LGBT sex education

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/LGBT_sex_education 

 

Poster carried in a gay pride march in Portland, Maine

LGBT sex education is a sex education program within a school, university, or community center that addresses prominent sexual health topics among LGBT groups. Within schools, topics on LGBT sexual health are usually integrated into the general sex education courses.

There is some debate about whether LGBT sex education should be included in sex education curricula. Advocates of LGBT sex education say that the inclusion of LGBT issues into sex education programs would reduce homophobic bullying, improve the health of LGBT people, and decrease instances of problems common in LGBT students such as depression and low self-esteem; opponents argue that LGBT sex education programs would force a political point of view on students, misuse tax money, and disrespect religious values. As of 2014, only 5% of middle and high school students in the United States reported receiving "positive discussions of LGBT-related topics" within their health classes.

Background

LGBT sex education is currently not covered in many schools. Research has also posited that students often do not find existing LGBT sex education programs to be effective. Teachers have differing views on the subject of homosexuality, and these personal opinions can impact LGBT sex education when it is implemented.

Research

Multiple studies have concluded that LGBT sex education is often not encompassed in school sex education courses and that most students do not receive effective instruction in LGBT sex issues. In a study conducted by Ellis and High in the UK (2004), 384 students were surveyed; they found that 24% had not received instruction in LGBT sex issues. The CDC Division of Adolescent and School Health's study revealed that 48% of schools in the US covered LGBT topics. According to research reported by Burston and Hart in 2001, 45% of surveyed students believed that they did not cover LGBT sex education sufficiently in school. Research has also shown that there can be an implicit assumption that all students are heterosexual in sex education classes. The LGBT students in Eleanor Formby's 2011 study of sex education said that they do not always feel welcomed by sex education classes or at school. Sex education courses commonly idealize marriage (not acknowledging that many countries outlaw same-sex marriage) thereby presenting a heterosexual view of sex and relationships as normal. Studies have suggested that sex education programs often do not cover safe sex practices for LGBT individuals.

However, there are some sex education curricula that do cover LGBT issues. For example, the Unitarian Universalist Association of Congregations provides a sex education program called Our Whole Lives, which includes discussion of sexual orientation and presents homosexuality and heterosexuality as equally valid. Our Whole Lives offers programs designed for a range of developmental stages, from Kindergarten-level through adulthood, and follows the "Guidelines for Comprehensive Sexuality Education" that the Sexuality Information and Education Council of the United States (SIECUS) endorses. The United Church of Christ also supports Our Whole Lives.

Issues with sex education programs

Research has illustrated that some sex education courses present LGBT issues in a negative light—portraying LGBT individuals and LGBT sexuality as something wrong, sick, and abnormal. According to the American Civil Liberties Union, "abstinence-only" approaches to sex education can also be alienating to LGBT students because these programs assume that marriage is a possibility and a desire for all students; however, same-sex marriage is illegal in many countries. Further, promoting marriage as the goal for LGBT students reproduces a homonormative standard, marginalizing those without access to or interest in marriage. Ellis and High's survey research in 2004 (including 384 students) revealed that 59% of young people who did receive LGBT sex education found it to be ineffective.

Teachers

Teachers have been identified as a hindrance to LGBT sex education in some studies. Teachers always have their own opinions about homosexuality, and, according to these studies, if teachers have negative views toward LGBT individuals this can come through in their teaching—causing LGBT students to feel unaccepted and unsafe. According to Ellis and High (2004), when LGBT students receive information about LGBT sexuality with negative undertones they are left feeling significantly worse and more unsafe than if homosexuality were left out of the curriculum. Researchers have documented multiple self-proclaimed "LGBT-friendly" teachers whose classrooms actually foster prejudiced lessons. Such teachers are also highly likely to ignore instances of homophobic bullying directed at LGBT youth within their classes. Burston and Hart (2001) reported that teachers sometimes believe that they should not take a side on the issue of homosexuality, and therefore should not interfere when homophobia occurs in the classroom. According to Formby (2011), even phrasing that subtly casts homosexuality in a negative light can have a detrimental effect on LGBT students' experience of sex education.

