=|G| nl=R1−α|αl.+βW|,ewrheecraell|Gth|e s:=tansduaprd re∈sQult|Gfo|r. In particular, (cid:107)G(cid:107)R(cid:48) < +∞ for all R(cid:48) < R.α,β R R (x,y) R R Throughout this paper we shall deal with perturbations satisfying the following conditions: 1. f is “at least” quadratic in x and “at least” linear in y: a property that we will denote with (QxLy), i.e. f (t) = 0 for all t ∈ R+ and for all (α, β) ∈ N2n \ Γ, where Γ := {(α, β) ∈ N2n : |α| ≥ 2, |αβ,β| ≥ 1}, 2. there exist M ∈ [1, +∞) and a ∈ [0, 1) such that5, for all (x, y, t) ∈ Q × R+, f R (cid:107) (cid:107) ≤ − f(x, y, t) M e at. (3) R f 3The exponential decay, see (3), is chosen for simplicity of discussion. The only necessary assumption is the summability in t of the perturbing function over the non-negative real semi-axis, see [FW15c]. · · · · · 4It is understood that xαyβ := xα1 . . . xαn yβ1 . . . yβn. ∈ 1 n 1 n ∈ 5The interval a [0, 1) is a compact way to denote either the time decay a (0, 1) or the boundedness a = 0. As in our previous paper we recall that we are interested in the case of small a (slow decay) and the upper bound ≥ a = 1 is set for simplicity. On the other hand, it is easy to realise that the case a 1 is straightforward. 3 ##LTLine##

text: On

text: the

text: other

text: hand,

text: the

text: ﬁrst

text: case,

text: has

text: a

text: more

text: general

text: character:

text: if

text: the

text: perturbation

text: in

text: time,

text: either

text: the

text: described

text: assumption

text: on

text: the

text: form

text: of

text: f

text: or

text: on

text: the

text: eigenvalues

text: turn

text: out

text: to

text: be

text: unnecessary.

text: Basically,

text: the

text: presence

text: of

text: resonance

text: phenomena

text: is

text: no

text: longer

text: an

text: obstruction

text: for

text: the

text: existence

text: of

text: the

text: normal

text: form,

text: see

text: also

text: The

text: paper,

text: based

text: on

text: the

text: Lie

text: series

text: formalism

text: developed

text: by

text: A.

text: Giorgilli

text: et

text: al.,

text: can

text: be

text: regarded,

text: at

text: the

text: same

text: time,

text: as

text: a

text: non-autonomous

text: version

text: of

text: [Gio].

label: section

text: Setting

text: Let

text: us

text: consider

text: the

text: following

text: Hamiltonian

text: n

text: H(x,

text: y,

text: η,

text: t)

text: h(x,

text: y,

text: η)

text: f(x,

text: y,

text: t),

text: h(x,

text: y,

text: η)

text: η

text: λ

text: x

text: y

text: l

text: l

text: l

text: where

text: (x,

text: y,

text: η)

text: D

text: [−r,

text: r]n×R,

text: with

text: n

text: and

text: r

text: λ

text: C

text: and

text: t

text: R+

text: l

text: The

text: assumptions

text: on

text: f

text: will

text: be

text: discussed

text: below.

text: The

text: system

text: is

text: nothing

text: but

text: the

text: “autonomous

text: H

text: equivalent”

text: of

text: (x,

text: y,

text: t)

text: n

text: λ

text: x

text: y

text: f(y,

text: x,

text: t),

text: once

text: η

text: has

text: been

text: deﬁned

text: as

text: the

text: conjugate

text: l

text: l

text: l

text: variable

text: to

text: t.

text: D

text: The

text: standard

text: use

text: of

text: the

text: analytic

text: tools

text: requires

text: the

text: complexiﬁcation

text: of

text: the

text: domain

text: as

text: follows.

text: D

text: Q

text: S

text: Given

text: R

text: set

text: where

text: R

text: R

text: R

text: Q

text: {(x,

text: y)

text: |x|,

text: |y|

text: R},

text: S

text: {η

text: C

text: R},

text: R

text: R

text: It

text: will

text: be

text: required

text: that,

text: for

text: all

text: t

text: R+,

text: f

text: belongs

text: to

text: the

text: space

text: of

text: real-analytic

text: functions

text: on

text: Q◦

text: Q

text: D

text: R

text: and

text: continuous

text: on

text: the

text: boundary,

text: which

text: we

text: denote

text: with

text: C(

text: In

text: such

text: a

text: way

text: H

text: C(

text: Q

text: R

text: R

text: In

text: particular,

text: the

text: space

text: of

text: all

text: the

text: G

text: C(

text: is

text: endowed

text: with

text: the

text: Taylor

text: norm

text: R

text: G(x,

text: y,

text: t)

text: g

text: (t)

text: R

text: α+β

text: R

text: ∈N

text: α,β

text: n

text: wwhheicrhe,Gif(Gx,∈y,Ct)(Q=:

text: foαr,βa∈lNl

text: nt

text: g∈α,Rβ(+t,)xthαeynβ

text: ≤:=|G|

text: s:=tansduaprd

text: re∈sQult|Gfo|r.

text: In

text: particular,

text: for

text: all

text: R.α,β

text: R

text: R

text: (x,y)

text: R

text: R

text: Throughout

text: this

text: paper

text: we

text: shall

text: deal

text: with

text: perturbations

text: satisfying

text: the

text: following

text: conditions:

text: f

text: is

text: “at

text: least”

text: quadratic

text: in

text: x

text: and

text: “at

text: least”

text: linear

text: in

text: y:

text: a

text: property

text: that

text: we

text: will

text: denote

text: with

text: (QxLy),

text: i.e.

text: f

text: (t)

text: for

text: all

text: t

text: R+

text: and

text: for

text: all

text: (α,

text: β)

text: Γ,

text: where

text: Γ

text: {(α,

text: β)

text: |α|

text: |αβ,β|

text: there

text: exist

text: M

text: and

text: a

text: such

text: for

text: all

text: (x,

text: y,

text: t)

text: Q

text: R+,

text: f

text: R

text: f(x,

text: y,

text: t)

text: M

text: e

text: at.

text: R

text: f

text: exponential

text: decay,

text: see

text: is

text: chosen

text: for

text: simplicity

text: of

text: discussion.

text: The

text: only

text: necessary

text: assumption

text: is

text: the

text: summability

text: in

text: t

text: of

text: the

text: perturbing

text: function

text: over

text: the

text: non-negative

text: real

text: semi-axis,

text: see

text: is

text: understood

text: that

text: xαyβ

text: xαn

text: yβn.

text: n

text: n

text: interval

text: a

text: is

text: a

text: compact

text: way

text: to

text: denote

text: either

text: the

text: time

text: decay

text: a

text: or

text: the

text: boundedness

text: a

text: As

text: in

text: our

text: previous

text: paper

text: we

text: recall

text: that

text: we

text: are

text: interested

text: in

text: the

text: case

text: of

text: small

text: a

text: (slow

text: decay)

text: and

text: the

text: upper

text: bound

text: a

text: is

text: set

text: for

text: simplicity.

text: On

text: the

text: other

text: hand,

text: it

text: is

text: easy

text: to

text: realise

text: that

text: the

text: case

text: a

text: is

text: straightforward.

