Spectral Theory for the Bloch-Torrey operator

Spectral Theory for the Bloch-Torrey operator

Bernard Helffer

(after Almog, Grebenkov, Helffer, Henry, Moutal,...)

27 février 2021

Introduction

We consider the semi-classical Bloch-Torrey operator inL²(Ω,R^k)where Ωis a domain in R^d andk ∈ {1, . . . ,3}.

If the simplest model is some closed realization of

−h²∆ +ix₁

the original model of Bloch-Torrey corresponds tok =d=3and reads

−∆⊗I3+~b(x)×

where~b(x)is the magnetic field.

After some general discussion of the qualitative spectral analysis of these models, our aim is to analyze, for various domains and in the limith→0, the left margin of the spectrum and to establish resolvent estimates.

The presented results have been obtained in collaboration with Y. Almog, D.S. Grebenkov and N. Moutal in continuation of earlier works by Y. Almog and R. Henry devoted to the semi-classical analysis of

−h²∆ +iV(x)whereV is aC^∞potential.

The Bloch-Torrey operator

We consider a simplified version of the Bloch-Torrey equation [26]

(Torrey 1956), that is commonly used to model Diffusion-Weighted Magnetic Resonance Imaging (DW-MRI).

It describes the diffusion-precession of spin-bearing particles in nuclear magnetic resonance experiments and helps in understanding the intricate relation between the geometric structure of a studied sample (domain) and the measured signal.

Here I just repeat what my physicist collaborator D. Grebenkov says ! It assumes the form

∂tm~ =−γ ~b×m~ +D∆m~. (1) This time-dependent equation describes the evolution in time of a vector fieldm~ onR³, representing the divergence free magnetization vector

~

To obtain any information on the semigroup associated with (1), we need to analyze the resolvent of a suitable realization of the differential operator−D∆⊗I3+γ~b(x)×.

After dilation and a change of notation we write

B(x,dx) :=−²∆ +~b(x)×, (2) and as→this can be seen as a semi-classical problem.

Considering what has been done for the simpler model−²∆ +iV(x), we should first analyzed toy models corresponding to linearized versions of the operator and typically

−∆ +i~v·~x inR^d or inR^d+.

The spectral properties are immediately related to the spectral properties of the realizations of the so called complex Airy operator inRandR⁺:

A(x, d

dx) :=−d² dx²+ix

Coming back to the general Torrey situation. The problem is much more complex and will not be solved in the most general situations.

We begin by considering the general problem inR³and associate with this differential operator a closed realizationB. It is natural to ask the following questions :

Is the operator maximally accretive.

Can we characterize the domain of the operators ? When do we have compact resolvent ?

Can we localize the spectrum ?

In view of tha analysis of the decay of the associated semi-group, can we control the resolvent ?

What can we do in the semi-classical limit ?

If there is a hudge litterature for−²∆ +iV(x), we are not aware of many contributions in this new context, except some generalities which do not take in account the specific structure of our matrix-valued potentialM (see nevertheless M. Kunze, L. Lorenzi, A. Maichine, and A. Rhandi [23, 24]).

A model (1D)

Let us first consider the case when~b(x)depends only on one variable (sayx1).

In this case, we apply a partial Fourier transform in thex2 andx3

direction which leads to the following family of(1D)operators depending on((ξ₂, ξ₃)∈R²)

B x₁, d

dx₁, ξ₂, ξ₃

:=−² d² dx₁²⊗I+





²(ξ₂²+ξ₃²) −b3 b2

b3 ²(ξ₂²+ξ₃²) −b1

−b2 b1 ²(ξ₂²+ξ₃²)



. (3) What can we say about this1D model ?

In the Airy approximation spirit, it is natural to consider the reduction to the caseξ2=ξ3=0) and to linearize~b

~b= (β0,0, β3x1). withβ36=0.

In the caseβ0=0, there is a big simplification (see below), the problem is non trivial forβ₀6=0and we will need a rather complicate analysis for answering to the general questions above.

A relatively simple 2D model

More generally, for~b= (0,0,b3(x1,x2))then the skew-symmetric matrix associated with~b×is given by

M=





0 −b3 0 b₃ 0 0

0 0 0



. (4)

In a new basis ofC³the operator becomes

Be:=





−²∆ +ib₃ 0 0 0 −²∆−ib₃ 0

0 0 −²∆



. (5)

Obviously, in this basis−²∆ +m~ can be considered as three separate scalar operators. The spectral properties of−²∆ +ib have been considered in Almog (2008), Henry (2015), Almog-Henry (2016), Almog-Grebenkov-Helffer (2016–2020).

