- Self-adjoint operator
In

mathematics , on a finite-dimensionalinner product space , a**self-adjoint operator**is one that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its ownconjugate transpose . By the finite-dimensionalspectral theorem such operators have anorthonormal basis in which the operator can be represented as adiagonal matrix with entries in thereal number s. In this article, we considergeneralization s of thisconcept to operators onHilbert space s of arbitrary dimension.Self-adjoint operators are used in

functional analysis andquantum mechanics . In quantum mechanics their importance lies in the fact that in the Dirac-von Neumann formulation of quantum mechanics, physicalobservable s such asposition ,momentum ,angular momentum and spin are represented by self-adjoint operators on a Hilbert space. Of particular significance is the Hamiltonian:$H\; psi\; =\; -\; frac\{hbar^2\}\{2\; m\}\; abla^2\; psi\; +\; V\; psi$

which as an observable corresponds to the total energy of a particle of mass "m" in a real potential field "V". Differential operators are an important class of unbounded operators.

The structure of self-adjoint operators on infinite dimensional Hilbert spaces essentially resembles the finite dimensional case, that is to say, operators are self adjoint if and only if they are unitarily equivalent to real-valued multiplication operators. With suitable modifications, this result can be extended to possibly unbounded operators on infinite dimensional spaces. Since an everywhere defined self adjoint operator is necessarily bounded, one needs be more attentive to the domain issue in the unbounded case. This is explained below in more detail

**ymmetric operators**A

partially-defined linear operator "A" on a Hilbert space "H" is called**symmetric**if and only if :$langle\; Ax\; mid\; y\; angle\; =\; lang\; x\; mid\; Ay\; ang$for all elements "x" and "y" in the domain of "A". More generally, a partially-defined linear operator "A" from atopological vector space "E" into itscontinuous dual space "E"^{∗}is said to be**symmetric**if:$langle\; Ax\; mid\; y\; angle\; =\; lang\; x\; mid\; Ay\; ang$for all elements "x" and "y" in the domain of "A". This usage is fairly standard in the functional analysis literature.A symmetric "everywhere defined" operator is

self-adjoint .By theHellinger-Toeplitz theorem , a symmetric "everywhere defined" operator is bounded.Bounded symmetric operators are also called

**Hermitian**.The previous definition agrees with the one for matrices given in the introduction to this article, if we take as "H" the Hilbert space

**C**^{"n"}with the standard dot product and interpret a square matrix as a linear operator on this Hilbert space. It is however much more general as there are important infinite-dimensional Hilbert spaces.The spectrum of any bounded symmetric operator is real; in particular all its eigenvalues are real, although a symmetric operator may not have any eigenvalues.

A general version of the

spectral theorem which also applies to bounded symmetric operators is stated below. If the set of eigenvalues for a symmetric operator is non empty, and the eigenvalues are nondegenerate, then it follows from the definition that eigenvectors corresponding to distinct eigenvalues are orthogonal.Contrary to what is sometimes claimed in introductory physics textbooks, it is possible for symmetric operators to have no eigenvalues at all (although the spectrum of any self adjoint operator is nonempty). The example below illustrates the special case when an (unbounded) symmetric operator does have a set of eigenvectors which constitute a Hilbert space basis. The operator "A" below can be seen to have a compact inverse, meaning that the corresponding differential equation "A" "f" = "g" is solved by some integral, therefore compact, operator "G". The compact symmetric operator "G" then has a countable family of eigenvectors which are complete in $L^2$. The same can then be said for "A".**Example**. Consider the complex Hilbert space L^{2}[0,1] and thedifferential operator : $A\; =\; -\; frac\{d^2\}\{dx^2\}$

defined on the subspace consisting of all complex-valued infinitely

differentiable functions "f" on [0,1] with the boundary conditions::$f(0)\; =\; f(1)\; =\; 0\; quad$

Then

integration by parts shows that "A" is symmetric. Its eigenfunctions are the sinusoids:$f\_n(x)\; =\; sin(n\; pi\; x)\; quad\; n=\; 1,2,\; ldots$

with the real eigenvalues "n"

^{2}π^{2}; the well-known orthogonality of the sine functions follows as a consequence of the property of being symmetric.We consider generalizations of this operator below.