text: ##LTLine##


text: 1.3 Main result In the described setting, the main result can be stated as follows Theorem 1.1. Suppose that one of the following conditions are satisﬁed: I. Time decay: a > 0. II. Linearity in y + non-resonance: a = 0 and the perturbation is linear in y, denoted by (Ly), · i.e. of the form f(x, y, t) = (cid:0)y g(x, t). In ad(cid:1)dition, the vector Λ := (λ1, . . . , λn), satisﬁes the non-resonance condition max |(cid:60)U(α, e , Λ)|−1 ≤ γ|α|τ, ∀α ∈ Nn, (4) l l=1,...,n U − · ≥ − where (α, β, Λ) := (α β) Λ, for some γ > 0 and τ n. e stands for the l th vector l R of the canonical basis of n. ≤ Then it is possible to determine R∗, R with 0 < R∗ < R R16 and a family of canonical M ∞ 0∞ ∞ M D 0 → D D ttr∈anRsf+o,rmcaasttiionngsth(xe,Hy,aηm)il=tonian(x((1))i,nyt(o t)h, ηe(st)r)o,ng no:rmaRl∗form R0, analytic on R∗ for all ∞ ∞ ∞ ∞ ∞ ∞ ∞ H( )(x( ), y( ), η( )) = h(x( ), y( ), η( )). (5) Remark 1.2. It is immediate to recognize the similarity between (4) and the standard Dio- phantine condition. Clearly, all the vectors Λ whose real part is a Diophantine vector, satisfy condition (4), no matter what the imaginary part is. Hence the set of vectors satisfying (4) is, a fortiori, a full-measure set. As anticipated in the introduction, we stress that condition (4) is stronger than the non-resonance condition imposed in [Pus74] and it is not satisﬁed in the case of purely imaginary Λ. M Remark 1.3. As usually done in the Lie series method, see e.g. [Gio03], the transformation will be constructed as the limit (deﬁned, at the moment, only at a formal level) M M ◦ M − ◦ ◦ M :=(cid:80)lim (j) (j 1) . . . (0), (6) →∞ j M L ≡ − L L {· } where (j) := exp( ) Id + ≥ (s!) 1 s and := , χ(j) . The generating sequence { } χ(j) s 1 χ(j) χ(j) χ(j) ∈N, where χ(j) = χ(j)(x, y, t), see [GZ92], is meant to be determined. j We will show (see the proof of Lemma 3.3) that in the case of a perturbation which is (Ly), it is possible to show that χ(j)(x, y, t) is (Ly) as well, for all j ∈ N. In such a case, it is easy to M check by induction that x(j) = (j)x(j+1) does not depend on the variable y, for all j. Hence ≡ M ∞ M ∞ ∞ the composition x x(0) = x( ) =: (x( ), t) does not depend on y( ) i.e. is an analytic map Mx : Q˜R∗ → Q˜R0 parametrised by tx, where Q˜R := {x ∈ Cn : |x| ≤ R}. This will play a key role in the next section. 1.4 The corollary Let us consider the following non-linear system v˙ = Av + g(v, t), (7) where v ∈ Rn, A is a n × n matrix with real entries and the function g is such that ∂νg(0, t) ≡ 0 for all ν ∈ Nn such that |ν| ≤ 1 i.e. g is at least quadratic in v. We restrict oursevlves to the 4

text: Main

text: result

text: In

text: the

text: described

text: setting,

text: the

text: main

text: result

text: can

text: be

text: stated

text: as

text: follows

text: Theorem

text: Suppose

text: that

text: one

text: of

text: the

text: following

text: conditions

text: are

text: satisﬁed:

text: I.

text: Time

text: decay:

text: a

text: II.

text: Linearity

text: in

text: y

text: non-resonance:

text: a

text: and

text: the

text: perturbation

text: is

text: linear

text: in

text: y,

text: denoted

text: by

text: (Ly),

text: i.e.

text: of

text: the

text: form

text: f(x,

text: y,

text: t)

text: g(x,

text: t).

text: In

text: the

text: vector

text: Λ

text: λn),

text: satisﬁes

text: the

text: non-resonance

text: condition

text: max

text: e

text: γ|α|τ,

text: ∀α

text: Nn,

text: l

text: U

text: where

text: (α,

text: β,

text: Λ)

text: (α

text: β)

text: Λ,

text: for

text: some

text: γ

text: and

text: τ

text: n.

text: e

text: stands

text: for

text: the

text: l

text: th

text: vector

text: l

text: R

text: of

text: the

text: canonical

text: basis

text: of

text: n.

text: Then

text: it

text: is

text: possible

text: to

text: determine

text: R∗,

text: R

text: with

text: R∗

text: R

text: and

text: a

text: family

text: of

text: canonical

text: M

text: 0∞

text: M

text: D

text: D

text: D

text: t)h,

text: ηe(st)r)o,ng

text: no:rmaRl∗form

text: analytic

text: on

text: R∗

text: for

text: all

text: H(

text: )(x(

text: y(

text: η(

text: h(x(

text: y(

text: η(

text: Remark

text: It

text: is

text: immediate

text: to

text: recognize

text: the

text: similarity

text: between

text: and

text: the

text: standard

text: Dio-

text: phantine

text: condition.

text: Clearly,

text: all

text: the

text: vectors

text: Λ

text: whose

text: real

text: part

text: is

text: a

text: Diophantine

text: vector,

text: satisfy

text: condition

text: no

text: matter

text: what

text: the

text: imaginary

text: part

text: is.

text: Hence

text: the

text: set

text: of

text: vectors

text: satisfying

text: is,

text: a

text: fortiori,

text: a

text: full-measure

text: set.

text: As

text: anticipated

text: in

text: the

text: introduction,

text: we

text: stress

text: that

text: condition

text: is

text: stronger

text: than

text: the

text: non-resonance

text: condition

text: imposed

text: in

text: and

text: it

text: is

text: not

text: satisﬁed

text: in

text: the

text: case

text: of

text: purely

text: imaginary

text: Λ.

text: M

text: Remark

text: As

text: usually

text: done

text: in

text: the

text: Lie

text: series

text: method,

text: see

text: e.g.

text: the

text: transformation

text: will

text: be

text: constructed

text: as

text: the

text: limit

text: (deﬁned,

text: at

text: the

text: moment,

text: only

text: at

text: a

text: formal

text: level)

text: M

text: M

text: M

text: M

text: (j)

text: (j

text: j

text: M

text: L

text: L

text: L

text: where

text: (j)

text: exp(

text: Id

text: (s!)

text: s

text: and

text: χ(j)

text: The

text: generating

text: sequence

text: χ(j)

text: s

text: χ(j)

text: χ(j)

text: χ(j)

text: ∈N,

text: where

text: χ(j)

text: χ(j)(x,

text: y,

text: t),

text: see

text: is

text: meant

text: to

text: be

text: determined.

text: j

text: We

text: will

text: show

text: (see

text: the

text: proof

text: of

text: Lemma

text: that

text: in

text: the

text: case

text: of

text: a

text: perturbation

text: which

text: is

text: (Ly),

text: it

text: is

text: possible

text: to

text: show

text: that

text: χ(j)(x,

text: y,

text: t)

text: is

text: (Ly)

text: as

text: well,

text: for

text: all

text: j

text: N.

text: In

text: such

text: a

text: case,

text: it

text: is

text: easy

text: to

text: M

text: check

text: by

text: induction

text: that

text: x(j)

text: does

text: not

text: depend

text: on

text: the

text: variable

text: y,

text: for

text: all

text: j.

text: Hence

text: M

text: M

text: the

text: composition

text: x

text: x(

text: (x(

text: t)

text: does

text: not

text: depend

text: on

text: y(

text: i.e.

text: is

text: an

text: analytic

text: map

text: Mx

text: Q˜R∗

text: parametrised

text: by

text: tx,

text: where

text: Q˜R

text: {x

text: Cn

text: |x|

text: R}.

text: This

text: will

text: play

text: a

text: key

text: role

text: in

text: the

text: next

text: section.

text: The

text: corollary

text: Let

text: us

text: consider

text: the

text: following

text: non-linear

text: system

text: v˙

text: Av

text: g(v,

text: t),

text: where

text: v

text: Rn,

text: A

text: is

text: a

text: n

text: n

text: matrix

text: with

text: real

text: entries

text: and

text: the

text: function

text: g

text: is

text: such

text: that

text: t)

text: for

text: all

text: ν

text: Nn

text: such

text: that

text: |ν|

text: i.e.

text: g

text: is

text: at

text: least

text: quadratic

text: in

text: v.