Note that if we define−²∆ +MonL²(R³,R³)for~b= (0,0,x1)we obtain that the spectrum isR+ (which is preciselyσ(−∆)on L²(R³)) given thatσ(−²∆ +ix1) =∅on L²(R³).

Properties of Schrödinger operators with matrix-valued potentials (after Helffer-Nourrigat, Almog-Helffer)

One can generalize (2) by considering the operator

P(x,dx) :=−∆⊗Id+M(x), (6) whereM∈C^∞(R^k,Md(R)).

With

Ms =1

2(M+M^t),Mas= 1

2(M−M^t). we assume

(Ass.1) Ms ≥0.

Accretivity

We can extend maximal accretivity results that have been established for the selfadjoint operator−∆ +V (see [11]) and also for two interesting non-selfadjoint operators : the Fokker-Planck operator [17] and the complex Schrödinger operator−∆ +iV [16].

Proposition A (maximal accretivity)

LetP (above) be defined onC₀^∞(R^k,R^d)and satisfy (Ass.1). Then, its closure, under the graph norm, denoted byP, is maximally accretive as an unbounded operator inL²(R^k,R^d). Moreover,

D(P)⊂H¹(R^k,R^d).

Essential spectrum

Ifdim kerM(x)>0 for allx∈R^k\K as in the caseMs ≡0 we can show that the resolvent is not compact.

The proof is based on the construction of an infinite orthonormal sequence of approximate solutions.

This is indeed true and as an application we get for the Bloch-Torrey model

Proposition ES

Suppose that for|x| ≥R it holds that~b6=0

|d_x^αb_j(x)| ≤C|~b(x)|,∀αs.t 1≤ |α| ≤2,j =1,2,3. (7) Then the resolvent ofB is not compact.

MoreoverR+⊆σ(P), whereR+:= [0,+∞).

Particular case

This applies for

~b=A x+f~,

whereA6=0is a d×k matrix andf~∈R^d. Note that ifA=0, then

σ(P) =R+∪ {R++i|f~|} ∪ {R+−i|f~|}.

Of particular interest is the case wherek =1andA= ˆi3and henceR+is in the essential spectrum ofB.

Maximal estimates

A natural question is the effective description ofD(P), currently defined as the closure ofC₀^∞(R^k,R^d)under the graph norm.

We can still determine the domain in various different cases. In particular, when looking at the Bloch-Torrey equation, we get as a corollary of a more general theorem

Theorem ME1

Letd=k=3 andM =~b×. Then, if∃C >0 and∃R>0s.t.∀|x| ≥R,

~b(x)6=0and

|d_x^αbj(x)| ≤C|~b(x)|,∀αs.t 1≤ |α| ≤2,j =1,2,3, (8) then

D(P) ={~u∈H²(R³,R³), ~b×~u∈L²(R³,R³)}.

Characterization of the domain when two components of ~ b are bounded

In this case, we can use the results in [18] (Helffer-Nourrrigat 2018), obtained for the scalar operator−∆ +iV(x).

Theorem ME2 (Almog-Helffer)

LetB ( >0) where b1,b2∈L^∞(R^k)andb3∈C^r+1(R^k)(r ≥1) satisfies

|β|=r+1max |D_x^βb3(x)| ≤C0

sX

|α|≤r

|D_x^αb3(x)|²+1. (9) Then

D(B₁) ={~u∈H²(R^k,R³), ~b×~u∈L²(R^k,R³)}. (10)

Proof

After a change of basisM assumes the form

M˜ :=





−ib3 0 −(b1−ib2)/√ 2

0 ib3 −(b1+ib2)/√

2 (b1+ib2)/√

2 (b1−ib2)/√

2 0



.

and

B˜1:=−∆⊗I3+ ˜M.

For~˜u∈L²(R^k,R^d)satisfyingB˜1~u˜=~˜f ∈L²(R^k,R³),

−∆ ˜u1−ib3u˜1= ˜g1

−∆ ˜u2+ib3u˜2= ˜g2

−∆ ˜u3= ˜g3,

(11)

where

˜

g1 = ˜f1+^b1√−ib2

2 u˜3,

˜

g2 = ˜f2+^b1√+ib2

2 u˜3

˜

g3 = ˜f3−^b1√+ib2

2 u˜1−^b1√−ib2

2 u˜2.