**Self-adjoint operators**Given a densely defined linear operator "A" on "H", its adjoint "A"* is defined as follows:

* The domain of "A"* consists of vectors "x" in "H" such that::$y\; mapsto\; langle\; x\; mid\; A\; y\; angle$

: (which is a densely defined "linear" map) is a continuous linear functional. By continuity and density of the domain of "A", it extends to a unique continuous linear functional on all of "H".

* By the

Riesz representation theorem for linear functionals, if "x" is in the domain of "A"*, there is a unique vector "z" in "H" such that::$langle\; x\; mid\; A\; y\; angle\; =\; langle\; z\; mid\; y\; angle\; quad\; forall\; y\; in\; operatorname\{dom\}\; A$:This vector "z" is defined to be "A"* "x". It can be shown that the dependence of "z" on "x" is linear.Notice that it is the denseness of the domain of the operator, along with the uniqueness part of Riesz representation, that ensures the adjoint operator is well defined.

A result of Hellinger-Toeplitz type says that an operator having a bounded adjoint is bounded. Therefore the adjoint of a unbounded operator is necessarily unbounded.

The condition for a linear operator on a Hilbert space to be "self-adjoint" is stronger than to be "symmetric".

For any densely defined operator "A" on Hilbert space one can define its adjoint operator "A"*.For a symmetric operator "A", the domain of the operator "A"* contains the domain of the operator "A", and the restriction of the operator "A"* on the domain of "A" coincides with the operator "A", i.e. $A\; subseteq\; A^*$, in other words "A"* is extension of "A". For a self-adjoint operator "A" the domain of "A"* is the same as the domain of "A", and "A"="A"*. See also

Extensions of symmetric operators .**Geometric interpretation**There is a useful

geometric al way of looking at the adjoint of an operator "A" on "H" as follows: we consider the graph G("A") of "A" defined by:$operatorname\{G\}(A)\; =\; \{(xi,\; A\; xi):\; xi\; in\; operatorname\{dom\}(A)\}\; subseteq\; H\; oplus\; H\; .$

**Theorem**. Let J be the symplectic mapping: $H\; oplus\; H\; ightarrow\; H\; oplus\; H$

given by

:$operatorname\{J\}:\; (xi,\; eta)\; mapsto\; (-eta,\; xi).$

Then the graph of "A"* is the

orthogonal complement of JG("A")::$operatorname\{G\}(A^*)\; =\; (operatorname\{J\}operatorname\{G\}(A))^perp=\; \{(x,y)\; in\; H\; oplus\; H\; :\; langle\; (x,y)|(-Axi,xi)\; angle\; =\; 0\; ;;forall\; xi\; in\; operatorname\{dom\}(A)\}$

A densely defined operator "A" is symmetric

if and only if :$A\; subseteq\; A^*.$where the subset notation $A\; subseteq\; A^*$ is understood to mean $G(A)\; subseteq\; G(A^*).$ An operator "A" is

**self-adjoint**if and only if $A=A^*$; that is, if and only if $G(A)=G(A^*).$**Example**. Consider the complex Hilbert space L^{2}(**R**), and the operator which multiplies a given function by "x"::$A\; f(x)\; =\; xf(x)$

The domain of "A" is the space of all L

^{2}functions for which the right-hand-side is square-integrable. "A" is a symmetric operator without any eigenvalues and eigenfunctions. In fact it turns out that the operator is self-adjoint, as follows from the theory outlined below.As we will see later, self-adjoint operators have very important spectral properties; they are in fact multiplication operators on general measure spaces.

**Spectral theorem**Partially defined operators "A", "B" on Hilbert spaces "H", "K" are

**unitarily equivalent**if and only if there is aunitary operator "U":"H" → "K" such that* "U" maps dom "A"

bijective ly onto dom "B",* $B\; U\; xi\; =\; U\; A\; xi\; ,quad\; xi\; in\; operatorname\{dom\}A.$

A

multiplication operator is defined as follows: Let $(X,\; Sigma,\; mu)$ be a countably additivemeasure space and "f" a real-valued measurable function on "X". An operator "T" of the form:$[T\; psi]\; (x)\; =\; f(x)\; psi(x)\; quad$

whose domain is the space of ψ for which the right-hand side above is in "L"

^{2}is called a multiplication operator.**Theorem**. Any multiplication operator is a (densely defined) self-adjoint operator. Any self-adjoint operator is unitarily equivalent to a multiplication operator.This version of the spectral theorem for self-adjoint operators can be proved by reduction to the spectral theorem for unitary operators. This reduction uses the "Cayley transform" for self-adjoint operators which is defined in the next section. We might note that if T is multiplication by f, then the spectrum of T is just the

essential range of f.**Borel functional calculus**Given the representation of "T" as a multiplication operator, it is easy to characterize the