text: We

text: restrict

text: oursevlves

text: to

text: the


text: I. INTRODUCTION Understanding chemical reaction dynamics in a microscopic perspective from ﬁrst prin- ciples has been a long standing goal in physical chemistry. Over many decades, molecular beams have been the prime tool to study gas phase reactions. Controlling the beams with electric or magnetic ﬁelds has recently rejuvenated this ﬁeld [1–3]. In particular, merging two beams with a magnetic ﬁeld [4–9] has allowed for unprecedented control over the col- lision energy, both in terms of accessible range and precision. Studying Penning ionization reactions with this technique comes with the additional advantage of nearly unit eﬃciency in detecting the ionic reaction products. This has allowed for observing orbiting resonances at sub-Kelvin temperatures [4] including an isotope eﬀect [5] and elucidating the role of internal rotation for the reaction [9]. However, a complete microscopic understanding of the reac- tion dynamics requires high-level ab initio theory in addition to precise experimental data. While the entrance channel of the reaction can be described with reasonable accuracy [10], a full model of Penning ionization reactions is faced with two major challenges. First, since it necessarily involves metastable states, the entrance channel is a highly excited electronic state, which is embedded in a continuum of ionized states. The corresponding electronic wave function does not belong to the space of square integrable functions and thus cannot be correctly described by standard electronic structure methods, designed for the description of bound states. The diﬃculty of coupling to a continuum of ionized states can be circumvented by replacement with an imaginary potential, deﬁned within the Fano-Feshbach formalism for an isolated resonance [11, 12]. In this approach, a projection operator is introduced, which allows to partition the total Hilbert space into discrete and continuum parts. The imaginary potential is obtained from the coupling matrix elements between wave functions in the two subspaces. However, choice and construction of the Feshbach projection operator are highly nontrivial. Second, even for diatomics, the best, currently available potential en- ergy curves are not suﬃciently accurate to correctly predict the scattering length and thus exact position and character of orbiting resonances [13]. Here, we suggest photoassociation spectroscopy of Penning ionization reactions to address these two issues. Photoassociation refers to the formation of a chemical bond upon laser excitation [14, 15]. In standard photoassociation spectroscopy, the laser excites two colliding ground state atoms to an electronically excited state. Detuning the laser frequency from 2

text: I.

text: INTRODUCTION

text: Understanding

text: chemical

text: reaction

text: dynamics

text: in

text: a

text: microscopic

text: perspective

text: from

text: ﬁrst

text: prin-

text: ciples

text: has

text: been

text: a

text: long

text: standing

text: goal

text: in

text: physical

text: chemistry.

text: Over

text: many

text: decades,

text: molecular

text: beams

text: have

text: been

text: the

text: prime

text: tool

text: to

text: study

text: gas

text: phase

text: reactions.

text: Controlling

text: the

text: beams

text: with

text: electric

text: or

text: magnetic

text: ﬁelds

text: has

text: recently

text: rejuvenated

text: this

text: ﬁeld

text: In

text: particular,

text: merging

text: two

text: beams

text: with

text: a

text: magnetic

text: ﬁeld

text: has

text: allowed

text: for

text: unprecedented

text: control

text: over

text: the

text: col-

text: lision

text: energy,

text: both

text: in

text: terms

text: of

text: accessible

text: range

text: and

text: precision.

text: Studying

text: Penning

text: ionization

text: reactions

text: with

text: this

text: technique

text: comes

text: with

text: the

text: additional

text: advantage

text: of

text: nearly

text: unit

text: eﬃciency

text: in

text: detecting

text: the

text: ionic

text: reaction

text: products.

text: This

text: has

text: allowed

text: for

text: observing

text: orbiting

text: resonances

text: at

text: sub-Kelvin

text: temperatures

text: including

text: an

text: isotope

text: eﬀect

text: and

text: elucidating

text: the

text: role

text: of

text: internal

text: rotation

text: for

text: the

text: reaction

text: However,

text: a

text: complete

text: microscopic

text: understanding

text: of

text: the

text: reac-

text: tion

text: dynamics

text: requires

text: high-level

text: ab

text: initio

text: theory

text: in

text: addition

text: to

text: precise

text: experimental

text: data.

text: While

text: the

text: entrance

text: channel

text: of

text: the

text: reaction

text: can

text: be

text: described

text: with

text: reasonable

text: accuracy

text: a

text: full

text: model

text: of

text: Penning

text: ionization

text: reactions

text: is

text: faced

text: with

text: two

text: major

text: challenges.

text: First,

text: since

text: it

text: necessarily

text: involves

text: metastable

text: states,

text: the

text: entrance

text: channel

text: is

text: a

text: highly

text: excited

text: electronic

text: state,

text: which

text: is

text: embedded

text: in

text: a

text: continuum

text: of

text: ionized

text: states.

text: The

text: corresponding

text: electronic

text: wave

text: function

text: does

text: not

text: belong

text: to

text: the

text: space

text: of

text: square

text: integrable

text: functions

text: and

text: thus

text: cannot

text: be

text: correctly

text: described

text: by

text: standard

text: electronic

text: structure

text: methods,

text: designed

text: for

text: the

text: description

text: of

text: bound

text: states.

text: The

text: diﬃculty

text: of

text: coupling

text: to

text: a

text: continuum

text: of

text: ionized

text: states

text: can

text: be

text: circumvented

text: by

text: replacement

text: with

text: an

text: imaginary

text: potential,

text: deﬁned

text: within

text: the

text: Fano-Feshbach

text: formalism

text: for

text: an

text: isolated

text: resonance

text: In

text: this

text: approach,

text: a

text: projection

text: operator

text: is

text: introduced,

text: which

text: allows

text: to

text: partition

text: the

text: total

text: Hilbert

text: space

text: into

text: discrete

text: and

text: continuum

text: parts.

text: The

text: imaginary

text: potential

text: is

text: obtained

text: from

text: the

text: coupling

text: matrix

text: elements

text: between

text: wave

text: functions

text: in

text: the

text: two

text: subspaces.

text: However,

text: choice

text: and

text: construction

text: of

text: the

text: Feshbach

text: projection

text: operator

text: are

text: highly

text: nontrivial.

text: Second,

text: even

text: for

text: diatomics,

text: the

text: best,

text: currently

text: available

text: potential

text: en-

text: ergy

text: curves

text: are

text: not

text: suﬃciently

text: accurate

text: to

text: correctly

text: predict

text: the

text: scattering

text: length

text: and

text: thus

text: exact

text: position

text: and

text: character

text: of

text: orbiting

text: resonances

text: Here,

text: we

text: suggest

text: photoassociation

text: spectroscopy

text: of

text: Penning

text: ionization

text: reactions

text: to

text: address

text: these

text: two

text: issues.

text: Photoassociation

text: refers

text: to

text: the

text: formation

text: of

text: a

text: chemical

text: bond

text: upon

text: laser

text: excitation

text: In

text: standard

text: photoassociation

text: spectroscopy,

text: the

text: laser

text: excites

text: two

text: colliding

text: ground

text: state

text: atoms

text: to

text: an

text: electronically

text: excited

text: state.

text: Detuning

text: the

text: laser

text: frequency

text: from


text: 3 He( P) + X + He + X 3 He( S) + X Inter-particle separation FIG. 1: Photoassociation spectroscopy in Penning ionization. The laser, indicated by the blue arrow, excites a scattering pair colliding with collision energy E in the He(3S)+X state into a bound rovibrational level in the potential energy curve dissociating into He(3P )+X. The grey shaded area at short inter-particle separations indicates the region where the probability for Penning ionization is unity. If the bound level extends into this region, ionization will happen faster than in the entrance channel, i.e., photoassociation will be obserevd in terms of an increased ionization rate. the atomic excitation energy probes molecular levels below the excited state dissociation threshold [14, 15]. Photoassociation rates are limited by the amplitude of collision wave- functions at the Condon radius, i.e., the inter-particle separation where the laser frequency matches the energy diﬀerence between ground and excited potential energy surface [16]. At sub-Kelvin temperatures, only few partial waves contribute to the collision and, due to the rotational barrier and quantum reﬂection, their wavefunctions have little amplitude at short inter-particle separations. Photoassociation is thus most eﬀecient at intermediate and large separations [17]. This implies excitation into weakly bound levels just below the dissociation threshold. When applied to a collision complex undergoing a Penning ionization reaction, the elec- tronic ground state as entrance channel is replaced by a metastable one, see Fig. 1. Such a conﬁguration of electronic states can also be used to produce metastable diatomic he- lium molecules via photoassociation [18]. While in that case, the giant bond length of the 3 ygrenE ##LTFigure## ##LTLine## ##LTLine## ##LTLine##

text: He(

text: P)

text: X

text: He

text: X

text: He(

text: S)

text: X

text: Inter-particle

text: separation

text: FIG.

text: Photoassociation

label: caption

text: spectroscopy

text: in

text: Penning

text: ionization.