(12)

Clearly,g˜i ∈L²(R^k)(i∈ {1,2,3}).

By standard elliptic estimates in the third lineu˜3∈H²(R^k).

For the two first lines the scalar theorem relative to−∆±ib3implies

Resolvent estimates

We can now obtain "rough" estimates for the resolvent ofB1, using known resolvent estimates obtained for−∆±ib3.

Better estimates are actually obtained for the particular case where~bis affine (see below).

The resolvent equation( ˜B1−λ) ˜u= ˜f, takes the form (dropping the accent in the sequel) :

f1= (−∆−ib3−λ)u1−b1−ib2

√2 u3 (13a)

f₂= (−∆ +ib₃−λ)u₂−b₁+ib₂

√2 u₃ (13b)

f3= (−∆−λ)u3+b1+ib2

√2 u1+b1−ib2

√2 u2. (13c)

Assuming thatλ /∈σ(−∆±ib₃)we write

R_±(λ) = (−∆∓ib₃−λ)⁻¹

to obtain foru3:

((−∆−λ)−cR₋(λ)¯c−cR¯ +(λ)c)u3=f3−cR+(λ)f1−¯cR₋(λ)f2, (14) where

c:= (b1+ib2)/√ 2.

Assuming furtherλ /∈R⁺ we can attempt to estimate the norm of the well-defined bounded operator

K_λ:= 1

2(−∆−λ)⁻¹(cR₋(λ)¯c+ ¯cR₊(λ)c). (15)

To proceed further we assume Assumption

For an intervalI,∃s<1,D₁>0andD₂>0 such that, ifReλ∈I and

|Imλ| ≥D1, then

kR_±(λ)k ≤D2|Imλ|^s.

The above bound applies forI = (−∞, τ)for everyτ∈R, and

b3(x) =x1 (in which cases=0) orb3(x) =x₁²(wheres=−¹₃) (see my Cambridge book (2013)).

Given the above assumption we can obtain the following resolvent estimate

Proposition RE

Letb3∈C^r(R^k)satisfy previous assumption, for someI. Then,∃C1>0 and∃C2>0 s. t.∀λ∈CsatisfyingReλ∈I and|Imλ| ≥C1, it holds thatλ /∈σ(B1)and

k(B1−λ)⁻¹k ≤C2|Imλ|^s. (16)

Point spectrum

Proposition PS

LetΩ =C\(R⁺∪σ(−∆±ib3)). Under the previous assumptions, it holds forλ∈Ω:

1 λ∈σ(B1)if and only if −1∈σ(K_λ).

2 If−∆±ib3has a compact resolvent, then σ(B1)∩Ωconsists in isolated eigenvalues ofB1 of finite multiplicity.

Remarks

One can establish Proposition PS as soon ascR¯ +(λ)c is compact.

One can go further in the case k=1 andb₃(x) =x.

In this case,σ(−²∆±ib₃) =∅ andR+⊂σ(B₁)and the proposition implies that :

σ(B)∩(C\R⁺)is discrete.

Assumingk =1,b1constant, b2=0, andb3(x) =x, we will obtain (Almog-Helffer) much more precise results forB in the asymptotic regime →0.

This is actually hard work ! We will only give the main results.

Main statements for a simplified model

In the case whereB is defined inRwe obtain Theorem : localization of the spectrum LetB be defined by

B x, d

dx

:=−² d²

dx²+~b×, (17) with~b= (1,0,x).

Let for >0,n∈N^∗,κ⁰_n() :=i+²ⁿ⁻¹₂ (1+i) . Then we have :

Λ∈σ(B)⇔Λ∈σ(B).

R+⊂σ(B).

Then ∀N∈N^∗ ∃0 andC, s.t.ˆ ∀0< ≤0,∃{κn()}^N_n=1⊂σ(B)

s.t.

κn()−κ⁰_n()

≤Cˆ ², forn=1. . . ,N. (18)

Theorem : estimates for the resolvent.

Under the same assumptions,

Let % >0,Rˆ>0andN%=h_2%+1

2

i ,and D( ˆR, %, )

={Λ∈C\ ∪^N_n=1^% B(κ⁰_n(),Rˆ ²)∪B(κ⁰_n(),Rˆ ²) }

∩{ReΛ≤%} ∩ {ImΛ6=0}.

(19)

Then, ∃C >0 and∃Rˆ₀>1s. t. ∀Rˆ₀<Rˆ<[√

2]⁻¹and Λ∈D( ˆR, %, )it holds that

k(B−Λ)⁻¹k ≤C

1+ ^2/3

|ImΛ|²+ 1 Rˆ ⁵³

. (20)

Part II : The Bloch-Torrey equation in the case with boundary.