There have also been issues around teachers feeling free to teach sex education that equally emphasizes both heterosexual and homosexual health information. Deana Morrow's study (1993) reported that some teachers said they were afraid they might be fired if they discussed LGBT issues. This fear is linked to the historical misconnection of homosexuality to molestation in the United States; this supposedly natural linkage has been debunked multiple times. Regardless, straight and LGBT teachers alike still experience allegations of molestation when they engage in discussions surrounding sexuality, particularly those discussions that are LGBT related. Teachers can also feel hindered because the school environment is inhospitable to homosexuality; in Burston and Hart's 2001 study, some even said that they were under the impression that the school would not allow them to teach LGBT sex education.

Classmates can also be non-receptive toward LGBT issues in current sex education courses, and students are often hesitant to talk about homosexuality, according to Buston and Hart (2001).

At home

Since sex education has been present in health education in schools, many parents expect their children to learn about sex there. Studies show that most families do not engage in conversation about sex in the home, and when they do it is often from a heteronormative perspective. The assumptions of being heterosexual can make LGBT people feel ashamed or lacking support from their family. Lack of conversation and knowledge received in the home for LGBT people can often lead them to receive their information for outside sources that contain false or misleading information. The same study showed that many parents don't have a solid knowledge base on same-sex or LGBT topics, nor do they know of resources to direct their children towards.

Proposed LGBT sex education programs

Advocates for LGBT sex education have suggested adjustments to current sex education practices in schools. One common place for improvement that researchers have identified is the angle from which sex education is approached in general. Buston and Hart (2001), Ellis and High (2004), and others have recommended that teachers frame sex education in terms of relationships rather than merely reproduction, which can lead to the exclusion of LGBT students. Ellis and High mention that sexual orientation might be more appropriately taught as "an aspect of culture and identity" (Ellis and High 2004, pg. 11). Other researchers such as Morrow (1993) believe that in order for sex education to be effective, it must present LGB as just as natural and legitimate as heterosexuality. Advocates for LGBT sex education ask that LGBT sexual health issues be given equal weight in the curriculum accordingly. They also say that more resources concerning LGBT sexual health issues need to be made available to students. According to UCLA's Center for the Study of Women's Policy Brief 11 (2012), LBGT students may not be willing to reach out for guidance themselves.

Researchers have recommended that teachers in sex education programs avoid framing homosexuality as something that is fundamentally connected to sexually transmitted diseases and refrain from practices that are potentially detrimental to LGBT students, such as referring to partners as specifically "him" or "her" (better to use the more flexible "they"). Gowen and Winges-Yanez (2014) suggest through their focus groups on LGBT teens that there are several problems with the way sex education is taught. The teens cited silencing, heterocentricity, and pathologizing of LGBT individuals as common practices. When asked how they would improve sex education, the group said inclusive sex education would include discussion of LGBT issues, learning how to access resources, STI or STD prevention, relationships, and anatomy. Advocates for LGBT rights also say that teachers need to abandon any reluctance to take a side in the debate about homosexuality.

There are also alternative sexual education programs for LGBT people, such as that of an online sexual education course. According to a study evaluating the effectiveness of an online, interactive sexual education program for LGBT people, all subsections recorded statistically significant improvement of knowledge. Some of the topics included safe sex practices, healthy relationships, pleasure and sexually transmitted infections. This type of program also created an online community for people taking the course to ask questions and interact with each other. This social aspect of the program also created a sense of normalcy and acceptance. Online programs could offer a means of education for those who cannot receive it in school. There are also various online LGBT sites on the internet that offer educational leaflets or information.