text: The

text: laser,

text: indicated

text: by

text: the

text: blue

text: arrow,

text: excites

text: a

text: scattering

text: pair

text: colliding

text: with

text: collision

text: energy

text: E

text: in

text: the

text: state

text: into

text: a

text: bound

text: rovibrational

text: level

text: in

text: the

text: potential

text: energy

text: curve

text: dissociating

text: into

text: )+X.

text: The

text: grey

text: shaded

text: area

text: at

text: short

text: inter-particle

text: separations

text: indicates

text: the

text: region

text: where

text: the

text: probability

text: for

text: Penning

text: ionization

text: is

text: unity.

text: If

text: the

text: bound

text: level

text: extends

text: into

text: this

text: region,

text: ionization

text: will

text: happen

text: faster

text: than

text: in

text: the

text: entrance

text: channel,

text: i.e.,

text: photoassociation

text: will

text: be

text: obserevd

text: in

text: terms

text: of

text: an

text: increased

text: ionization

text: rate.

text: the

text: atomic

text: excitation

text: energy

text: probes

text: molecular

text: levels

text: below

text: the

text: excited

text: state

text: dissociation

text: threshold

text: Photoassociation

text: rates

text: are

text: limited

text: by

text: the

text: amplitude

text: of

text: collision

text: wave-

text: functions

text: at

text: the

text: Condon

text: radius,

text: i.e.,

text: the

text: inter-particle

text: separation

text: where

text: the

text: laser

text: frequency

text: matches

text: the

text: energy

text: diﬀerence

text: between

text: ground

text: and

text: excited

text: potential

text: energy

text: surface

text: At

text: sub-Kelvin

text: temperatures,

text: only

text: few

text: partial

text: waves

text: contribute

text: to

text: the

text: collision

text: and,

text: due

text: to

text: the

text: rotational

text: barrier

text: and

text: quantum

text: reﬂection,

text: their

text: wavefunctions

text: have

text: little

text: amplitude

text: at

text: short

text: inter-particle

text: separations.

text: Photoassociation

text: is

text: thus

text: most

text: eﬀecient

text: at

text: intermediate

text: and

text: large

text: separations

text: This

text: implies

text: excitation

text: into

text: weakly

text: bound

text: levels

text: just

text: below

text: the

text: dissociation

text: threshold.

text: When

text: applied

text: to

text: a

text: collision

text: complex

text: undergoing

text: a

text: Penning

text: ionization

text: reaction,

text: the

text: elec-

text: tronic

text: ground

text: state

text: as

text: entrance

text: channel

text: is

text: replaced

text: by

text: a

text: metastable

text: one,

text: see

text: Fig.

text: Such

text: a

text: conﬁguration

text: of

text: electronic

text: states

text: can

text: also

text: be

text: used

text: to

text: produce

text: metastable

text: diatomic

text: he-

text: lium

text: molecules

text: via

text: photoassociation

text: While

text: in

text: that

text: case,

text: the

text: giant

text: bond

text: length

text: of

text: the

text: ygrenE

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: −11 10 4 10−12 E = 0.25 K 2 3 1 0 1 −13 2 4 10 3 0 −14 1 5 10 l’ = 0 10−15 5 6 −16 10 6 −17 10 7 −18 10 8 v’ = 0 v’ = 1 v’ = 2 −19 10 −16 −14 −12 −10 −8 −6 −4 −2 0 −1 Detuning (cm ) → FIG. 4: Photoassociation rates as a function of laser detuning from the He 3S 3 P atomic line for rovibrational levels of the (2)3Σ+ state at a collision energy of 0.25 Kelvin and a laser intensity of 1 W/cm2. The vibrational and rotational quantum numbers are indicated for each peak. The (cid:48) peaks for l = 2, 3, 4 are enhanced due to the l = 3 shape resonance as initial state at 0.25 K, whereas for all other peaks photoassociation starts from regular scattering states. bound level. Photoassociation rates for levels in the (1)3Σ+ and 3Π electronic states are obtained separately from single-channel calculations since we neglect nonadiabatic couplings. Figure 4 presents the spectrum for photoassociation into the (2)3Σ+ state at a collision energy of 0.25 Kelvin. A rotational progression for three vibrational levels is clearly visible. Note that our potential energy curves are not suﬃciently accurate to predict the exact detunings of these peaks. Reﬁnement of the potentials based on spectroscopic information would be required to this end. However, the spacing between rotational and vibrational levels can be predicted correctly. ≤ (cid:48) ≤ The highest rates in Fig. 4 occur for 2 l 4, i.e., for bound levels which are directly accessible from the l = 3 shape resonance. However, the observed enhancement of the rates compared to the other levels is one order of magnitude at most. This relatively minor resonance eﬀect is easily explained by the above-the-barrier character of this shape resonance, and the corresponding diﬀuse nature of its wavefunction, cf. Fig. 3(a). The widths of the 10 1− 3 ) s mc( K AP ##LTFigure##

text: E

text: K

text: l’

text: v’

text: v’

text: v’

text: −8

text: −6

text: −4

text: −2

text: −1

text: Detuning

text: (cm

text: FIG.

text: Photoassociation

text: rates

text: as

text: a

text: function

text: of

text: laser

text: detuning

text: from

text: the

text: He

text: P

text: atomic

text: line

text: for

text: rovibrational

text: levels

text: of

text: the

text: state

text: at

text: a

text: collision

text: energy

text: of

text: Kelvin

text: and

text: a

text: laser

text: intensity

text: of

text: The

text: vibrational

text: and

text: rotational

text: quantum

text: numbers

text: are

text: indicated

text: for

text: each

text: peak.

text: The

text: peaks

text: for

text: l

text: are

text: enhanced

text: due

text: to

text: the

text: l

text: shape

text: resonance

text: as

text: initial

text: state

text: at

text: K,

text: whereas

text: for

text: all

text: other

text: peaks

text: photoassociation

text: starts

text: from

text: regular

text: scattering

text: states.

text: bound

text: level.

text: Photoassociation

text: rates

text: for

text: levels

text: in

text: the

text: and

text: 3Π

text: electronic

text: states

text: are

text: obtained

text: separately

text: from

text: single-channel

text: calculations

text: since

text: we

text: neglect

text: nonadiabatic

text: couplings.

text: Figure

text: presents

text: the

text: spectrum

text: for

text: photoassociation

text: into

text: the

text: state

text: at

text: a

text: collision

text: energy

text: of

text: Kelvin.

text: A

text: rotational

text: progression

text: for

text: three

text: vibrational

text: levels

text: is

text: clearly

text: visible.

text: Note

text: that

text: our

text: potential

text: energy

text: curves

text: are

text: not

text: suﬃciently

text: accurate

text: to

text: predict

text: the

text: exact

text: detunings

text: of

text: these

text: peaks.

text: Reﬁnement

text: of

text: the

text: potentials

text: based

text: on

text: spectroscopic

text: information

text: would

text: be

text: required

text: to

text: this

text: end.

text: However,

text: the

text: spacing

text: between

text: rotational

text: and

text: vibrational

text: levels

text: can

text: be

text: predicted

text: correctly.

text: The

text: highest

text: rates

text: in

text: Fig.

text: occur

text: for

text: l

text: i.e.,

text: for

text: bound

text: levels

text: which

text: are

text: directly

text: accessible

text: from

text: the

text: l

text: shape

text: resonance.

text: However,

text: the

text: observed

text: enhancement

text: of

text: the

text: rates

text: compared

text: to

text: the

text: other

text: levels

text: is

text: one

text: order

text: of

text: magnitude

text: at

text: most.

text: This

text: relatively

text: minor

text: resonance

text: eﬀect

text: is

text: easily

text: explained

text: by

text: the

text: above-the-barrier

text: character

text: of

text: this

text: shape

text: resonance,

text: and

text: the

text: corresponding

text: diﬀuse

text: nature

text: of

text: its

text: wavefunction,

text: cf.

text: Fig.