Consider a bounded smooth open setO ⊂R^k and the Dirichlet realization ofP inO denoted byP^O. ForM ∈C²(O,Md(R)), P^O is a bounded perturbation of−∆ and hence

D(P^O) =H²(O,R^d)∩H₀¹(O,R^d), andP^O has a compact resolvent.

Nevertheless, in the presence of a small parameter, i.e. when P^O=−²∆ +M

the behaviour of spectrum and the resolvent in the limit→0could probably be understood from the analysis of linearized operators acting onR^k.

This is precisely the case in the two dimensional setting whenP^O is equivalent to the Dirichlet realization of−²∆ +iV (considered inn the

The scalar Bloch-Torrey equation

We would like to analyze the behavior ash→0 of the spectrum of the semi-classical realizationA^D_h of−h²∆ +ix.

The main result is : Theorem

We have

h→0lim 1 h^2/3inf

Reσ(A^D_h) =|a₁|

2 , (21)

wherea1<0 is the rightmost zero of the Airy functionAi.

Moreover, for everyε >0, there existhε>0 andCε>0 such that

∀h∈(0,hε), sup

γ≤^|a₂^1| ν∈R

k(A^D_h −(γ−ε)h^2/3−iν)⁻¹k ≤ C_ε

h^2/3. (22)

This holds for the interior problem and the exterior problem. One can also consider Neumann, Robin or transmission problems.

In all these studies it appears that there is a localization of the "first"

eigenfunctions at the boundary and more specifically at points where∇V is parallel to the normal.

This can be understood starting from the analysis of our toy models

−∆ +i~v·~x in R² andR²+.

Note, and this explains the above mentioned localization, in the second case that the spectrum is empty except in the case when~v = (1,0)i.e. is parallel to the normal to the boundary{x1=0}.

Bloch-Torrey operator in periodic structures

We consider the Bloch-Torrey operator in a periodically perforated domain (d=2),

Ω =R^d\ {∪_γ∈ZdHγ}, where

Hγ={x∈R² : x−γ∈H0} and

H₀⊂(−1/2,1/2)²

is a domain with a piecewise smooth boundary such that all the H_γ are disjoint.

A typical example ofH0 is a ballB(0,r)of radiusr <¹₂ centered at the origin.

One of the major difficulties in the definition and study of such non-self-adjoint operators is that the potentialix₁is not periodic, unbounded and changing sign. We focus on the Dirichlet realizationA^D.

Periodic problems in the non self adjoint case

1 What about the spectral application of the Floquet theory for non selfadjoint operators. If the Floquet decomposition exists, do we have always

σ(A^D) =∪σ(A^D_q) ?

2 Do we haveρ(A^D) =C? Isρ(A^D)connected ?

3 Is the link between the existence of a Weyl’s sequence and spectrum correct in the non self-adjoint case ?

4 Another difficulty is that the Torrey operator in a periodic structure is NOT periodic in the two directions.

5 Is the spectrum non empty ?

Partial answers

1 For the first item, we can start with a Floquet theory in the x2-direction.

We can only prove Proposition F1

∪q₂∈Rσ(A^D_q2)⊂σ(A^D),

2 The questions are open but probably yes (in 2D). In the self-adjoint case the property is evidently true.

An instructive model in1D is to consider the2-steps model on the line −_dx^d22 +V(x), whereV(x)has period2 and is defined on[0,2[

by V(x) =0, forx∈[0,1[,V(x) =L, forx ∈[1,2[. For this model with ReL>0large, one can show by explicit

[3] No in general but yes in our particular case.

[5] Yes in the semi-classical result.

Partial answer to item 4

We observe that

τ1◦ A^D_q2 = (A^D_q2−i)◦τ1. (23) Hence

Kq2 :=exp(−2πA^D_q2), (24) commutes with the first translationτ1.

One can then do the Floquet theory forKq2 leading a priori to a family K_q2,q1.

Proposition

Ifµis an eigenvalue ofK_q2,0with corresponding eigenfunctionu0, then

1

2πlogµ+iZbelongs to the spectrum ofA^D_q2 and, for eachk, we can construct starting fromu0 an eigenfunctionuλ_k ofA^D_q2 associated with λk := _2π¹ logµ+ik.

The heuristics behind the proof.

Heuristically, we start from an eigenfunctionu0and associate with it uq₁ =exp

−q1(B_q^D₂−λ0)

u0, (25)

which satisfies the Floquet condition relative toq1. Definingu by

u= 1 2π

Z 2π 0

uq₁dq1. (26)

we obtain formally :

(B_q^D₂−λ0)u = _2π¹ R2π

0 (B_q^D₂−λ0)exp(−p(B^D_q2−λ0))dq1

u0

= _2π¹ R2π

0 (−_dp^d exp(−q1(B^D_q2−λ0)))dq1

u0

= _2π¹ I −exp(−2π(B^D_q2−λ0)) u0

=0.

The end

Thanks to the organizers.

M. Abramowitz and I. A. Stegun,Handbook of Mathematical Functions. Dover, 1972.

Y. Almog, D. Grebenkov, and B. Helffer,On a Schrödinger operator with a purely imaginary potential in the semiclassical limit. Comm.

Partial Differential Equations, 44 (2019), 1542–1604.

Y. Almog, B. Helffer and X. B. Pan,

Superconductivity near the normal state in a half-plane under the action of a perpendicular electric current and an induced magnetic field.

Trans. AMS 365 (2013), 1183-1217.

Y. Almog,The stability of the normal state of superconductors in the presence of electric currents. SIAM Journal on Mathematical Analysis, 40 (2008), pp. 824–850.

Y. Almog and B. Helffer,On the stability of laminar flows between plates. ArXiv :1908.06328, (2019).

,Superconductivity near the normal state in a half-plane under the action of a perpendicular electric current and an induced magnetic field. Trans. Amer. Math. Soc., 365 (2013), 1183–1217.

Y. Almog and R. Henry.

Spectral analysis of a complex Schrödinger operator in the semiclassical limit.

SIAM J. Math. Anal. 44 (2016), pp. 2962–2993.

P. T. Callaghan,Principles of nuclear magnetic resonance microscopy, Oxford University Press on Demand, 1993.

E. B. Davies,

Linear Operators and their Spectra. vol. 106 of Cambridge Studies in Advanced Mathematics,

Cambridge University Press, Cambridge, 2007.

B. Helffer,Semiclassical analysis, Witten Laplacians, and statistical mechanics.

Series on Partial Differential Equations and Applications, Vol. 1 (2002).

NMR Survey of Reflected Brownian Motion.

Rev. Mod. Phys. 79 (2007), pp. 1077–1137.

D. S. Grebenkov,

Diffusion MRI/NMR at high gradients : challenges and perspectives.

(accepted to Micro. Meso. Mater., 2017).

D. S. Grebenkov, B. Helffer.

On spectral properties of the Bloch-Torrey operator in two dimensions.

Submitted. Preprint : http ://arxiv.org/abs/1608.01925.

D.S. Grebenkov, B. Helffer and N. Moutal.

On the spectral properties of the Bloch-Torrey equation in infinite periodically perforated domains.

ArXiv 2020.

B. Helffer,Spectral Theory and its Applications. vol. 139 of

Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2013.

Lecture Notes in Mathematics, 1862. Springer-Verlag, Berlin, 2005.

B. Helffer, J. Nourrigat,

On the domain of a magnetic Schrödinger operator with complex electric potential.

Analysis and operator theory, 149–165, Springer Optim. Appl., 146, Springer, Cham, 2019.

R. Henry,On the semiclassical analysis of Schrödinger operators with purely imaginary electric potentials in a bounded domain. ArXiv preprint arXiv :1405.6183.

T. Kato,Perturbation Theory for Linear Operators. Springer, third edition, 1980.

P. Kuchment.

Floquet Theory for Partial Differential Equations.

Operator Theory : Advances and Applications, vol. 60, Birkhäuser Verlag, Basel, 1993.

P. Kuchment.

An overview of periodic elliptic operators.

M. Kunze, L. Lorenzi, A. Maichine, and A. Rhandi,L^p-theory for Schrödinger systems, Math. Nachr., 292 (2019), pp. 1763–1776.

M. Kunze, A. Maichine, and A. Rhandi,Vector-valued Schrödinger operations inL^p-spaces, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), pp. 1529–1541.

N. Moutal, A. Moutal, and D. S. Grebenkov.

Diffusion NMR in periodic media : efficient computation and spectral properties.

J. Phys. A : Math. Theor.53 (2020), 325201.

H.C. Torrey,

Bloch equations with diffusion terms.

Phys. Rev. Vol. 104, no 3 (Nov. 1956), 563–565.