: If "h" is a bounded real-valued Borel function onBorel functional calculus **R**, then "h"("T") is the operator of multiplication by the composition $h\; circ\; f$. In order for this to be well-defined, we must show that it is the unique operation on bounded real-valued Borel functions satisfying a number of conditions.**Resolution of the identity**It has been customary to introduce the following notation

:$operatorname\{E\}\_T(lambda)\; =\; mathbf\{1\}\_\{(-infty,\; lambda]\; \}\; (T)$

where $mathbf\{1\}\_\{(-infty,\; lambda]\; \}$ denotes the function which is identically 1 on the interval $(-infty,\; lambda]$. The family of projection operators E

_{"T"}(λ) is called**resolution of the identity**for "T". Moreover, the followingStieltjes integral representation for "T" can be proved::$T\; =\; int\_\{-infty\}^\{+infty\}\; lambda\; d\; operatorname\{E\}\_T(lambda).$

The definition of the operator integral above can be reduced to that that of a scalar valued Stieltjes integral using the weak operator topology. In more modern treatments however, this representation is usually avoided, since most technical problems can be dealt with by the functional calculus.

**Formulation in the physics literature**In physics, particularly in quantum mechanics, the spectral theorem is expressed in a way which combines the spectral theorem as stated above and the

Borel functional calculus usingDirac notation as follows:If "H" is Hermitian (the name for self-adjoint in the physics literature) and "f" is a

Borel function ,:$f(H)=\; int\; dE\; midPsi\_\{E\}\; angle\; f(E)\; langle\; Psi\_\{E\}\; mid$

with

:$H\; mid\; Psi\_\{E\}\; angle\; =\; E\; midPsi\_\{E\}\; angle$

where the integral runs over the whole spectrum of "H". The notation suggests that "H" is diagonalized by the eigenvalues Ψ

_{"E"}. Such a notation is purelyformal . One can see the similarity between Dirac's notation and the previous section. The resolution of the identity(sometimes called projection valued measures) formally resembles the rank-1 projections $|\; Psi\_\{E\}\; angle\; langle\; Psi\_\{E\}\; |$.In the Dirac notation, (projective) measurements are described viaeigenvalues andeigenstates , both purely formal objects. As one would expect, this does not survive passage to the resolution of the identity. In the latter formulation, measurements are described using thespectral measure of $|\; Psi\; angle$, if the system is prepared in $|\; Psi\; angle$ prior to the measurement. Alternatively, if one would like to preserve the notion of eigenstates and make it rigorous, rather than merely formal, one can replace the state space by a suitablerigged Hilbert space .If "f"=1, the theorem is referred to as resolution of unity:

:$I\; =\; int\; dE\; mid\; Psi\_\{E\}\; angle\; langle\; Psi\_\{E\}\; mid$

In the case $H\_\{mathit\; eff\}=H-iGamma$ is the sum of an Hermitian "H" and a skew-Hermitian (see

skew-Hermitian matrix ) operator $-iGamma$, one defines the biorthogonal basis set:$H^*\_\{mathit\; eff\}\; mid\; Psi\_\{E\}^*\; angle\; =\; E^*\; mid\; Psi\_\{E\}^*\; angle$

and write the spectral theorem as:

:$f(H\_\{mathit\; eff\})=\; int\; dE\; mid\; Psi\_\{E\}\; angle\; f(E)\; langle\; Psi\_\{E\}^*\; mid$

(See

Feshbach–Fano partitioning method for the context where such operators appear inscattering theory ).**Extensions of symmetric operators**The following question arises in several contexts: if an operator "A" on the Hilbert space "H" is symmetric, when does it have self-adjoint extensions? One answer is provided by the

of a self-adjoint operator and the deficiency indices. (We should note here that it is often of technical convenience to deal withCayley transform closed operator s. In the symmetric case, the closedness requirement poses no obstacles, since it is known that all symmetric operators are closable.)**Theorem**. Suppose "A" is a symmetric operator. Then there is aunique partially defined linear operator: $operatorname\{W\}(A)\; colon\; operatorname\{ran\}(A+i)\; ightarrow\; operatorname\{ran\}(A-i)$

such that

: $operatorname\{W\}(A)(Ax\; +\; ix)\; =\; Ax\; -\; ix\; quad\; x\; in\; operatorname\{dom\}(A).$

Here, "ran" and "dom" denote the range and the domain, respectively. W("A") is isometric on its domain. Moreover, the range of 1 − W("A") is dense in "H".