text: The

text: widths

text: of

text: the

text: 1−

text: s

text: mc(

text: K

text: AP

text: ##LTFigure##


text: 16 MARIANNE AKIAN, RAVINDRA BAPAT, AND STE´PHANE GAUBERT (cid:54) ∈ { − } P = 0, and p − = 0 for all i 1, . . . , n 1 such that c < c , where 0 (cid:54) · · · (cid:54) n i i i+1 (c c ) = R(P). 1 n P (cid:62) P P When these assertions hold, we have e( ) P, e( ) = P, and R(e( )) = R(P) = (cid:54) · · · (cid:54) (cid:54) · · · (cid:54) ∈ R (c c ) = (Y Y ). Moreover, if c is a root of P with 1 n 1 · · · n multiplicity m and c = = c = c, then y , . . . , y are precisely the i+1 i+m i+1 i+m non-zero roots of the polynomial p(c) of (23), counted with multiplicities. Theorem 6.2 is a “precise large deviation” version of the Newton-Puiseux theo- rem: we assume only the existence of asymptotic equivalents for the coeﬃcients of P P , and derive the existence of asymptotic equivalents for the branches of ((cid:15), y) = 0. The Newton-Puiseux algorithm is sometimes presented for asymptotic expansions, as in [Die68]. The interest of the statements of Theorems 6.1 and 6.2, is to show that if some coeﬃcients are known to be negligible, the asymptotics of the roots is P determined only from the asymptotics of those coeﬃcients such that (i, P ) is an i i exposed point of the epigraph of P. P − P Example 6.3. Consider = Y3+(cid:15)5Y2 (cid:15)6Y+(cid:15)13. Then, is a polynomial over the K P K ﬁeld of complex Puiseux series, hence the roots of are elements of . Moreover, P ⊕ ⊕ the min-plus polynomial P = ν( ) = Y3 5Y2 6Y + 13 is the one of Figure 1, hence its roots are c = c = 3 and c = 7. In that case, Proposition 4.3 says that 1 2P 3 the valuations of roots of coincide with the roots of P. If now P = Y3 + (cid:15)3 Y2 −(cid:15)6Y+(cid:15)13, the coeﬃcient P of P is not in K. Moreover, log(cid:15) P C C† P2 although all coeﬃcients of belong to and even , does not have a ﬁrst order P (cid:39) 2 ⊕ ⊕ ⊕ asymptotics−. Howev∈erC, j((cid:15)) pj(cid:15)Pj, j = 0, . . . , 3, with P = Y3 3Y2 6Y 13 and p = Y3 Y + 1 [Y]. The min-plus polynomial P has same roots as the one (cid:54) (cid:54) deﬁned above: c = c = 3 < c = 7 and since p = 0, p = 0, p = 1 and P = 1, 1 2 3 0 1 3 P3 Assertion (2) of Theorem 6.2 holds. Hence, by Theorem 6.2, the roots of form Y Y ∈ C 3 continuous branches around 0: , . . . , with ﬁrst order asymptotics, and 1 3 respective exponents c = c = 3 < c = 7. 1 2 3 In both cases above, Theorem 6.2 gives the additional information that the roots Y Y P Y ∼ 1, . . . , 3 of sa−tisfy j((cid:15)) yj(cid:15)cj, j = 1, . . . , 3, where y−1, y2 are the non-zero roots of p(3) = Y3 Y, and y is the non-zero root of p(7) = Y + 1. This gives for − 3 Y ∼ Y ∼ − Y ∼ instance y = 1, y = 1 and y = 1, so that (cid:15)3, 2 (cid:15)3 and (cid:15)7. 1 2 3 1 3 In order to prove Theorem(cid:80)s 6.1 and 6.2, we need the following standard result. Q Q Q Lemma 6.4. Let ((cid:15), Y) = n ((cid:15))Yj, where the are continuous functions ∈ Q · (cid:54) i=0 j Q · j(cid:62) of (cid:15) [0, (cid:15) ), assume (0, ) = 0 and let d = deg (0, ) 0. Then, for any open 0 Q · Z Z ball B containing the roots of (0, ), there are d continuous branches , . . . , (cid:54) Z Z1 d deﬁned in some interval [0, (cid:15) ), with 0 < (cid:15) (cid:15) , such that ((cid:15)), . . . , ((cid:15)) are Q · 1 1 0 1 d exactly the roots of ((cid:15), ) in B counted with multiplicities. Moreover, the roots of Q · ((cid:15), ) that are outside B tend to inﬁnity when (cid:15) goes to 0. (cid:82) Proof. We only sketch the proof, which is classical. By the Cauchy index theorem, C Q · Q · if γ is any circle in containing no roots of ((cid:15), ), the number of roots of ((cid:15), ) − Q Q − (cid:55)→ Q · inside γ is (2πi) 1 ∂ ((cid:15), z)( ((cid:15), z)) 1 dz. By continuity of (cid:15) ((cid:15), ), the Qγ (cid:48)z · (cid:48) number of roots of ((cid:15) , ) inside γ (counted with multiplicities) is constant for (cid:15) ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: MARIANNE

text: AKIAN,

text: RAVINDRA

text: BAPAT,

text: AND

text: STE´PHANE

text: GAUBERT

label: list

text: P

text: and

text: p

text: for

text: all

text: i

text: n

text: such

text: that

text: c

text: c

text: where

text: n

text: i

text: i

text: (c

text: c

text: R(P).

text: n

text: P

text: P

text: P

text: When

text: these

text: assertions

text: hold,

text: we

text: have

text: e(

text: P,

text: e(

text: P,

text: and

text: R(e(

text: R(P)

text: R

text: (c

text: c

text: (Y

text: Y

text: Moreover,

text: if

text: c

text: is

text: a

text: root

text: of

text: P

text: with

text: n

text: n

text: multiplicity

text: m

text: and

text: c

text: c

text: c,

text: then

text: y

text: y

text: are

text: precisely

text: the

text: i+m

text: i+m

text: non-zero

text: roots

text: of

text: the

text: polynomial

text: p(c)

text: of

text: counted

text: with

text: multiplicities.

text: Theorem

text: is

text: a

text: “precise

text: large

text: deviation”

text: version

text: of

text: the

text: Newton-Puiseux

text: theo-

text: rem:

text: we

text: assume

text: only

text: the

text: existence

text: of

text: asymptotic

text: equivalents

text: for

text: the

text: coeﬃcients

text: of

text: P

text: P

text: and

text: derive

text: the

text: existence

text: of

text: asymptotic

text: equivalents

text: for

text: the

text: branches

text: of

text: y)

text: The

text: Newton-Puiseux

text: algorithm

text: is

text: sometimes

text: presented

text: for

text: asymptotic

text: expansions,

text: as

text: in

text: The

text: interest

text: of

text: the

text: statements

text: of

text: Theorems

text: and

text: is

text: to

text: show

text: that

text: if

text: some

text: coeﬃcients

text: are

text: known

text: to

text: be

text: negligible,

text: the

text: asymptotics

text: of

text: the

text: roots

text: is

text: P

text: determined

text: only

text: from

text: the

text: asymptotics

text: of

text: those

text: coeﬃcients

text: such

text: that

text: (i,

text: P

text: is

text: an

text: i

text: i

text: exposed

text: point

text: of

text: the

text: epigraph

text: of

text: P.

text: P

text: P

text: Example

text: Consider

text: Then,

text: is

text: a

text: polynomial

text: over

text: the

text: K

text: P

text: K

text: ﬁeld

text: of

text: complex

text: Puiseux

text: series,

text: hence

text: the

text: roots

text: of

text: are

text: elements

text: of

text: Moreover,

text: P

text: the

text: min-plus

text: polynomial

text: P

text: ν(

text: is

text: the

text: one

text: of

text: Figure

text: hence

text: its

text: roots

text: are

text: c

text: c

text: and

text: c

text: In

text: that

text: case,

text: Proposition

text: says

text: that

text: the

text: valuations

text: of

text: roots

text: of

text: coincide

text: with

text: the

text: roots

text: of

text: P.

text: If

text: now

text: P

text: the

text: coeﬃcient

text: P

text: of

text: P

text: is

text: not

text: in

text: K.

text: Moreover,

text: P

text: C

text: C†

text: although

text: all

text: coeﬃcients

text: of

text: belong

text: to

text: and

text: even

text: does

text: not

text: have

text: a

text: ﬁrst

text: order

text: P

text: asymptotics−.

text: Howev∈erC,

text: j

text: with

text: P

text: and

text: p

text: Y

text: [Y].

text: The

text: min-plus

text: polynomial

text: P

text: has

text: same

text: roots

text: as

text: the

text: one

text: deﬁned

text: above:

text: c

text: c

text: c

text: and

text: since

text: p

text: p

text: p

text: and

text: P

text: Assertion

text: of

text: Theorem

text: holds.