Novels including LGBT relationships can be a useful tool in an LGBT inclusive sexual education course or as a way for youth and teens to learn about LGBT relationships and issue in a different type of way. Novels that include LGBT relationships can aid in normalizing queer relationships potentially creating a more accepting and inclusive atmosphere for LGBT youth. It can also supplement information learned by reinforcing it in the form of a story. Many LGBT youth use young adult novels that include LGBT relationships as sources of information, especially if they don't receive sex education in school. Sexually explicit young adult novels can provide details about sexual intercourse, intimacy and sexual identity that LGBT youth can relate to allowing them to explore their own sexual identity.

A study of LGBT youth asked them what their current curriculum is teaching them and what they would want to see in a new curriculum. Some of the responses included a more inclusive curriculum that described different people with different gender identities, sexual orientations and ethnicity, "how-to" information that related to LGBT people relationships, and specific sections related to LGBT risks, problems and behaviors. They also mentioned the use of internet information and resources as a way of creating a community for extended education and support.

Support for LGBT sex education

Proponents of incorporating LGBT sex education into school curricula commonly present several arguments. According to the Huffington Post, some supporters claim that failing to include LGBT issues in sex education programs will overlook a significant number of students who identify as LGBT; the Center for American Progress (CAP) says that this can cause them to feel marginalized and removed from the lesson because it doesn't pertain to them. LGBT sex education advocates also argue that leaving LGBT safe sex instruction out of the curriculum will increase the likelihood of health problems for LGBT students. Supporters say that since LGBT people are particularly at risk for HIV/AIDS, it is especially important to provide them with sexual health information. According to researcher Eleanor Formby (2011), lesbian women are a high-risk group for sexually transmitted diseases (STDs), because many do not know that they can be susceptible to STDs or how to engage in safe sex. Therefore, it is important that they receive lesbian sex education. LGBT sex education advocates suggest that because LGBT students aren't taught sex education that pertains to them in school, they feel unprepared for sex, unable to talk about it openly, and have to learn about it by themselves—which can result in negative health outcomes. Sanchez (2012) argues that LGBT students are unlikely to reach out to resources that could give them good information on their own, which furthers the need for LGBT sex education in schools. LGBT youth are also at higher risk of engaging in high-risk behavior such as higher rates of suicide attempts, substance use and high risk sexual behavior. Since many of these high risk actions among LGBT youth have been correlated with depression, emotional distress and victimization experiences from non- LGBT people, LGBT sensitive sex and HIV education in schools could reduce this high-risk behavior by normalizing LGBT people and also providing support services to LGBT youth.

Parade attendees wave rainbow flags at the 2012 Washington, D.C. Capital Pride parade

LGBT sex education supporters have also argued that the inclusion of LGBT topics in the curriculum can decrease instances of bullying in schools by familiarizing students with the range of sexual orientations and reducing harmful stereotypes. The Center for American Progress argues that LGBT sex education results in a decrease in homophobic comments. According to the Huffington Post, supporters say that educating young people about LGBT individuals could help them have a more positive attitude toward their gay peers. The Center for American Progress (2013) says that LGBT sex education would therefore reduce common problems LGBT students face as a result of negative attitudes; these include mental health issues like depression, the risk of suicide, drug abuse, self-esteem issues, and poorer academic performance due to stress caused by discrimination. They argue that covering homosexuality in sex education programs helps students feel more secure at school.

Several studies have also shown that heteronormative and negative attitudes toward LGBT people are associated with lower rates of academic success. In schools, heteronormative and non inclusive culture can poorly effect motivation, health, and learning habits in students who identify as LGBTQ+. Andreas Gegenfurtner and Markus Gebhardt have shared findings which suggest that tolerance and acceptance toward sexual minorities were reported to be more positive when people are more highly educated and less religious. Similar findings within their study have shown a positive correlation between academic success among LGBTQ+ students and inclusive school environments.

