A survey of topics related to Functional Analysis and Applied Sciences

: This survey is the result of investigations suggested by recent publications on functional analysis and applied sciences. It contains short accounts of the above theories not usually combined in a single document and completes the work of D. Huet 2017. The main topics which are dealt with involve spectrum and pseudospectra of partial diﬀerential equations, Steklov eigenproblems, harmonic Bergman spaces, rotation number and homeomorphisms of the circle, spectral ﬂow, homogenization. Applications to diﬀerent types of natural sciences such as echosystems, biology, elasticity, electromagnetisme, quantum mechanics, are also presented. It aims to be a useful tool for advanced students in mathematics and applied sciences.


INTRODUCTION
The article is divided into several sections, presented in the alphabetical order, as follows

Formation of patterns in reaction-diffusion systems
Set-up (This section is taken from A. Doelman [20]).

Definition 1.
An N -component reaction-diffusion system for U = (U 1 , U 2 , ...U N ) ∈ R N is a system of the form where U(x, t) depends on (x, t) ∈ Ω × R + , with Ω ⊂ R n , D is a diffusion matrix i.e. a diagonal N × N matrix with strictly positives entries, ∆ is the Laplace/diffusion operator, µ ∈ R m represents parameters and the vector field F(U, µ) : R N → R N represents the nonlinear reaction terms.
Definition 2. In [58], A.M. Turing wrote "a system of chemical substances, called morphogens, reacting together and difffusing through a tissue, is adequate to account for the main phenomena of morphogenesis. Such a system, although it may originally be quite homogeneous, may later develop a pattern or structure due to an instability of the homogeneous equilibrium.

The onset of patterns formation: the Turing/Ginsburg-Landau bifurcation
It is assumed that L < 0. In fact, the author ( [20]) shows that no (small amplitude) Turing and dropping the hats, equation (16) becomes This equation has a family of stationary spatially periodic solutions A(ξ, τ ) = Re iKξ) , R > 0 with K 2 + R 2 = 1 and − 1 < K < 1.

Remark 1.
In [20], Section 2.5, the author investigates the case of Hopf bifurcation in (9) i.e. the generation of patterns as µ passes through the critical value µ c .
1. 4 The particular case of the 1D-Gray-Scott model and chemistry (Cf. David S. Morgan et al. [42]). The Gray-Scott model, related to chemical rections between two species U and V, consists of the reaction-diffusion system Where U = U (x, t), x ∈ R, V = V (x, t), are the concentrations of U (inhibitor) and V (activtator) , A and B are rate constants, D U and D V are diffusivities. Here, for convenience, D U = 1 and D V = δ 2σ , 0 < δ 1, σ > 0.The stationary states (trivial patterns) solutions of the system are U ≡ 1, V ≡ 0 and, when 4B 2 < A, The state (U ≡ 1, V ≡ 0) is is linearly stable for all A, B > 0. On the other hand, it is proved ( [42], p.117) that the state (U + , V + ) cannot be marginally stable (cf. Definition 6) Therefore the authors focus only on (U − , V − ).
Linear stability of (U − , V − ), Turing/Ginsburg-Landau bifurcation. Linearizing (22) around the stationary with A, B rescaled as A = δ α a, B = δ β b, α, β ≥ 0. The analysis of the eigenvalues of M shows that (U − , V − ) is linearly stable if and only if 2α ≤ 3β and determines the values a c and k c of the parameter a and the wavenumber k such that Remark 2. 1) With the above scaling for A and B, one has, to leading order 2) For the Gray-Scott model, the Ginsburg-Landau equation (cf. (16)) has the form After setting A(ξ, τ ) = Re iκξ , solutions R and κ satisfy the equation The main result. Finally, the following result is obtained ( [42], Theorem 3.2): Let a = a c − γ 2 and 3β = 2(σ + α).
For 0 < γ 1 small enough, there exists a one parameter family of stationary spatially-periodic solutions of (22)(cf definition 4) that are close to the stationary state (U − , V − ): where R and κ are related by (30).

Initial Kausmeier model
In [37], C. A. Klausmeier considers the nondimensionalized system: for water w and plant biomass n, defined on an infinite two-dimensional domain indexed by x and y. In (32), a controls water input, m measures plant losses and ν contols the rate at which water flows downhill. The corresponding nonspatial model is a − w − wn 2 = 0 He shows that this model has a bare stable stateŵ = a,n = 0 and an other one vegetated. These stable states correspond to spatially homogeneous equilibria of (32). Then he uses linear stabiliy analysis, for system (32) where u(x, t), v(x, t) : R × R + → R, and k i ≥ 0, i = 0, ..., 5, d ≥ 0.The flow of water is denoted by u t , the slope of the aera by k o u x , the constant precipitation rate by k 1 , an evaporation rate by −k 2 u, and an infiltration rate by −k 3 k 5 uv 2 .
The change of biomass is assumed to be controlled by a diffusion term d v v xx . The death rate is denoted by −k 4 v and the infiltration feedback by k 5 uv 2 . In [59], system (34) is completed by the nonlinear diffusion term d u (u γ ) xx : where γ ≥ 1 and 0 < d v d u , and rescaled as with 0 < δ 1 and Remark 3. For ecosystems without a slope k 0 = 0 and therefore Definition 4. Spatially periodic solutions (patterns) or wave trains are solutions u(x, t) that can be written that satisfy u p (ξ) = u p (ξ + 2π). Here κ is called the nonlinear wave number. It is pointed out that the ecologically relevant parameter values of γ are γ = 1 or γ = 2. For γ = 1 and C = 0, system (36) is the gray-Scott system (22). In [42] Section 6.2, the existence of a Busse balloon, for the Gray-Scott model, is investigated.

The background states and the Turing-hopf instability
The model (36) has the same homogeneous background states as the Gray-Scott model for A > 4B 2 , namely (cf (23) and (24)).
The state (U 0 , V 0 ) = (1, 0) represents the desert since, in this case, v = k1 k2k3 V 0 = 0. The state (U − , V − ) does not represent a homogeneously vegetated state. By linearization of (36) about the state u + = (U + , V + ) we have : , and suitable matrices C and D. Let M be the matrix defined by where 0 < δ 1 and with a, b, c, = O(1) with respect to δ. Here k is refered to as the linear wavenumber.
Remark 5. With the above scaling, (U + , V + ) can be written out to leading order in δ The L 2 -spectrum of (41) is the set of λ ∈ C such that Definition 6. L(∂ x ) is called marginally stable with critical Fourier mode u 0 e ik * x associated to the critical eigenmode iω * , up to complex conjugaison, if d(iω * , ik * ) = 0, d(iω * , ik) = 0 for all k = ±k * (46) and d(λ, ik) = 0, for all k ∈ R , all λ ∈ C with λ = iω * and Reλ ≥ 0. In this case, we also say that the background state (U + , V + ) of (36) is marginally stable.
Main results:1) when C=0, (U + , V + ) is marginally stable for σ, a = a * , k = k * satisfying, to leading order in δ, and drop the tilde onk. Then, the stationary state (U + , V + ) undergoes a Turing-Hopf instability at a uniquely defined critical parameter a = a * and a critical wavenumber k = k * that satisfy and, if c 1 to leading order in c and δ ([59] Proposition 2).

Ginzburg-Landau equation
If |a − a * | = r 2 (51) and is small enough, the Ginzburg-Landau equation (16) associated to (36) has the form whose coefficients are functions of b, c, and γ ( [59], Proposition 3) .The Turing-Hopf instability of (U + , V + ) is super- The Benjamin-Feir-Newell criterion. If the Turing-Hopf bifurcation is supercritical, there exists a band of stable spatially periodic patterns if and only if By means of the computing system Matematica, the authors evaluated the coefficients of (52), for γ = (1, 2). These 3) For c = 0, all coefficients of (52) are real and L 1 = L 1 (γ) becomes positive for large γ and equals 0 for γ ≈ 13.0446.

Remark 6.
In Section 3 of [59], existence of stable patterns if a is not closed to a * is studied. Busse Ballons (cf. Definition 5) are numerically constructed for different values of a and fixed values of b, c, γ

Gierer-Meinhardt system and biology
The Gierer-Meinhardt system (cf. [27]) is one of the most popular models in biological pattern formation. Let (S, g) be a compact two-dimensional Riemannian manifold without boundary.(For definitions related to Riemannian manifolds, see W. Kühnel [39]). In [57], W. Tse et al. consider the system in S. Here, A(p, t), H(p, t) > 0 represent the activator and the inhibitor concentrations, repectively, at a point p ∈ S and at time t; their corresponding diffusivities are denoted by 2 and 1 β 2 ; τ is the time-relaxation constant of the inhibitor and ∆ g is the Laplace-Beltrami operator with respect to the metric g. Their assumptions on the parameters and β are: is small enough and lim β 2 2 = κ > 0. Let G 0 be the Green function defined by K(p) denote the Gauss curvature on S, and w be the solution of the problem They introduce the function F (p) = c 1 K(p) + c 2 R(p) where R(p) denotes the diagonal of the regular part of the Green function, and

Existence
If p 0 ∈ S is a non-degenerate critical point of F (p), i.e.

Remark 7.
It is assumed that ξ ,p is the height of the peak and that p ∈ Λ δ is the location of the peak, where is the ball with center p o and radius δ, with respect to the metric g).

Linear stability
Linearizing (54) around the equilibrium states (A + φ e λ t , H + ψ e λ t, ) the following eigenvalue problem L is where (A , H ) is the above solution of system (54).
let (A , H ) be the above single peaked solution, whose peak approches p o , then, there exists a unique τ 1 > 0 such that
With the rescaled amplitudes an equilibrium solution (a,h) solves the rescaled Gierer-Meinhardt system on a compact two-dimensional Riemannian manifold (M, g) without boundary. They assume It is assumed that there is a non-degenerate local maximum point of the Gaussian curvature function K(p) of M at p o = 0, i.e. ∇K(0) = 0 and Let p o be a non-degenerate local maximum point of the gaussian curvature K(p) of M. They show that, under stronger conditions on and D, namely the system (67) has, at least, two different 2-spike cluster solutions (A i , H i ) for i=1,2. Moreover, one of the solutions is stable and the other one is unstable.

Introduction
This section is related to Steklov eigenproblems. Cf. F. Gazzola, H-C. Grunau, G. Sweers [26]. Let Ω be a bounded domain in R n , n ≥ 2 with Lipschitz boundary ∂Ω, and a ∈ R. The classical Dirichlet biharmonic Steklov eigenproblem is the boundary value eigenvalue problem where ν(x) is the unit outward normal at x ∈ ∂Ω. (Since ∂Ω is Lipschitzian, the tangent hyperplane and the unit outward normal ν = ν(x) are well-defined for (a.e) x ∈ ∂Ω).

Orthogonal decomposition of H
The bounded domain Ω is assumed to have a C 2 -boundary, and the Hilbert space Let Z be the space and V be the completion of Z with respect to the scalar product (73). The following orthogonal decomposition is obtained and

The least positive eigenvalue of (71)
Let Ω be a bounded domain with C 2 boundary.The least positive eigenvalue of (71) is characterised by In [26] (Theorem 3.17), it is proved that the minimum in (78) is achieved, and, up to a multiplicative constant, the minimiserū is unique and solves (71) when a = δ 1 . Furthermore,ū ∈ C ∞ (Ω), and, up to the boundary,ū is as smooth as the boundary permits. An alternative characterisation of δ 1 (Ω) Here Ω is a bounded domain with Lipschitz boundary. Let Let It is proved that σ 1 (Ω) admits a minimiser. If, moreover, Ω satisfies a uniform outer ball condition, then, the minimiser is positive, unique up to a constant multiplier, and σ 1 (Ω) = δ 1 (Ω) ( [26], Theorem 3.23).

Subspaces of H 1
(Ω) related to the usual Laplacian ∆. Real harmonic Bergman space (Cf. G. Auchmuty [8]). Let Ω be a sufficiently smooth open subset of R N .
Assumption (B 1 ) on Ω: Ω is a bounded connected open set whose boundary is the union of a finite number of disjoint closed Lipschitz surfaces, each surface having finite surface area.
The Hausdorff (N-1)-dimensional measure and integration with respect to this measure are denoted, respectively, by All functions, in this section, are real valued. The following equivalent inner The spaces H(∆, Ω) and H 0 (∆, Ω) With the inner product it is a real Hilbert space.
Space H(Ω) The class of all H 1 -harmonic functions on Ω is denoted by H(Ω). The space H 0 (∆, Ω) is the orthogonal complement of H(Ω), with respect to the inner product (85). It is equipped with the inner product which generates an equivalent norm to that of (85)( [8], Lemma 3.3).
From now on, assumption (B 1 ) on Ω is replaced by :

The Dirichlet Biharmonic Steklov (DBS) eigenproblem
Here, the DBS eigenproblem is to find where ν is an outward unit normal, defined at σ a.e. point of ∂Ω. Here q is the DBS eigenvalue which appears only in the boundary condition.
By means of a suitable algorithm, the author shows the existence of a maximal countable sequence of ∆-orthonormal Moreover, he obtains the ∆-orthogonal ((87)) decomposition where Let {q k , k ≥ 1} be the sequence of eigenvalues corresponding to B. The following sets are introduced: It is proved that W is an orthonormal basis for L 2 (∂Ω, dσ).

An orthonormal basis for the real harmonic Bergman space
Definition 11. The real harmonic Bergman space L 2 H (Ω) is the space of functions u ∈ L 2 (Ω) that satisfy The h j = ∆b j for j ≥ 1 are harmonic and L 2 -orthonormal. It is proved that B H is an ortonormal basis of L 2 H (Ω), the orthogonal projection P H of L 2 (Ω) onto L 2 H (Ω) has the representation

Complex harmonic Bergman spaces
(Cf. [9] p.172). Let Ω denote an open subset of R n , and p a number satisfying 1 ≤ p < ∞. The harmonic Bergman For closed subspace of L p (Ω), and, therefore, it is a Banach space.

Cas p=2, reproducing kernel of Ω
As a closed subspace of linear functional on the Hilbert space b 2 (Ω). By the Riesz representation theorem, there exists a unique function Therefore, b 2 (Ω) is a reproducing kernel Hilbert space with reproducing kernel R Ω defined on Ω × Ω. is an orthonormal basis of b 2 (Ω), then for all x, y ∈ Ω, and Complex and real Bergman spaces have the same reproducing kernel Indeed, let L 2 H (Ω) be the real harmonic Bergman space (cf. section 4.2) where Ω satisfies conditions (B 2 ). Thanks to properties of harmonic functions, Therefore, the two spaces b 2 (Ω) and L 2 H (Ω) have the same reproducing kernel given by (98).

Homeomorphisms of the circle. Rotation number
There are several definitions of rotation number in the literature.

Simple examples
Coddington and N. Levinson [13]. Consider the differential equation where f is a real continuous function, x) and through every point of the (t, x) plane there passes a unique solution of (105). In R 3 , with rectilinear coordinates (u, v, w), let J be the torus given by where a and b are constants with 0 < b < a. The function f may be considered as a function on J , whose points can be described by Cartesian coordinates (t, x), where two points (t 1 , x 1 ) and (t 2 , x 2 ) are regarded as identical if Through every point P of J , there exists a unique solution path (t, ϕ(t)) of (105).
In the general case, the rotation number µ of system (111) (or (114)) is defined by 5.2 Rotation number of an orientation-preserving homeomorphism f : S 1 → S 1 (cf. L. Wen [62] p.24). Here S 1 denotes the unit circle. Let f : S 1 → S 1 be an orientation-preserving homeomorphism, and F : R → R be a lifting of f . For any t ∈ R, the limit exists, is independent of t, and is denoted by ρ(F ).
In some sens, the rotation number of f measures the average rotation of points under f.
where π : t ∈ R → e 2πit ∈ S 1 is the projection of modulo integer parts.
The homeomorphism f is orientation preserving [resp. reversing] if any lifting of f to R is strictly increasing [resp. decreasing].
Example 3. If f α is a rotation by the angle α, then F α (x) = x + α and F n α = x + nα, which is periodic if α is rational.

Arnold's Theorem
(V.I. Arnold [5], pp.112-115). Set and, for an analytic and bounded function a in this strip, An irrational number µ is of type for any integers p and q = 0.
For a number µ of type (K, σ), we have the "small denominator estimate" (cf [4] ) Then the author states the following theorem: Theorem. There exists (K, µ, σ) > 0, such that, if a is a 2π-periodic analytic function, real on the real axis, with ||a|| ρ < and such that the transformation is the lifting to R of a diffeomorphism of the circle A, with rotation number µ of type (K, σ), then A is analytically equivalent to the rotation R 2πµ by the angle 2πµ (cf. Example 3) i.e. there exists an analytic diffeomorphism H : R → R such that In this case, A is said to be analytically conjugate to the rotation of angle 2πµ, or analytically linearized, and H is called the conjugacy or the linearization.
Definition 13. α is said to verify the Brjuno condition if and only if This is also a condition on the α ∈ R/Z. The set of such α is denoted by B.

The condition and the set H
Let A : (0, 1) → (0, 1) be the map defined by where 1 x denotes the integer part of 1 x , and, for α ∈ R/Q, let (α n ) n≥0 be defined by and (a n ) n≥0 defined by a 0 = α α −1 n−1 = a n + α n , for all n ≥ 1.
Then α = a 0 + 1 and p n q n = a 0 + 1 Let For α ∈ R/Q, the Brjuno function B : R/Q → R + ∪ {∞} is defined by Condition H For α ∈ (0, 1), x ∈ R, let r α the function defined by and set, for α ∈ R/Q, and k > 0, (cf. (130)) Let α ∈ B and define, for n ≥ k ≥ 0, Definition of the set H It is proved that where the inclusions are proper. In [63], J-C Yoccoz proves the following result (Theorem 6 Homogenization and linear elasticity

Introduction
This subsection is an introduction to the theory of homogenization. Complements will be found in the references.
and also where α, β > 0 and αβ ≤ 1, and M s N is the space of real symmetric matrices of order N.
where u is the solution of the homogenized equation in Ω, and u = 0 on ∂Ω, 3-For any f ∈ V , 4-For each sequence f ∈ H such that sup f H < ∞, one can extract a subsequence, still denoted f , such that, for The following spectral problem, for the operators A and A o , is considered with, for > 0 or = o, It is proved that there is a sequence {β k } such that β k → 0 as → 0, 0 < β k < µ k o , and This definition is valid for vector valued functions.
Main result Let A ∈ L ∞ (Ω; M s α,β ) be a sequence of symmetric matrices which H-converges to an homogenized matrix A * . Let ρ be a sequence of positive functions, such that which converges weakly * in L ∞ (Ω) to a limit ρ(x). Let (λ m )m ≥ 1 be the eigenvalues, labeled by increasing order, and (u m ) m≥1 be associated normalized, in L 2 (Ω), eigenvectors of the spectral problem in Ω and u m = 0 on ∂Ω.
Then, for any fixed m ≥ 1, and, up to a subsequence, u m converges weakly in H 1 0 (Ω), to a normalized eigenvector associated to λ m , which are solutions of the homogenized eigenvalue problem and (λ m ) m≥1 is the complete family of eigenvalues of (160), labeled in increasing order., ([2], Theorem 1.3.16).
A similar result is obtained in O. A. Oleinik et al. [44], Chapter III Theorem 2.1.

Periodic homogenization
Cf. [2], Section 1.3.4. Let Y = (0, 1) N be the unit periodic cell which is identified with the unit N -dimensional torus.
Remark 17. Here the conductivity A( x ) of Ω shows that the domain Ω is highly heterogeneous with periodic heterogeneities of lengthscale .
For each unit vector e i , in the canonical basis of R N , let w i be a solution of the following problem in Y : Then, the sequence A (x) = A( x ), H-converges to A * ∈ M α,β defined by its entries The starting point of the proof of this result is to write the solution u of equation (162) on the form of a two-scale asymptotic expansion : where u i (x, y) is a function of x and y, periodic in y with period Y .

The linear elasticity operator
In [44], the usual linear elasticity operator, in a domain Ω ⊂ R n , is defined, with the usual convention of repeated indices, as where u = (u 1 , ..., u n ) is the displacement vector, and A kh (x) are (n × n)-matrices whose elements a hk ij (x) are bounded measure functions such that and for any symmetric matrix with real elements {η ih }, with positive constants κ 1 , κ 2 .

Homogenization in perforated domains
Notations and assumptions (Cf. [44], chapter 1). Let ω be a smooth unbounded domain of R n invariant under the shifts by any z = (z 1 , z 2 , ..., z n ) ∈ Z n , Q = {x|0 < x j < 1, j = 1, 2, ..., n}, G = {x| −1 x ∈ G} satisfying the conditions: ω ∩ Q is a domain with Lipschitz boundary, the sets Q\ω and V δ ∩ ∂Q consist of a finite number of Lipshitz domains separated from each other and from the edges of Q by a positive distance, where V δ is a δ-neigborhhod (δ < 1 4 ) of ∂Q.

Definition 19.
Let Ω be a bounded smooth domain in R n . A domain Ω is of type I if The boundary of Ω can be represented as Example (Cf. [44], p. 119-134). Let Ω be a domain of type (170). Consider the following mixed problem: where the coefficients a ij kl of matrices A ij are sufficiently smooth, satisfont (167)-(168) and are 1-periodic in ξ, with the boundary conditions Here, f ∈ L 2 (Ω ) and Φ ∈ H 1 (Ω ). The homogenized operator corresponding to L has the form where the coefficients matrices A ij (i, j = 1, ..., n) are given by the formula and matrices N j (ξ) are 1-periodic in ξ with Q∩ω N j (ξ)dξ = 0 and are solutions of the following boundary problem: Let u o (x) be a weak solution of the problem where f 0 ∈ H 1 (Ω), and Φ o ∈ H 3 (Ω). If f is the restriction to Ω of f o , and Φ is the restriction to Γ of Φ o , then (178) is the homogenized system corresponding to system (172) and (173) and ||u − u o || L 2 (Ω ) → 0 as tends to 0.
Here, is a small, positive, parameter, Γ consists of N sets whose diameter is less than or equal to and the distance between then is greater than or equal to 2 , and γ = ∂Ω\(γ 1 ∪Γ ). The boundary value problem for the elasticity system (E): and is considered. Here, the elements a ij kl of the d × d-matrices A ij are bounded measurable functions, with where κ 1 , κ 2 are constants > 0, ν = (ν 1 , ν 2 , ..; ν d ) is an outward normal vector to the boundary ∂Ω, g(x) ∈ (L 2 (∂Ω)) d .
The following estimate is established, for sufficiently small, and where C is a constant independent of , ( [12], Theorem 5).The proof is based on an adaptation of the general abstract result (subsection 6.1.2), thanks to the above results R1 and R2. Definition 20. We recall that , for a three dimensional vector distribution, v ∈ D (Ω) 3 , equipped with the graph norm equipped with the norm of L 2 (∂Ω) 3 , where ν is the unit outward normal to Ω.

Maxwell's equations
Cf. [41]. The electric and magnetic field vectors are respectively denoted by E and H. The. vector functions electric displacement and magnetic induction are denoted, respectively, by D and B. They are functions of the position x ∈ R 3 and time t, and are related by Maxwell's equations: where ρ is a scalar charge density function and J is the current vector density function.

Time-harmonic Maxwell's system
If the radiation has a frequency ω > 0, in time, the electromagnetic field is said to be time-harmonic, provided And, also Substituting these relations into Maxwell's equations (subsection 7.2), leads to the time harmonic Maxwell's system:

The cavity problem.
Cf. [41]. In this section, the domain Ω ⊂ R 3 is bounded and simply connected. Its boundary ∂Ω consists of at most two connected components Σ and Γ. Define the space: where ν is the unit outward normal to Ω and u T = (ν × u) × u on ∂Ω. We also need the following definitions: Definition 24. The electric permittivity, the magnetic permeability and the conductivity are denoted by o , µ o , and σ, respectively.Then the relative permittivity and relative permeability are denoted by r and µ r .
and let F be a given current density. The elimination of H in the system (204)-(207) leads to the problem: find the time electric field E corresponding to F such that ( [41], Section 1.4) with the boundary conditions where g is a given tangential vector field on Σ, κ = ω √ o µ o and the impedance λ is a positive function on the surface of the material.

Variational formulation
Let (u, v) and u, v be the inner products in L 2 (Ω) 3 and L 2 (Σ) 3 , respectively and set for all u, v ∈ X. Then the variational cavity problem is to find E ∈ X such that

The Bean model in superconductivity
Cf. L. Prigozhin [46] and [47]. The superconductor occupies a three dimensional domain Ω ⊂ R 3 . Let ω denote the exterior space. In Maxwell's equations, the displacement current,very small, can be omitted.These equations read: where J is the current density and B = µ o H where µ o is the permeability of the vacuum. The density of the external current J e (x, t) which satisfies the condition divJ e = 0 is known: Only the case where the vectors of current density J and electric field E are collinear is considered. This is the case for two dimensional problems and three dimensional problems with axial symmetry. Jointly with the above Maxwell's equations, the Bean model presented in [47] consists of the following equations (Ohm's law):: where ρ is an unknown function and J is the effective resistivity, where the critical value J c is constant in the Bean model, moreover with the initial condition: and the boundary conditions: the tangential component H τ of H is assumed to be continuous i.e.
where [ . ] denotes the jump across the boundary, and The following variational relation is valid for all ϕ ∈ V : The above Bean model is equivalent to the problem: find h ∈ K(h) such that Existence and uniqueness of solution to this problem is proved ( [47], p.193).

The variational formulation of the Bean model (229) is used to solve the problem numerically ([47] Section 4), and
examples are presented.
The degenerate evolution system studied by H-M Yin et al. [64] is: with the conditions and where H is the magnetic field, p ≥ 2 is fixed and ν is the outward normal to ∂Ω. In type-II superconductors, the electromagnetic field concerns an alloy-type conductive medium where the displacement current is small in comparaison with eddy currents J = σE (Ohm's law). Let E be the electric field and J be the current density. In Bean's model, there exists a critical current J c such that |J| ≤ J c and Under suitable conditions on the data F and H o , the authors prove the existence of a weak solution to the system (230)-(232), whose limit H (∞) , as p → ∞, satisfies in the sense of distributions, where a(x, t) is a nonnegative, bounded and measurable function. Moreover supp a(x) ⊂ N t for a.e. every t ∈ (0, T ] where N t = {(x, t) ∈ Q T }. Therefore, H (∞) is a solution to the Bean's model. 8 Pseudospectra and non hermitian, one dimensional, quantum mechanics As usual, operator means, always, linear operator.
Pseudospectra are related to closed operators.
Notation In a Banach space X, the norm is denoted by || . ||, or, if a confusion is possible, by || . || X .
Given A ∈ C(X), the resolvent set ρ(A) is the set of z ∈ C for which the inverse (z − A) −1 exists and is in B(X).
The spectrum σ(A) of A ∈ C(X) is the complement of ρ(A) in C, i.e. σ(A) = C/ρ(A), with the convention that, for
Remark 21. Let H, K be two Hilbert spaces. The product H × K with the scalar product is a Hilbert space. On the other hand, the topological direct sum H ⊕ K is the Hilbert space {h ⊕ k = (h, k) ∈ H × K} with the scalar product and the map (h, k) ∈ H × K → h ⊕ k ∈ H ⊕ K is an isomorphism (cf. [19], p.112) Metrics for closed linear subspaces of a Hilbert space (Cf. Gohberg-Krein [28] and Cordes-Labrousse [14]).
Let H be a Hilbert space and S, T two closed subspaces of H.
defines a metric on the totality of closed linear subsets of H. Equivalent metrics are defined by where P S , P T are the orthogonal projections on S and T, respectively, and by

Definition of the -pseudospectrum σ (A)
Let A ∈ C(X) and > 0 be arbitrary.
In [56], p. 31, the authors give the following equivalent definitions.
Definition 26. The -pseudospectrum of A is the set of z ∈ C satisfying any of the conditions z ∈ σ(A), or ||(z − A)u|| < for some u ∈ D(A) with ||u|| = 1.
If z and u satisfy the last equation they are called -pseudoeigenvalue and -pseudoeigenvector, respectively, for the operator A. The pseudospectra of A is the family Importance of the pseudospectra. In accordance with formula (249), the notion of pseudospectra provides information about the instability mentionned in remark 26. The size of the pseudospectral regions provides a clear indication of the instability of typical Hamiltonians in quantum mechanic.

Trivial pseudospectra
A closed operator T is said to have a trivial pseudospectra if, for some positive constant κ, (cf [32], section 2.3).

Examples
Example 4. The virtual eigenvalues of J. Arazy and L. Zelenko [6] are in the pseudospectra of (−∆) , in Indeed, they consider the operator (see also [33], p.132): where V (x) ≥ 0 is assumed to be continuous and to satisfy lim as soon as Therefore, for > 0, σ(H γ ) is the -pseudospectra of (−∆) l when the inequality (254) is satisfied.
We recall that a function F (x) is absolutely continuous in an interval (a, b) if it is the indefinite integral of a function f ∈ L 1 loc (cf. [54] Section 11.7). The spectrum σ(A) is empty, since, for z ∈ C, u(d) = e zd = 0. Nevertheless, the pseudospectra of A are "enormous". The resolvent (z − A) −1 exists as a bounded operator, and, for any z ∈ C, The equation (257) means that (z − A) −1 v(x) is the restriction to (0, d) of the convolution product v * g where v and g are regarded as functions in L 2 (−∞, +∞) with By means of the Fourier transform in L 2 (R), (257) leads to where || . || denotes the norm in L 2 (−∞, +∞). Then for Rez > 0, and for Rez < 0. These results imply ( [56], Theorem 5.1) that the pseudospctra of A are half-planes of the form Example 6. The ghost solution of D. Domokos and P. Holmes (Cf. [21]). In [56], p. 98-99, this ghost solution is presented in the following way. The author considers the linear differential equation acting on sufficiently smooth functions in L 2 (−L, +L) and associated to the boundary conditions The function satisfies the boundary conditions (265) and the equation i.e. (264) for all x, up to an error no greater than Le − L 2 2 . Therefore and 0 belongs to the -pseudospectrum of A for a value of that decreases exponentially as L → ∞.
Example 7. The non-self-adjoint (NSA) harmonic oscillator The harmonic oscillator The harmonic oscillator is the closure, in L 2 (R), of the operator H a defined by for f in the L. Schwartz's space S(R), with a > 0. The operator H a is essentialy self-adjoint on S (i.e. its closure is self-adjoint in L 2 (R)), and the resolvent operators are compact. Moreover, the spectrum of H a is {(2n + 1)a 1/2 : n = 0, 1, ...}, each eigenvalue λ is of multiplicity 1, and the corresponding eigenfunctions are where H n is the hermite polynomial of degree n. After normalization, the eigenfunctions provide a complete orthonormal set in L 2 (R).

Reminder: Definitions
Cf. [17]. Let X be a Hilbert space with inner product (f, g) → f, g . A sequence {x j }, in X, is a normalized basis if it is a basis with ||x j || = 1 for each j.
An unconditional basis is a basis with the property that every permutation of the sequence is also a basis.
A sequence {f n } ∞ n=1 , in X, is said to be an Abel-Lidskii basis in X, if it is a part of a biorthogonal pair {f n }, {φ n } such that, for all f ∈ X, one has f = lim He also proves that the eigenfunctions of H a do not form an unconditional basis in L 2 (R).
Example 8. In [32], R. Henry and D.Krejčiřík consider, in L 2 (R), the operator with domain It is closed and densely defined, but, neither self-adjoint nor normal.
where H * denotes the adjoint of H and T and P are, respectively, defined by T ψ =ψ, (Pψ)(x) = ψ(−x). Its numerical range Num(H) (i.e. the set of all complex numbers (ψ, Hψ) with ψ ∈ D(H) and ||ψ|| = 1), is Moreover, They show that H cannot have trivial pseudospectra. For that, they set z = τ + iδ and they construct a function f 0 such that as τ → ∞, where φ(τ, δ) is a suitable function. For z real, positive (δ = 0) and dist(z, σ(H)) = 1. The equation (280) shows that, for any positive constant C, there exists a z 0 ∈ C/(σ(H), real, positive, such that ever, H has a non-trivial pseudospectra. Indeed, thanks to (280), given > 0, there exist z ∈ C/σ(H) such that Remark 24. In the references of this section, the authors present nice figures of the pseudospectra.

Introduction to quantum mechanics
In one dimensional motion of a single particule restricted to motion along a line, M. Schechter [50] postulates: there is a function ψ(x, t) of position x ∈ R at the time t such that the probability that the particule is in an interval I, at the time t, is given by ψ(x, t) is called the state function, and satisfies ∞ ∞ |ψ(x, t)| 2 dx = 1, forall t. Set (with the notations of [50]) p = m dx dt where m is the mass of the particule, and define the operator L by where is the Planck's constant. (No confusion is possible between the square L 2 of the operator L ans the space L 2 (R)). The total energy of the particule is the sum of the kinetic energy T = 1 2 mv 2 = p 2 2m and the potential energy V : x ∈ R → R. The corresponding energy-operator or hamiltonian is the operator Remark 25. If H is the generator of a C o semi-group e −tH on L 2 (R), ψ(x, t) = e −tH ψ(x, 0). Therefore, it will be convenient to estimate ||e −tH ||.
Definition 27. Let w be a measurable quantity which can take on the values w 1 , ..., w N and suppose the probability that w takes the value w k is P k , k = 1, ..., N . The quantitȳ is called the average value or he mathematical expectation of w.
In [50], this definition is justified by a Feller's theorem namely: If a sequence of identical experiments is performed and the values S 1 , ..., S n , ... are observed (The numbers S n are among the values w 1 , ..., w N ), then the average value converges tow, in the sense of probability, as n → ∞ ( [22]).
The kinetic energy of the particule is E = p 2 2m . The average value of E is Any quantity, that can be measured, is called an observable. In formula (287),Ē is an observable.
Remark 26. Importance of the spectrum :An observable can assume values only in the spectrum of its corresponding operator. Moreover, even in the cases in which the eigenfunctions can be determined explicitely, they often do not form a basis. This is closely related to a high degree of instability of the eigenfunctions under small perturbations of the operator (cf [16], Abstract).

Examples
This paragraph is devoted to examples of pseudospectra related to models of non-hermitian quantum mechanics (cf. Krejčiřík et al.) [38] Example 9. The rotated harmonic oscillator that is the quantum hamiltonian of the harmonic oscillator See operator (269), with a = 1.

Remark 27.
It is proved, in [38], that the shifted harmonic oscillator has the same spectrum as the above H, but large pseudospectra in parabolic regions of the complex plane ( .
Remark 28. In [38], the authors point out the differences between the above operator (294) and the convectiondiffusion operator: Example 13. (Cf. [48]). Consider the operator Main results 1) For any H satisfying the above assumption, we have: 2) If H is normal we have ||exp(tH)|| = e tα(H ), ∀t > 0 If H is not normal But, when H has a basis of eigenvectors, the last inequality, in (308), may be affined in terms of the condition number of this basis.

Symmetrizability
Definition 32. An operator L is symmetrizable if it is similar, by a diagonal similarity transformation, to a self-adjoint operator with the same real eigenvalues.
For example, in (cf [48]), the authors consider the convection-diffusion operator

The gap topology
Let T ∈ C sa , and P denotes the orthogonal projection onto the graph of T in H × H. The gap metric is δ(T 1 , T 2 ) = ||P 1 − P 2 || (where ||.|| denotes the norm in the space B of bounded operators acting in H)

The "Cayley transform" metric
For a densely defined operator T in H, the Cayley transform κ is defined by Let U be the subspace of B of unitary operators H → H. It is proved that moreover the metricδ, on CF sa , defined byδ is equivalent to the metric δ.

Homotopy
(Cf. J. Dieudonné [19]). Let L 1 and L 2 two paths defined in the same intervalle I = [a, b], and A an open set in C, such that L 1 (I) ⊂ A and L 2 (I) ⊂ A. An homotopy of L 1 to L 2 , in A, is a continuous map ϕ : (t, ξ) ∈ I × [α, β](α < β ∈ R) → A such that ϕ(t, α) = L 1 (t) and ϕ(t, β) = L 2 (t) in I. Then the two paths L 1 and L 2 are said to be homotopes in A Derivative of second order (Cf. [19]. Let f be a continuously differentiable function in an open set A of a Banach space E to a Banach space F. Then Df is a continuous map from A to L(E, F ). If Df is differentiable at the point a ∈ A, f is said to be twice continuously diffferentiable at a ∈ A and the derivative of Df at the point a is called the second derivative of f at a, and is denoted by f (a) or D 2 f (a).
Partial derivatives Let E 1 , E 2 , F be Banach spaces, E = E 1 × E 2 , A be an open set of E, and f a differentialbe map from A to F and a = (a 1 , a 2 ) ∈ A Definition 34. The map f is said differentiable, at the point (a 1 , a 2 ), with respect to the first [resp. second variable] if the partial map x 1 → f (x 1 , a 2 ) [resp. x 2 → f (a 1 , x 2 )] is differentiable in a 1 [resp. a 2 ]. These derivatives are called partial derivative with respect to the first variable [resp. the second variable] at the point (a 1 , a 2 ) and are denoted by D x1 f (a 1 , a 2 )(∈ L(E 1 , F )) [resp. D x2 f (a 1 , a 2 )(∈ L(E 2 , F ))]. Moreover the gradient of f at the point (a 1 , a 2 ) is defined by ∇f (a 1 , a 2 ) = D x1 f (a 1 , a 2 ) × D x2 f (a 1 , a 2 ) ∈ L(E 1 , F ) × L(E 2 , F ) Remark 30. The above definition can be extented, in the same way, when E is a product of more than two spaces. is a point λ * ∈ I such that every neighborhood of (λ * , 0) contains nontrivial solutions of this equation. Let, for λ ∈ I, L λ = D x (f (λ, x))(λ, 0) ∈ L(X, X) be the derivative of f (λ, x) with respect to x, at the trivial solution. By the implicit function theorem, bifurcation can occur only at points where L λ is singular i.e. is noninvertible.
The following result is presented in P. M. Fitzpatrick et al. [24] Theorem A. Let I = [a, b] be an interval of real numbers, X be a real separable Hilbert space and U be a neighborhood of I × {0} in R × X on which the C 2 function ψ : (λ, x) ∈ U → ψ(λ, x) ∈ R has the property that, for each λ ∈ I, 0 is a critical point of ψ λ ≡ ψ(λ, . ). It is assumed that the Hessian L λ of ψ λ , at 0, is Fredholm, and that L a and L b are nonsingular. Then, if the spectral flow of {L λ } on the interval I is nonzero, every neigborhood of I × {0} contains points of the form (λ, x) where x = 0 is a critical point of ψ λ .

Example 14. ([24])
Let I = [a, b] and H = (λ, t, u) ∈ I × R × R 2n → H(λ, t, u) ∈ R be a twice continuously differentiable function, 2π-periodic in t with H(λ, t, 0) ≡ 0. The following Hamiltonian system for the differentiable function u : R → R 2n , is considered: where J = 0 −Id n Id n 0 (328) is the symplectic 2n × 2n matrix. The authors make assumptions under which they can apply their previous results [23] and show that bifurcation of 2π-periodic orbits from the branch of equilibrium arises. Here is Fredholm. The spectral flow sf (L, I) of the path L = {L λ } λ∈I is nonzero.
Let Ω be a bounded domain in R N , N ∈ N, with a smooth boundary ∂Ω, I = [0, 1]. Let a, b, c : I ×Ω → R and G : I ×Ω × R 2 → R be C 2 -functions. Systems of elliptic partial differential equations of the form are considered. Here G u (λ, x, 0, 0) and G v (λ, x, 0, 0) are assumed to be 0 for all (λ, x) ∈ I × Ω. Aditionnal assumptions are made on G and its derivatives such that the following results are justified. Let D 2 G(λ, x, u, v) denote the Hessian matrix of G(λ, x, . , . ) : R 2 → R, at the point (u, v) ∈ R 2 with D 2 G(λ, x, 0, 0) = 0. Let E be the Hilbert space H 1 0 (Ω, R) × H 1 0 (Ω, R) with the corresponding scalar product . , . E , and for z = (u, v) ∈ E, define the map which is C 2 . Moreover, f λ (0) = 0 for all λ ∈ I. For z = (u, v) ∈ E, the mapz = (ũ,ṽ) ∈ E to is a continuous linear form on E. From the Riesz representation theorem, there exists L λ (z) ∈ E such that L λ z,z E = D 2 0 (f λ (z,z)) z,z ∈ E, and L λ ∈ Φ S (E). Therefore, the path L = {L λ |λ ∈ I} is a path of bounded linear self-adjoint, Fredholm operators and the spectral flow sf (L, I) is well defined. It is proved that if the linearized systems

Conflict of Interests
Th author declares that there is no conflict of interests regarding the publication of this paper.