text: Hence,

text: by

text: Theorem

text: the

text: roots

text: of

text: form

text: Y

text: Y

text: C

text: continuous

text: branches

text: around

text: with

text: ﬁrst

text: order

text: asymptotics,

text: and

text: respective

text: exponents

text: c

text: c

text: c

text: In

text: both

text: cases

text: above,

text: Theorem

text: gives

text: the

text: additional

text: information

text: that

text: the

text: roots

text: Y

text: Y

text: P

text: Y

text: of

text: sa−tisfy

text: j

text: where

text: are

text: the

text: non-zero

text: roots

text: of

text: Y,

text: and

text: y

text: is

text: the

text: non-zero

text: root

text: of

text: Y

text: This

text: gives

text: for

text: Y

text: Y

text: Y

text: instance

text: y

text: y

text: and

text: y

text: so

text: that

text: and

text: In

text: order

text: to

text: prove

text: and

text: we

text: need

text: the

text: following

text: standard

text: result.

text: Q

text: Q

text: Q

text: Lemma

text: Let

text: Y)

text: n

text: where

text: the

text: are

text: continuous

text: functions

text: Q

text: j

text: Q

text: of

text: assume

text: and

text: let

text: d

text: deg

text: Then,

text: for

text: any

text: open

text: Q

text: Z

text: Z

text: ball

text: B

text: containing

text: the

text: roots

text: of

text: there

text: are

text: d

text: continuous

text: branches

text: Z

text: d

text: deﬁned

text: in

text: some

text: interval

text: with

text: such

text: that

text: are

text: Q

text: d

text: exactly

text: the

text: roots

text: of

text: in

text: B

text: counted

text: with

text: multiplicities.

text: Moreover,

text: the

text: roots

text: of

text: Q

text: that

text: are

text: outside

text: B

text: tend

text: to

text: inﬁnity

text: when

text: goes

text: to

text: Proof.

text: We

text: only

text: sketch

text: the

text: proof,

text: which

text: is

text: classical.

text: By

text: the

text: Cauchy

text: index

text: theorem,

text: C

text: Q

text: Q

text: if

text: γ

text: is

text: any

text: circle

text: in

text: containing

text: no

text: roots

text: of

text: the

text: number

text: of

text: roots

text: of

text: Q

text: Q

text: Q

text: inside

text: γ

text: is

text: z)(

text: z))

text: dz.

text: By

text: continuity

text: of

text: the

text: Qγ

text: number

text: of

text: roots

text: of

text: inside

text: γ

text: (counted

text: with

text: multiplicities)

text: is

text: constant

text: for

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: NON-ARCHIMEDEAN VALUATIONS OF EIGENVALUES OF MATRIX POLYNOMIALS 21 A (cid:39) (cid:54) (cid:54) (29) ( (cid:15),k)ij (ak)ij(cid:15)(Ak)ij , for all 1 i, j n . A We shall also assume that the matrix polynomial is regular, which means A A (cid:15) that the characteristic polynomial det( ) (or det( )) of the matrix polynomial A (cid:15) A A is non identically zero. Then, the eigenvalues of (or ) are by deﬁnition (cid:15) A (cid:15) ∞ the roots of this polynomial. If deg det( ) < nd, then we also say that is an A − A A A − eigenvalue of with multiplicity nd deg det( ). When = YI, these are A 0 the usual eigenvalues of . We shall study here the ﬁrst order asymptotics of the A 0,(cid:15) eigenvalues of in the same spirit as in the previous section. (cid:15) A To the matrix polynomial , one can associate the min-plus matrix polynomial A A ⊕ A ⊕· · ·⊕ A e( ) := e( ) Ye( ) Yde( ). Here, we shall rather consider the min-plus 0 1 d matrix polynomial associated to the asymptotics (29): ⊕ ⊕ · · · ⊕ ∈ R × (30) A = A YA YdA n n[Y] . 0 1 d min The min-plus matrix polynomial A can (cid:98)be seen either as a (formal) polynomial R × witRh coeﬃcients in (cid:98) nminn, namely the Ak, k = 0, . . . , d, or as a matrix with entrie∈s in [Y], denoted A . We denote by A the function which associates to y R min ij ⊕ ⊕ · · · ⊕ ∈ R × , the matrix A(y) := A yA ydA n n. We call min-plus min 0 1 d min cha(cid:99)racteristic po(cid:98)lynomial of the matrix polynomial A, the permanent P = per A. A This is a formal polynomial, the associated polynomial function of which is equal to P (y) = per A(y). A We shall say that the matrix polynomial A is regular if P is not identically A zero. Then, the min-plus roots of P will be called the algebraic eigenvalues of the A min-plus matrix polynomial A. Note that the valuation val P can be computed ∈ R × A by introducing the matrix val A n n, such that (val A) = val A , where A min ij ij ij denotes the (i, j) entry of the min-plus polynomial A. Then, val P is equal to the A min-plus permanent of the matrix val A. By symmetry, the degree deg P is equal ∈ R × A to the max-plus permanent of the matrix deg A n n, such that (deg A) = ∞ max ij deg A . When val P > 0, 0 = + is an eigenvalue of A with multiplicity val P . ij A R A When deg P < nd, P has only deg P eigenvalues (bel(cid:99)onging to (cid:98) ). One A −∞ A R A min can−say that (the inﬁnity of min) ∈isRan eigenvalue of A w(cid:98)ith multiplicity nd deg P . Recall that, for any scalar y , we have P (y) = per A(y) which A min A i(cid:98)s the value of the optimal assignment associated to the matrix A(y). Hence, the algebraic eigenvalues of the matrix polynomial A (and so, the polynomial function P ) can be computed in O(n4d) time by adapting the method of Burkard and A Butkoviˇc [BB03]. Moreover, by adapting the parametric method of Gassner and Klinz [GK10], Hook [HT] showed that this can be reduced to a O(n3d2) time. The following result generalizes Theorem 7.1. A Theorem 8.1. Let be as in (28) and (29) and denote by A the min-plus matrix A polynomial (30) with coeﬃcients A as in (29). Assume also that = I, the C × k R × d identity matrix in n n so that A = I the identity matrix in n n. Then, A has d R min(cid:54) · · · (cid:54) nd min-plus algebraic eigenvalues (belonging to ). Let Γ = (γ γ ) be min L1 L nd A the sequence of these eigenvalues. Assume that the eigenvalues 1, . . . , nd of (cid:15) (cid:15) (cid:15)

text: NON-ARCHIMEDEAN

text: VALUATIONS

text: OF

text: EIGENVALUES

text: OF

text: MATRIX

text: POLYNOMIALS

text: A

label: equation

text: for

text: all

text: i,

text: j

text: n

text: A

text: We

text: shall

text: also

text: assume

text: that

text: the

text: matrix

text: polynomial

text: is

text: regular,

text: which

text: means

text: A

text: A

text: that

text: the

text: characteristic

text: polynomial

text: det(

text: (or

text: det(

text: of

text: the

text: matrix

text: polynomial

text: A

text: A

text: A

text: is

text: non

text: identically

text: zero.

text: Then,

text: the

text: eigenvalues

text: of

text: (or

text: are

text: by

text: deﬁnition

text: A

text: the

text: roots

text: of

text: this

text: polynomial.

text: If

text: deg

text: det(

text: nd,

text: then

text: we

text: also

text: say

text: that

text: is

text: an

text: A

text: A

text: A

text: A

text: eigenvalue

text: of

text: with

text: multiplicity

text: nd

text: deg

text: det(

text: When

text: YI,

text: these

text: are

text: A

text: the

text: usual

text: eigenvalues

text: of

text: We

text: shall

text: study

text: here

text: the

text: ﬁrst

text: order

text: asymptotics

text: of

text: the

text: A

text: eigenvalues

text: of

text: in

text: the

text: same

text: spirit

text: as

text: in

text: the

text: previous

text: section.