According to Jen Gilbert, associate professor of education at York University, LGBT kids often do not have queer parents who they can ask for sexuality-related advice, nor access to LGBT adults. LGBT sex education could potentially fill this gap and provide LGBT students with elders well-versed in their specific needs and equipped with affirming information, that students are otherwise unable to receive at home or in school.

Finally, proponents of LGBT sex education have said that curricula that explore all facets of sexuality would be beneficial to straight students as well, because they claim that it presents a more accurate picture of the world and human sexuality. A study of Gay/Straight Alliances in Utah found that peer-facilitated discussions concerning the spectrum of sexuality and gender identities benefited both straight and LGBT students. It exposed them to the reality of relationships outside of the heteronormative images that dominate media (as well as sex education), and even positively impacted all involved students' academic performance. Proponents also argue that offering LGBT-inclusive sex education can be of major assistance to any questioning students that might be in the class.

According to the Center for American Progress (2013), the majority of parents support including homosexuality in the sex education curriculum; they report that 73% of high school parents think LGBT issues should be taught. The CAP claims that this high percentage of support indicates that LGBT topics should be incorporated.

Support within the United States

Eight states have a sexual health curriculum that affirms LGBTQ+ people. These states are California, Oregon, Nevada, Colorado, Illinois, New Jersey, Hawaii, and Connecticut.

California Healthy Youth Act

In 2016, the California Healthy Youth Act stated that all sexual health education in California districts must be inclusive of LGBTQ students. This requires schools to discuss examples of same-sex relationships and teach about different gender identities. School districts must protect LGBTQ students from harassment and prevent discrimination against them. Parents are not allowed to "opt-out" specifically from the LGBTQ-related content because this would be considered discrimination.

Opposition to LGBT sex education

Opponents of LGBT sex education argue that it is wrong to teach students about the issue of homosexuality because it is too contentious. They say that parents should have control over what their children are exposed to and taught, and allowing public schools to cover LGBT sex education would undermine this right, forcing a particular political view on students. Furthermore, many opponents of inclusive sex ed programs argue that parents are forced to lose control of what their children learn in school. This belief is especially common in households that are religiously affiliated, or identify politically with views against LGBT rights. According to The Christian Post, some parents do not want their children to study homosexuality. Critics often cite a misuse of citizens' tax dollars, claiming citizens should not have to pay for children to learn about other lifestyles that their parents do not agree with. Parents and guardians within these families commonly argue that lesbian, gay, bisexual, or transgender activity is immoral, abnormal, and unnatural.

According to Formby (2011), opponents have also argued that LGBT sex education is harmful to students because they say it exposes them to damaging information. They claim that the students should not learn about LGBT issues until they are older. Some opponents of LGBT sex education have argued that including LGBT issues in sex education programs will encourage more young people to practice homosexuality as well. LGBT sex education has also been accused of being disrespectful to certain families’ religious beliefs. The Christian Post argued that if schools elect to teach about LGBT people while neglecting religious topics, the curriculum would be unfairly balanced.

There have also been concerns that LGBT sex education wouldn't be effective because it is difficult for homophobic students to accept homosexuality, which may prevent them from being receptive to the instruction.

Opposition within the United States

There are laws prohibiting the "promotion of sexuality" (referred to as "No Promo Homo" laws"). Four states (Louisiana, Mississippi, Oklahoma, and Texas) mandate pointedly negative messages regarding all LGBT identities, when sex education is provided. Eight states (the four previously mentioned, Alabama, Arizona, South Carolina, and Utah) prohibit discussion of any topics deemed LGBT-related. According to the Guttmacher Institute's findings in 2017, "If HIV education is taught in Arizona it cannot 'promote' a 'homosexual lifestyle' or portray homosexuality in a positive manner. Mandated HIV education in Oklahoma teaches that among other behaviors that 'homosexual activity' is considered to be 'responsible for contact with the AIDS virus'." Utah, Alabama and Arizona used to have "No Homo Promo" laws but they were since repealed.

Laws and legal battles

Section 28

Main article: Section 28

Section 28 was a controversial law in the United Kingdom that barred schools from presenting homosexuality as a viable sexual orientation or basis for relationships (though the law was never used in court). It was enacted in 1988 and repealed throughout the UK by 2003. Critics of Section 28 say that the law prevented teachers from intervening in instances of homophobic bullying and greatly hindered the development of gay rights in Great Britain. According to Moran (2001), proponents of the law argued that it protected students from being harmed by gay propaganda. Recently, LGBT advocates have raised concerns that policies similar to Section 28 are appearing again in British schools. Wales has sought to challenge the implications of this section by implementing a new education curriculum Relationships and Sexuality Education (RSE) by 2022. The goals of this new curriculum will be to broaden traditional sex education and include information relating to relationships and a greater understanding of sexuality. It will also include LGBTQI topics such as gender identity and bring up issues of consent and sexual violence. The new curriculum will be required in primary and secondary schools with each containing different curriculum focuses, but it will not be required by religious schools.

Croatian textbook

In 2009, the European Committee of Social Rights found several statements in a Croatian mandatory Biology course textbook, including: "Many individuals are prone to sexual relations with persons of the same sex.... It is believed that parents are to blame because they impede their children's correct sexual development with their irregularities in family relations. Nowadays it has become evident that homosexual relations are the main culprit for increased spreading of sexually transmitted diseases (e.g. AIDS)," and "The disease [AIDS] has spread amongst promiscuous groups of people who often change their sexual partners. Such people are homosexuals because of sexual contacts with numerous partners, drug addicts...and prostitutes." The European Committee of Social Rights deemed these statements discriminatory and in violation of Croatia's obligations under the European Social Charter.

Among minority groups

CDC findings

A 2018 CDC study has maintained that Latino and black youth and young adult men who have sex with men often face stigma, discrimination, and language barriers that hinder their ability to access STD education, prevention, and treatment. As a result, they are vulnerable to high rates of HIV and other health disparities. In 2017, African Americans accounted for 43% of all new HIV diagnoses. Additionally, Hispanic/Latinos are also strongly affected. They accounted for 26% of all new HIV diagnoses. In 2017, gay and bisexual men accounted for 66% of all HIV diagnoses in the United States and 6 dependent areas.

Structural barriers

One case study has demonstrated that homophobia, racism, coupled with financial hardship and social support were associated with higher exposure to HIV among homosexual men of color. In the United States, Latino men who have sex with men (MSM) are disproportionately affected by HIV. Another study demonstrated that in a multivariable analysis, increasing age, low income, and queer identity. Additionally, people Living With HIV, MSM and transgender women are considered the "most in need" due to the stigma that prevents them from accessing high-quality health care, prevention, and sex education. According to Mattew E. Levy of The George Washington University, many systematic factors have led to the disproportionate rates of HIV among Black and Latino Men who have sex with men, including insufficient healthcare, social stigma and discrimination, incarceration, and poverty. Men of color who have sex with men experience inadequate access to culturally competent services, stigma and discrimination that impede access to services, a deficiency of services in correctional institutions, and limited services in areas where they live.

Eigenfunction

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Eigenfunction 

 

This solution of the vibrating drum problem is, at any point in time, an eigenfunction of the Laplace operator on a disk.

In mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function ${\displaystyle f}$ in that space that, when acted upon by D, is only multiplied by some scaling factor called an eigenvalue. As an equation, this condition can be written as

$${\displaystyle Df=\lambda f}$$

for some scalar eigenvalue ${\displaystyle \lambda .}$ The solutions to this equation may also be subject to boundary conditions that limit the allowable eigenvalues and eigenfunctions.

An eigenfunction is a type of eigenvector.

Eigenfunctions

In general, an eigenvector of a linear operator D defined on some vector space is a nonzero vector in the domain of D that, when D acts upon it, is simply scaled by some scalar value called an eigenvalue. In the special case where D is defined on a function space, the eigenvectors are referred to as eigenfunctions. That is, a function f is an eigenfunction of D if it satisfies the equation

$${\displaystyle Df=\lambda f,}$$

(1)

where λ is a scalar. The solutions to Equation (1) may also be subject to boundary conditions. Because of the boundary conditions, the possible values of λ are generally limited, for example to a discrete set λ₁, λ₂, … or to a continuous set over some range. The set of all possible eigenvalues of D is sometimes called its spectrum, which may be discrete, continuous, or a combination of both.

Each value of λ corresponds to one or more eigenfunctions. If multiple linearly independent eigenfunctions have the same eigenvalue, the eigenvalue is said to be degenerate and the maximum number of linearly independent eigenfunctions associated with the same eigenvalue is the eigenvalue's degree of degeneracy or geometric multiplicity.

Derivative example

A widely used class of linear operators acting on infinite dimensional spaces are differential operators on the space C^∞ of infinitely differentiable real or complex functions of a real or complex argument t. For example, consider the derivative operator ${\textstyle {\frac {d}{dt}}}$ with eigenvalue equation

$${\displaystyle {\frac {d}{dt}}f(t)=\lambda f(t).}$$

This differential equation can be solved by multiplying both sides by ${\textstyle {\frac {dt}{f(t)}}}$ and integrating. Its solution, the exponential function

$${\displaystyle f(t)=f_{0}e^{\lambda t},}$$

is the eigenfunction of the derivative operator, where f₀ is a parameter that depends on the boundary conditions. Note that in this case the eigenfunction is itself a function of its associated eigenvalue λ, which can take any real or complex value. In particular, note that for λ = 0 the eigenfunction f(t) is a constant.

Suppose in the example that f(t) is subject to the boundary conditions f(0) = 1 and ${\textstyle \left.{\frac {df}{dt}}\right|_{t=0}=2}$. We then find that

$${\displaystyle f(t)=e^{2t},}$$

where λ = 2 is the only eigenvalue of the differential equation that also satisfies the boundary condition.

Link to eigenvalues and eigenvectors of matrices

Eigenfunctions can be expressed as column vectors and linear operators can be expressed as matrices, although they may have infinite dimensions. As a result, many of the concepts related to eigenvectors of matrices carry over to the study of eigenfunctions.

Define the inner product in the function space on which D is defined as

$${\displaystyle \langle f,g\rangle =\int _{\Omega }\ f^{*}(t)g(t)dt,}$$

integrated over some range of interest for t called Ω. The * denotes the complex conjugate.

Suppose the function space has an orthonormal basis given by the set of functions {u₁(t), u₂(t), …, u_n(t)}, where n may be infinite. For the orthonormal basis,

$${\displaystyle \langle u_{i},u_{j}\rangle =\int _{\Omega }\ u_{i}^{*}(t)u_{j}(t)dt=\delta _{ij}={\begin{cases}1&i=j\\0&i\neq j\end{cases}},}$$

where δ_ij is the Kronecker delta and can be thought of as the elements of the identity matrix.

Functions can be written as a linear combination of the basis functions,

$${\displaystyle f(t)=\sum _{j=1}^{n}b_{j}u_{j}(t),}$$

for example through a Fourier expansion of f(t). The coefficients b_j can be stacked into an n by 1 column vector b = [b₁ b₂ … b_n]^T. In some special cases, such as the coefficients of the Fourier series of a sinusoidal function, this column vector has finite dimension.

Additionally, define a matrix representation of the linear operator D with elements

$${\displaystyle A_{ij}=\langle u_{i},Du_{j}\rangle =\int _{\Omega }\ u_{i}^{*}(t)Du_{j}(t)dt.}$$

We can write the function Df(t) either as a linear combination of the basis functions or as D acting upon the expansion of f(t),

$${\displaystyle Df(t)=\sum _{j=1}^{n}c_{j}u_{j}(t)=\sum _{j=1}^{n}b_{j}Du_{j}(t).}$$

Taking the inner product of each side of this equation with an arbitrary basis function u_i(t),

$${\displaystyle {\begin{aligned}\sum _{j=1}^{n}c_{j}\int _{\Omega }\ u_{i}^{*}(t)u_{j}(t)dt&=\sum _{j=1}^{n}b_{j}\int _{\Omega }\ u_{i}^{*}(t)Du_{j}(t)dt,\\c_{i}&=\sum _{j=1}^{n}b_{j}A_{ij}.\end{aligned}}}$$

This is the matrix multiplication Ab = c written in summation notation and is a matrix equivalent of the operator D acting upon the function f(t) expressed in the orthonormal basis. If f(t) is an eigenfunction of D with eigenvalue λ, then Ab = λb.

Eigenvalues and eigenfunctions of Hermitian operators

Many of the operators encountered in physics are Hermitian. Suppose the linear operator D acts on a function space that is a Hilbert space with an orthonormal basis given by the set of functions {u₁(t), u₂(t), …, u_n(t)}, where n may be infinite. In this basis, the operator D has a matrix representation A with elements

$${\displaystyle A_{ij}=\langle u_{i},Du_{j}\rangle =\int _{\Omega }dt\ u_{i}^{*}(t)Du_{j}(t).}$$

integrated over some range of interest for t denoted Ω.

By analogy with Hermitian matrices, D is a Hermitian operator if A_ij = A_ji*, or:

$${\displaystyle {\begin{aligned}\langle u_{i},Du_{j}\rangle &=\langle Du_{i},u_{j}\rangle ,\\[-1pt]\int _{\Omega }dt\ u_{i}^{*}(t)Du_{j}(t)&=\int _{\Omega }dt\ u_{j}(t)[Du_{i}(t)]^{*}.\end{aligned}}}$$

Consider the Hermitian operator D with eigenvalues λ₁, λ₂, … and corresponding eigenfunctions f₁(t), f₂(t), …. This Hermitian operator has the following properties:

• Its eigenvalues are real, λ_i = λ_i*
• Its eigenfunctions obey an orthogonality condition, ${\displaystyle \langle f_{i},f_{j}\rangle =0}$ if i ≠ j

The second condition always holds for λ_i ≠ λ_j. For degenerate eigenfunctions with the same eigenvalue λ_i, orthogonal eigenfunctions can always be chosen that span the eigenspace associated with λ_i, for example by using the Gram-Schmidt process. Depending on whether the spectrum is discrete or continuous, the eigenfunctions can be normalized by setting the inner product of the eigenfunctions equal to either a Kronecker delta or a Dirac delta function, respectively.

For many Hermitian operators, notably Sturm–Liouville operators, a third property is

• Its eigenfunctions form a basis of the function space on which the operator is defined

As a consequence, in many important cases, the eigenfunctions of the Hermitian operator form an orthonormal basis. In these cases, an arbitrary function can be expressed as a linear combination of the eigenfunctions of the Hermitian operator.

Applications

Vibrating strings

The shape of a standing wave in a string fixed at its boundaries is an example of an eigenfunction of a differential operator. The admissible eigenvalues are governed by the length of the string and determine the frequency of oscillation.

Let h(x, t) denote the transverse displacement of a stressed elastic chord, such as the vibrating strings of a string instrument, as a function of the position x along the string and of time t. Applying the laws of mechanics to infinitesimal portions of the string, the function h satisfies the partial differential equation

$${\displaystyle {\frac {\partial ^{2}h}{\partial t^{2}}}=c^{2}{\frac {\partial ^{2}h}{\partial x^{2}}},}$$

which is called the (one-dimensional) wave equation. Here c is a constant speed that depends on the tension and mass of the string.

This problem is amenable to the method of separation of variables. If we assume that h(x, t) can be written as the product of the form X(x)T(t), we can form a pair of ordinary differential equations:

$${\displaystyle {\frac {d^{2}}{dx^{2}}}X=-{\frac {\omega ^{2}}{c^{2}}}X,\qquad {\frac {d^{2}}{dt^{2}}}T=-\omega ^{2}T.}$$

Each of these is an eigenvalue equation with eigenvalues ${\textstyle -{\frac {\omega ^{2}}{c^{2}}}}$ and −ω², respectively. For any values of ω and c, the equations are satisfied by the functions

$${\displaystyle X(x)=\sin \left({\frac {\omega x}{c}}+\varphi \right),\qquad T(t)=\sin(\omega t+\psi ),}$$

where the phase angles φ and ψ are arbitrary real constants.

If we impose boundary conditions, for example that the ends of the string are fixed at x = 0 and x = L, namely X(0) = X(L) = 0, and that T(0) = 0, we constrain the eigenvalues. For these boundary conditions, sin(φ) = 0 and sin(ψ) = 0, so the phase angles φ = ψ = 0, and

$${\displaystyle \sin \left({\frac {\omega L}{c}}\right)=0.}$$

This last boundary condition constrains ω to take a value ω_n = ncπ/L, where n is any integer. Thus, the clamped string supports a family of standing waves of the form

$${\displaystyle h(x,t)=\sin \left({\frac {n\pi x}{L}}\right)\sin(\omega _{n}t).}$$

In the example of a string instrument, the frequency ω_n is the frequency of the n-th harmonic, which is called the (n − 1)-th overtone.

Schrödinger equation

In quantum mechanics, the Schrödinger equation

$${\displaystyle i\hbar {\frac {\partial }{\partial t}}\Psi (\mathbf {r} ,t)=H\Psi (\mathbf {r} ,t)}$$

with the Hamiltonian operator

$${\displaystyle H=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V(\mathbf {r} ,t)}$$

can be solved by separation of variables if the Hamiltonian does not depend explicitly on time. In that case, the wave function Ψ(r,t) = φ(r)T(t) leads to the two differential equations,

$${\displaystyle H\varphi (\mathbf {r} )=E\varphi (\mathbf {r} ),}$$

(2)

$${\displaystyle i\hbar {\frac {\partial T(t)}{\partial t}}=ET(t).}$$

(3)

Both of these differential equations are eigenvalue equations with eigenvalue E. As shown in an earlier example, the solution of Equation (3) is the exponential

$${\displaystyle T(t)=e^{{-iEt}/{\hbar }}.}$$

Equation (2) is the time-independent Schrödinger equation. The eigenfunctions φ_k of the Hamiltonian operator are stationary states of the quantum mechanical system, each with a corresponding energy E_k. They represent allowable energy states of the system and may be constrained by boundary conditions.

The Hamiltonian operator H is an example of a Hermitian operator whose eigenfunctions form an orthonormal basis. When the Hamiltonian does not depend explicitly on time, general solutions of the Schrödinger equation are linear combinations of the stationary states multiplied by the oscillatory T(t), ${\textstyle \Psi (\mathbf {r} ,t)=\sum _{k}c_{k}\varphi _{k}(\mathbf {r} )e^{{-iE_{k}t}/{\hbar }}}$ or, for a system with a continuous spectrum,

$${\displaystyle \Psi (\mathbf {r} ,t)=\int dE\,c_{E}\varphi _{E}(\mathbf {r} )e^{{-iEt}/{\hbar }}.}$$

The success of the Schrödinger equation in explaining the spectral characteristics of hydrogen is considered one of the greatest triumphs of 20th century physics.

Signals and systems

In the study of signals and systems, an eigenfunction of a system is a signal f(t) that, when input into the system, produces a response y(t) = λf(t), where λ is a complex scalar eigenvalue.