Conversely, given any partially defined operator "U" which is isometric on its domain (which is notnecessarily closed) and such that 1 − "U" is dense, there is a (unique) operator S("U")

: $operatorname\{S\}(U)\; colon\; operatorname\{ran\}(1\; -\; U)\; ightarrow\; operatorname\{ran\}(1+U)$

such that

: $operatorname\{S\}(U)(x\; -\; Ux)=\; i(x\; +\; U\; x)\; quad\; x\; in\; operatorname\{dom\}(U).$

The operator S("U") is densely defined and symmetric.

The mappings W and S are inverses of each other.

The mapping W is called the

**Cayley transform**. It associates a partially defined isometry to any symmetric densely-defined operator. Note that the mappings W and S aremonotone : This means that if "B" is a symmetric operator that extends the densely defined symmetric operator "A", then W("B") extends W("A"), and similarly for S.**Theorem**. A necessary and sufficient condition for "A" to be self-adjoint is that its Cayley transform W("A") be unitary.This immediately gives us a necessary and sufficient condition for "A" to have a self-adjoint extension, as follows:

**Theorem**. A necessary and sufficient condition for "A" to have a self adjoint extension is that W("A") have a unitary extension.A partially defined isometric operator "V" on a Hilbert space "H" has a unique isometric extension to the norm closure of dom("V"). A partially defined isometric operator with closed domain is called a

partial isometry .Given a partial isometry "V", the

**deficiency indices**of "V" are defined as the dimension of theorthogonal complement s of the domain and range::$n\_+(V)\; =\; operatorname\{dim\}\; operatorname\{dom\}(V)^\{perp\}$

:$n\_-(V)\; =\; operatorname\{dim\}\; operatorname\{ran\}(V)^\{perp\}$

**Theorem**. A partial isometry "V" has a unitary extension if and only if the deficiency indices are identical. Moreover, "V" has a "unique" unitary extension if and only if the both deficiency indices are zero.We see that there is a bijection between symmetric extensions of an operator and isometric extensions of its Cayley transform. An operator which has a unique self-adjoint extension is said to be

**essentially self-adjoint**. Such operators have a well-definedBorel functional calculus . Symmetric operators which are not essentially self-adjoint may still have acanonical self-adjoint extension. Such is the case for "non-negative" symmetric operators (or more generally, operators which are bounded below). These operators always have a canonically definedFriedrichs extension and for these operators we can define a canonical functional calculus. Many operators that occur in analysis are bounded below (such as the negative of theLaplacian operator), so the issue of essential adjointness for these operators is less critical.**elf adjoint extensions in quantum mechanics**In quantum mechanics, observables correspond to self-adjoint operators. By

Stone's theorem , self-adjoint operators are precisely the infinitesimal generators of unitary groups oftime evolution operators. However, many physical problems are formulated as a time-evolution equation involving differential operators for which the Hamiltonian is only symmetric. In such cases, either the Hamiltonian is essentially self-adjoint, in which case the physical problem has unique solutions or one attempts to find self-adjoint extensions of the Hamiltonian corresponding to different types of boundary conditions or conditions at infinity.**Von Neumann's formulas**Suppose "A" is symmetric; any symmetric extension of "A" is a restriction of "A"*; Indeed if "B" is symmetric

:$A\; subseteq\; B\; implies\; B\; subseteq\; B^*\; subseteq\; A^*$

**Theorem**. Suppose "A" is a densely defined symmetric operator. Let:$N\_+\; =\; operatorname\{ran\}(A+i)^\{perp\}$

:$N\_-\; =\; operatorname\{ran\}(A-i)^\{perp\}$

Then

:$N\_+\; =\; operatorname\{ker\}(A^*-i)$

:$N\_-\; =\; operatorname\{ker\}(A^*+i)$

and

:$operatorname\{dom\}(A^*)\; =\; overline\{operatorname\{dom\}(A)\}\; oplus\; N\_+\; oplus\; N\_-$

where the decomposition is orthogonal relative to the graph inner product of dom("A"*):

:$langle\; xi\; |\; eta\; angle\_mathrm\{graph\}\; =\; langle\; xi\; |\; eta\; angle\; +\; langle\; A^*xi\; |\; A^*\; eta\; angle.$

These are referred to as von Neumann's formulas in the Akhiezer and Glazman reference.

**Examples**We first consider the differential operator

: $D:\; phi\; mapsto\; frac\{1\}\{i\}\; phi\text{'}$

defined on the space of complex-valued C

^{∞}functions on [0,1] vanishing near 0 and 1. "D" is a symmetric operator as can be shown byintegration by parts . The spaces "N"_{+}, "N"_{−}are given respectively by the distributional solutions to the equation:$u\text{'}\; =\; i\; u\; quad$

:$u\text{'}\; =\; -\; i\; u\; quad$

which are in "L"

^{2}[0,1] . One can show that each one of these solution spaces is 1-dimensional, generated by the functions"x" → "e"^{"ix"}and "x" → "e"^{−"ix"}respectively. This shows that "D" is not essentially self adjoint, but does have self-adjoint extensions. These self-adjoint extensions are parametrized by the space of unitary mappings: $N\_\{+\}\; ightarrow\; N\_\{-\}\; ,!$

which in this case happens to be the unit circle

**T**.This simple example illustrates a general fact about self-adjoint extensions of symmetric differential operators "P" on an open set "M". They are determined by the unitary maps between the eigenvalue spaces

: $N\_pm\; =\; \{u\; in\; L^2(M):\; P\_\{operatorname\{dist\; u\; =\; pm\; i\; u\}$

where "P"

_{dist}is the distributional extension of "P".We next give the example of differential operators with

constant coefficient s. Let:$P(vec\{x\})\; =\; sum\_alpha\; c\_alpha\; x^alpha$

be a polynomial on

**R**^{"n"}with "real" coefficients, where α ranges over a (finite) set of multi-indices. Thus: $alpha\; =\; (alpha\_1,\; alpha\_2,\; ldots,\; alpha\_n)\; ,!$

and

: $x^alpha\; =\; x\_1^\{alpha\_1\}\; x\_2^\{alpha\_2\}\; cdots\; x\_n^\{alpha\_n\}.$

We also use the notation

:$D^alpha\; =\; frac\{1\}\{i^\{|alpha|\; partial\_\{x\_1\}^\{alpha\_1\}partial\_\{x\_2\}^\{alpha\_2\}\; cdots\; partial\_\{x\_n\}^\{alpha\_n\}.$

Then the operator "P"(D) defined on the space of infinitely differentiable functions of compact support on

**R**^{"n"}by:$P(operatorname\{D\})\; phi\; =\; sum\_alpha\; c\_alpha\; D^alpha\; phi$

is essentially self-adjoint on "L"

^{2}(**R**^{"n"}).**Theorem**. Let "P" a polynomial function on**R**^{"n"}with real coefficients,**F**the Fourier transform considered as a unitary map "L"^{2}(**R**^{"n"}) → "L"^{2}(**R**^{"n"}). Then**F*** "P"(D)**F**is essentially self-adjoint and its unique self-adjoint extension is the operator of multiplication by the function "P".More generally, consider linear differential operators acting on infinitely differentiable complex-valued functions of compact support. If "M" is an open subset of

**R**^{"n"}:$P\; phi(x)\; =\; sum\_alpha\; a\_alpha\; (x)\; [D^alpha\; phi]\; (x)\; quad$

where "a"

_{α}are (not necessarily constant) infinitely differentiable functions. "P" is a linear operator:$C\_0^infty(M)\; ightarrow\; C\_0^infty(M).\; quad$

Corresponding to "P" there is another differential operator, the

of "P"formal adjoint :$P^\{mathrm\{*form\; phi\; =\; sum\_alpha\; D^alpha\; (overline\{a\_alpha\}\; phi)\; quad$

**Theorem**. The operator theoretic adjoint "P"* of "P" is a restriction of the distributional extension of the formal adjoint. Specifically::$operatorname\{dom\}\; P^*\; =\; \{u\; in\; L^2(M):\; P^\{mathrm\{*formu\; in\; L^2(M)\}.$

**Spectral multiplicity theory**The multiplication representation of a self-adjoint operator, though extremely useful, is not a canonical representation. This suggests that it is not easy to extract from this representation a criterion to determine when self-adjoint operators "A" and "B" are unitarily equivalent. The finest grained representation which we now discuss involves spectral multiplicity. This circle of results is called the "Hahn-Hellinger theory of spectral multiplicity".

We first define "uniform multiplicity":

**Definition**. A self-adjoint operator "A" has uniform multiplicity "n" where "n" is such that 1 ≤ "n" ≤ ωif and only if "A" is unitarily equivalent to the operator M_{"f"}of multiplication by the function "f"(λ) = λ on:$L^2\_\{mu\}(mathbb\{R\},\; mathbf\{H\}\_n)=\; \{psi:\; mathbb\{R\}\; ightarrow\; mathbf\{H\}\_n:\; psi\; mbox\{\; measurable\; and\; \}\; int\_\{mathbb\{R\; |psi(t)|^2\; d\; mu(t)\; <\; infty\}$

where

**H**_{"n"}is a Hilbert space of dimension "n". The domain of M_{"f"}consists of vector-valued functions ψ on**R**such that:$int\_\{mathbb\{R\; |lambda|^2\; |\; psi(lambda)|^2\; ,\; d\; mu(lambda)\; <\; infty.$

Non-negative countably additive measures μ, ν are

**mutually singular**if and only if they are supported on disjoint Borel sets.**Theorem**. Let "A" be a self-adjoint operator on a "separable" Hilbert space "H". Then there is an ω sequence of countably additive finite measures on**R**(some of which may be identically 0):$\{mu\_ell\}\_\{1\; leq\; ell\; leq\; omega\}$

such that the measures are pairwise singular and "A" is unitarily equivalent to the operator of multiplication by the function "f"(λ) = λ on

:$igoplus\_\{1\; leq\; ell\; leq\; omega\}\; L^2\_\{mu\_ell\}(mathbb\{R\},\; mathbf\{H\}\_ell).$

This representation is unique in the following sense: For any two such representations of the same "A", the corresponding measures are equivalent in the sense that they have the same sets of measure 0.

The spectral multiplicity theorem can be reformulated using the language of

direct integral s of Hilbert spaces:**Theorem**. Any self-adjoint operator on a separable Hilbert space is unitarily equivalent to multiplication by the function λ → λ on:$int\_mathbb\{R\}^oplus\; H\_x\; d\; mu(x).$

The measure equivalence class of μ (or equivalently its sets of measure 0) is uniquely determined and the measurable family{"H"

_{"x"}}_{"x"}is determined almost everywhere with respect to μ.**Example: structure of the Laplacian**The Laplacian on

**R**^{"n"}is the operator:$Delta\; =\; sum\_\{i=1\}^n\; partial\_\{x\_i\}^2.$

As remarked above, the Laplacian is diagonalized by the Fourier transform. Actually it is more natural to consider the "negative" of the Laplacian - Δ since as an operator it is non-negative; (see

elliptic operator ).**Theorem**. If "n"=1, the - Δ has uniform multiplicity**mult**=2, otherwise - Δ has uniform multiplicity**mult**=ω. Morover, the measure μ_{mult}is Borel measure on[ 0, ∞).**Pure point spectrum**A self-adjoint operator "A" on "H" has pure point spectrum if and only if "H" has an orthonormal basis {"e"

_{"i"}}_{"i" ∈ I}consisting of eigenvectors for "A".**Example**. The Hamiltonian for the harmonic oscillator has a quadratic potential "V", that is:$-Delta\; +\; |x|^2\; quad$

This Hamiltonian has pure point spectrum; this is typical for bound state Hamiltonians in quantum mechanics. As was pointed out in a previous example, a sufficient condition that an unbounded symmetric operator has eigenvectors which form a Hilbert space basis is that it has a compact inverse.

**See also***

Compact operator on Hilbert space

*Theoretical and experimental justification for the Schrödinger equation **References*** N.I. Akhiezer and I. M. Glazman, "Theory of Linear Operators in Hilbert Space" (two volumes), Pitman, 1981.

* T. Kato, "Perturbation Theory for Linear Operators", Springer, New York, 1966.

* M. Reed and B. Simon, "Methods of Mathematical Physics" vol 2, Academic Press, 1972.

* G. Teschl, "Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger Operators", http://www.mat.univie.ac.at/~gerald/ftp/book-schroe/

* K. Yosida, "Functional Analysis", Academic Press, 1965.

*Wikimedia Foundation.
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