text: A

text: To

text: the

text: matrix

text: polynomial

text: one

text: can

text: associate

text: the

text: min-plus

text: matrix

text: polynomial

text: A

text: A

text: A

text: A

text: e(

text: e(

text: Ye(

text: Yde(

text: Here,

text: we

text: shall

text: rather

text: consider

text: the

text: min-plus

text: d

text: matrix

text: polynomial

text: associated

text: to

text: the

text: asymptotics

text: R

text: A

text: A

text: YA

text: YdA

text: n

text: n[Y]

text: d

text: min

text: The

text: min-plus

text: matrix

text: polynomial

text: A

text: can

text: seen

text: either

text: as

text: a

text: (formal)

text: polynomial

text: R

text: witRh

text: coeﬃcients

text: in

text: nminn,

text: namely

text: the

text: Ak,

text: k

text: d,

text: or

text: as

text: a

text: matrix

text: with

text: entrie∈s

text: in

text: [Y],

text: denoted

text: A

text: We

text: denote

text: by

text: A

text: the

text: function

text: which

text: associates

text: to

text: y

text: R

text: min

text: ij

text: R

text: the

text: matrix

text: A(y)

text: A

text: yA

text: ydA

text: n

text: n.

text: We

text: call

text: min-plus

text: min

text: d

text: min

text: of

text: the

text: matrix

text: polynomial

text: A,

text: the

text: permanent

text: P

text: per

text: A.

text: A

text: This

text: is

text: a

text: formal

text: polynomial,

text: the

text: associated

text: polynomial

text: function

text: of

text: which

text: is

text: equal

text: to

text: P

text: (y)

text: per

text: A(y).

text: A

text: We

text: shall

text: say

text: that

text: the

text: matrix

text: polynomial

text: A

text: is

text: regular

text: if

text: P

text: is

text: not

text: identically

text: A

text: zero.

text: Then,

text: the

text: min-plus

text: roots

text: of

text: P

text: will

text: be

text: called

text: the

text: algebraic

text: eigenvalues

text: of

text: the

text: A

text: min-plus

text: matrix

text: polynomial

text: A.

text: Note

text: that

text: the

text: valuation

text: val

text: P

text: can

text: be

text: computed

text: R

text: A

text: by

text: introducing

text: the

text: matrix

text: val

text: A

text: n

text: n,

text: such

text: that

text: (val

text: A)

text: val

text: A

text: where

text: A

text: min

text: ij

text: ij

text: ij

text: denotes

text: the

text: (i,

text: j)

text: entry

text: of

text: the

text: min-plus

text: polynomial

text: A.

text: Then,

text: val

text: P

text: is

text: equal

text: to

text: the

text: A

text: min-plus

text: permanent

text: of

text: the

text: matrix

text: val

text: A.

text: By

text: symmetry,

text: the

text: degree

text: deg

text: P

text: is

text: equal

text: R

text: A

text: to

text: the

text: max-plus

text: permanent

text: of

text: the

text: matrix

text: deg

text: A

text: n

text: n,

text: such

text: that

text: (deg

text: A)

text: max

text: ij

text: deg

text: A

text: When

text: val

text: P

text: is

text: an

text: eigenvalue

text: of

text: A

text: with

text: multiplicity

text: val

text: P

text: ij

text: A

text: R

text: A

text: When

text: deg

text: P

text: nd,

text: P

text: has

text: only

text: deg

text: P

text: eigenvalues

text: to

text: One

text: A

text: A

text: R

text: A

text: min

text: can−say

text: that

text: (the

text: inﬁnity

text: of

text: min)

text: ∈isRan

text: eigenvalue

text: of

text: A

text: multiplicity

text: nd

text: deg

text: P

text: Recall

text: that,

text: for

text: any

text: scalar

text: y

text: we

text: have

text: P

text: (y)

text: per

text: A(y)

text: which

text: A

text: min

text: A

text: the

text: value

text: of

text: the

text: optimal

text: assignment

text: associated

text: to

text: the

text: matrix

text: A(y).

text: Hence,

text: the

text: algebraic

text: eigenvalues

text: of

text: the

text: matrix

text: polynomial

text: A

text: (and

text: so,

text: the

text: polynomial

text: function

text: P

text: can

text: be

text: computed

text: in

text: time

text: by

text: adapting

text: the

text: method

text: of

text: Burkard

text: and

text: A

text: Butkoviˇc

text: Moreover,

text: by

text: adapting

text: the

text: parametric

text: method

text: of

text: Gassner

text: and

text: Klinz

text: Hook

text: [HT]

text: showed

text: that

text: this

text: can

text: be

text: reduced

text: to

text: a

text: time.

text: The

text: following

text: result

text: generalizes

text: Theorem

text: A

text: Theorem

text: Let

text: be

text: as

text: in

text: and

text: and

text: denote

text: by

text: A

text: the

text: min-plus

text: matrix

text: A

text: polynomial

text: with

text: coeﬃcients

text: A

text: as

text: in

text: Assume

text: also

text: that

text: I,

text: the

text: C

text: k

text: R

text: d

text: identity

text: matrix

text: in

text: n

text: n

text: so

text: that

text: A

text: I

text: the

text: identity

text: matrix

text: in

text: n

text: n.

text: Then,

text: A

text: has

text: d

text: R

text: nd

text: min-plus

text: algebraic

text: eigenvalues

text: (belonging

text: to

text: Let

text: Γ

text: (γ

text: γ

text: be

text: min

text: L

text: nd

text: A

text: the

text: sequence

text: of

text: these

text: eigenvalues.

text: Assume

text: that

text: the

text: eigenvalues

text: nd

text: of


text: 22 MARIANNE AKIAN, RAVINDRA BAPAT, AND STE´PHANE GAUBERT (counted with multiplicities) have ﬁrst order asymptotics, L ∼ i λ (cid:15)Λi , (cid:15) i (cid:54) · · · (cid:54) ≺ and let Λ = (Λ Λ ). Then, Λ w Γ. 1 n Q A ∈ C A Proof. Let us denote by = det( ) [Y] the characteristic polynomial of A and by P = per A the min-plus characteristic polynomial of A. Since = I, we Q L L A d have = 1 and P = 1. The eigenvalues 1, . . . , nd of are the roots of nd Q nd (cid:15) (cid:15) (cid:15) the polynomial . Assume that they have ﬁrst order asymptotics with exponents (cid:54) · · · (cid:54) C† Λ Λ . From (20), these are elements of and their images by e (counted 1 nd (cid:54) · · · (cid:54) with multiplicities) are equal to Λ Λ . By Proposition 5.1 applied to the C (cid:54) · · · (cid:54) 1 nd ring , Λ = (Λ Λ ) coincide with the roots of the min-plus polynomial Q 1 nd Q Q = e( ). Then, computing the coeﬃcients of the formal polynomial , we can (cid:62) ≺ deduce that Q P. Since Q = P = 1, we get that Λ = R(Q) w R(P) = Γ. nd nd Note that another way to prove this result is to apply a block companion trans- A B formation to the matrix polynomials and A, yielding linear pencils and B with B leading terms and B equal to the identity matrix, then to apply Theorem 7.1 B 1 1 (cid:3) to . 0 ∈ R × (cid:54) For any matrix B n n such that per B = 0, we deﬁne the graph Opt(B) as min the set of arcs belonging to optimal assignments: the set of nodes of Opt(B) is [n] and there is an arc from i to j if there is a permutation σ such that j = σ(i) and | | σ = per B. One can compute Opt(B) by using the following construction. B R R \{ } We shall say that two vectors U, V of dimension n with entries in = 0 (cid:62) min form a Hungarian pair with respect to B if, for all i, j, we have B U V , and · · · · · · ij i j U U V V = per B, the products being understood in the min-plus sense 1 n 1 n R (since U, V are seen as elements of n ). Thus, (U, V ) coincides with the optimal min dual variable in the linear programming formulation of the optimal assignment problem. In particular, a Hungarian pair always exists if the optimal assignment (cid:54) problem is feasible, i.e., if per B = 0, and it can be computed in O(n3) time by § the Hungarian algorithm (see for instance [Sch03, 17]). For any Hungarian pair (U, V ), we now deﬁne the saturation graph, Sat(B, U, V ), which has set of nodes [n] and an arc from i to j if B = U V . We shall see in the following section that ij i j Opt(B) can be computed easily from Sat(B, U, V ). ∈ R (cid:98) For any min-plus matrix polynomial A and (cid:98)any scalar γ , we denote by G ∈min (A, γ) the graph with set of nodes [n], and an arc from i to j [n] if γk(A ) = k (cid:54) k(cid:48) ij A (γ) = 0. This is a subgraph of the graph of A(γ). For any graphs G and G , the ij ∩ (cid:48) intersection G G is the graph whose set of nodes (resp. arcs) is the intersection (cid:48) of the sets of nodes (resp. arcs) of G and G . Finally, if G is any graph with set of ∈ C × ∈ nodes [n], and if b n n, we deﬁne the matrix bG by (bG) = b if (i, j) G, ij ij and (bG) = 0 otherwise. The following results generalize Theorem 6.1 and 7.4, ij respectively. Their proof will be given in the next section. A C Theorem 8.2. Let be a regular matrix polynomial over as in (28) satisfy- ing (29) and denote by A the min-plus matrix polynomial (30) with coeﬃcients A k as in (29). Then A is a regular matrix polynomial and we have:

text: MARIANNE

text: AKIAN,

text: RAVINDRA

text: BAPAT,

text: AND

text: STE´PHANE

text: GAUBERT

text: (counted

text: with

text: multiplicities)

text: have

text: ﬁrst

text: order

text: asymptotics,

text: L

text: i

text: λ

text: i

text: and

text: let

text: Λ

text: (Λ

text: Λ

text: Then,

text: Λ

text: w

text: Γ.

text: n

text: Q

text: A

text: C

text: A

text: Proof.

text: Let

text: us

text: denote

text: by

text: det(

text: [Y]

text: the

text: characteristic

text: polynomial

text: of

text: A

text: and

text: by

text: P

text: per

text: A

text: the

text: min-plus

text: characteristic

text: polynomial

text: of

text: A.

text: Since

text: I,

text: we

text: Q

text: L

text: L

text: A

text: d

text: have

text: and

text: P

text: The

text: eigenvalues

text: nd

text: of

text: are

text: the

text: roots

text: of

text: nd

text: Q

text: nd

text: the

text: polynomial

text: Assume

text: that

text: they

text: have

text: ﬁrst

text: order

text: asymptotics

text: with

text: exponents

text: C†

text: Λ

text: Λ

text: From

text: these

text: are

text: elements

text: of

text: and

text: their

text: images

text: by

text: e

text: (counted

text: nd

text: with

text: multiplicities)

text: are

text: equal

text: to

text: Λ

text: Λ

text: By

text: Proposition

text: applied

text: to

text: the

text: C

text: nd

text: ring

text: Λ

text: (Λ

text: Λ

text: coincide

text: with

text: the

text: roots

text: of

text: the

text: min-plus

text: polynomial

text: Q

text: nd

text: Q

text: Q

text: e(

text: Then,

text: computing

text: the

text: coeﬃcients

text: of

text: the

text: formal

text: polynomial

text: we

text: can

text: deduce

text: that

text: Q

text: P.

text: Since

text: Q

text: P

text: we

text: get

text: that

text: Λ

text: R(Q)

text: w

text: R(P)

text: Γ.

text: nd

text: nd

text: Note

text: that

text: another

text: way

text: to

text: prove

text: this

text: result

text: is

text: to

text: apply

text: a

text: block

text: companion

text: trans-

text: A

text: B

text: formation

text: to

text: the

text: matrix

text: polynomials

text: and

text: A,

text: yielding

text: linear

text: pencils

text: and

text: B

text: with

text: B

text: leading

text: terms

text: and

text: B

text: equal

text: to

text: the

text: identity

text: matrix,

text: then

text: to

text: apply

text: Theorem

text: B

text: to

text: R

text: For

text: any

text: matrix

text: B

text: n

text: n

text: such

text: that

text: per

text: B

text: we

text: deﬁne

text: the

text: graph

text: Opt(B)

text: as

text: min

text: the

text: set

text: of

text: arcs

text: belonging

text: to

text: optimal

text: assignments:

text: the

text: set

text: of

text: nodes

text: of

text: Opt(B)

text: is

text: [n]

text: and

text: there

text: is

text: an

text: arc

text: from

text: i

text: to

text: j

text: if

text: there

text: is

text: a

text: permutation

text: σ

text: such

text: that

text: j

text: σ(i)

text: and

text: σ

text: per

text: B.

text: One

text: can

text: compute

text: Opt(B)

text: by

text: using

text: the

text: following

text: construction.

text: B

text: R

text: R

text: We

text: shall

text: say

text: that

text: two

text: vectors

text: U,

text: V

text: of

text: dimension

text: n

text: with

text: entries

text: in

text: min

text: form

text: a

text: Hungarian

text: pair

text: with

text: respect

text: to

text: B

text: if,

text: for

text: all

text: i,

text: j,

text: we

text: have

text: B

text: U

text: V

text: and

text: ij

text: i

text: j

text: U

text: U

text: V

text: V

text: per

text: B,

text: the

text: products

text: being

text: understood

text: in

text: the

text: min-plus

text: sense

text: n

text: n

text: R

text: (since

text: U,

text: V

text: are

text: seen

text: as

text: elements

text: of

text: n

text: Thus,

text: (U,

text: V

text: coincides

text: with

text: the

text: optimal

text: min

text: dual

text: variable

text: in

text: the

text: linear

text: programming

text: formulation

text: of

text: the

text: optimal

text: assignment

text: problem.

text: In

text: particular,

text: a

text: Hungarian

text: pair

text: always

text: exists

text: if

text: the

text: optimal

text: assignment

text: problem

text: is

text: feasible,

text: i.e.,

text: if

text: per

text: B

text: and

text: it

text: can

text: be

text: computed

text: in

text: time

text: by

text: the

text: Hungarian

text: algorithm

text: (see

text: for

text: instance

text: For

text: any

text: Hungarian

text: pair

text: (U,

text: V

text: we

text: now

text: deﬁne

text: the

text: saturation

text: graph,

text: Sat(B,

text: U,

text: V

text: which

text: has

text: set

text: of

text: nodes

text: [n]

text: and

text: an

text: arc

text: from

text: i

text: to

text: j

text: if

text: B

text: U

text: V

text: We

text: shall

text: see

text: in

text: the

text: following

text: section

text: that

text: ij

text: i

text: j

text: Opt(B)

text: can

text: be

text: computed

text: easily

text: from

text: Sat(B,

text: U,

text: V

text: R

text: For

text: any

text: min-plus

text: matrix

text: polynomial

text: A

text: and

text: scalar

text: γ

text: we

text: denote

text: by

text: G

text: ∈min

text: (A,

text: γ)

text: the

text: graph

text: with

text: set

text: of

text: nodes

text: [n],

text: and

text: an

text: arc

text: from

text: i

text: to

text: j

text: [n]

text: if

text: γk(A

text: k

text: ij

text: A

text: (γ)

text: This

text: is

text: a

text: subgraph

text: of

text: the

text: graph

text: of

text: A(γ).

text: For

text: any

text: graphs

text: G

text: and

text: G

text: the

text: ij

text: intersection

text: G

text: G

text: is

text: the

text: graph

text: whose

text: set

text: of

text: nodes

text: (resp.

text: arcs)

text: is

text: the

text: intersection

text: of

text: the

text: sets

text: of

text: nodes

text: (resp.

text: arcs)

text: of

text: G

text: and

text: G

text: Finally,

text: if

text: G

text: is

text: any

text: graph

text: with

text: set

text: of

text: C

text: nodes

text: [n],

text: and

text: if

text: b

text: n

text: n,

text: we

text: deﬁne

text: the

text: matrix

text: bG

text: by

text: (bG)

text: b

text: if

text: (i,

text: j)

text: G,

text: ij

text: ij

text: and

text: (bG)

text: otherwise.

text: The

text: following

text: results

text: generalize

text: Theorem

text: and

text: ij

text: respectively.

text: Their

text: proof

text: will

text: be

text: given

text: in

text: the

text: next

text: section.

text: A

text: C

text: Theorem

text: Let

text: be

text: a

text: regular

text: matrix

text: polynomial

text: over

text: as

text: in

text: satisfy-

text: ing

text: and

text: denote

text: by

text: A

text: the

text: min-plus

text: matrix

text: polynomial

text: with

text: coeﬃcients

text: A

text: k

text: as

text: in

text: Then

text: A

text: is

text: a

text: regular

text: matrix

text: polynomial

text: and

text: we

text: have: